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Jose Wudka

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Contents

Contents

From Antiquity to Einstein

o 1 . Introduction

Overview

The scientific method

What is the "scientific method"?

What is the difference between a fact, a theory and a

hypothesis?

Truth and proof in science.

If scientific theories keep changing, where is the Truth?

What is Ockham's Razor?

How much fraud is there in science?

Are scientists wearing blinkers?

Why should we worry?
Large numbers

Times (in seconds)
Distances (in meters)
Velocities (in meters per second)
Masses (in kilograms)
Temperatures (in deg. Kelvin)
Monies (in 1994 US dollars)

2. Greek cosmology

■ Egypt and Babylon

■ Babylon
" Egypt

■ Other nations

■ India

■ China

■ Early Greeks

■ Mythology

■ Early cosmology

■ The Pythagoreans

■ Early heliocentric systems

■ Aristotle and Ptolemy

■ Aristotelian Cosmology

■ The motion according to Aristotle

■ Ptolemy

3. From the Middle Ages to
Heliocentrism

■ The Middle Ages.

■ The Copernican Revolution

■ Aristotle in the 16th century

■ Kepler

o 4. Galileo and Newton

■ Introduction

■ Galileo Galilei

■ Galilean relativity

■ Mechanics

- The motion of falling bodies

■ The motion of projectiles

■ Astronomy

■ Galileo and the Inquisition

■ Isaac Newton

■ Mechanics.

■ 1st Law and Newtonian space and time.

■ 2nd Law

■ 3rd Law

■ Optics

■ Gravitation.

o 5. The Clouds Gather

■ Electricity and magnetism

■ Electricity

■ Magnetism

■ Waves vs. particles

■ Light

■ Problems

■ Ether

■ Galilean Relativity

■ Prelude to relativity
Einstein's Relativity and Modern Cosmology

o 6. The Special Theory of Relativity

■ Introduction

■ Enter Einstein

■ The first prediction: the speed of light and the demise of
Newton's mechanics

■ The second prediction: Simultaneity is relative

■ The first murder mystery (ca. 1890)

■ The second murder mystery (ca. 2330)

■ The third prediction: The demise of Universal Time

■ Length contraction

■ Paradoxes.

■ Space and Time

■ The top speed.

■ Mass and energy.

7. The General Theory of Relativity

■ The happiest thought of my life.

■ Newton vs. Einstein
Gravitation vs acceleration
Light

Clocks in a gravitational force-
Black holes
Gravitation and energy
Space and time-
Properties of space and time-
Curvature
Waves
Summary-
Tests of general relativity.

■ Precession of the perihelion of Mercury

■ Gravitational red- shift.

■ Light bending

■ The double pulsar

8. The universe: size, origins,
contents

■ Introduction

■ Light revisited

■ The inverse- square law

■ The Doppler effect

■ Emission and absorption lines

■ A happy marriage

■ Cosmic distance ladder

■ Step 1: distances up to 100 l.y.

• Step 2: distances up to 300,000 l.y.

- Step 3: distances up to 13,000,000 l.y.

• Step 4: distances up to 1,000,000,000 l.y.

• Step 5: distances beyond 1,000,000,000 l.y.
' The relativistic universe

■ The expanding universe

■ And now what?

■ The Microwave Background Radiation

■ Nucleosynthesis

■ At the cutting edge

■ Dark matter

■ Neutrinos

■ The cosmological constant

Homogeneity and isotropy

■ Inflation
Summary

9. The lifes of a star

Introduction-
Stellar Power
The lifes of a star

In the beginning
A rising star
A Giant appears
And so it goes

■ Light stars

■ Medium-size stars
The heavyweights

T*hTjSicS 7

Chapter 1
Introduction

1.1 Overview

These notes cover the development of the current scientific concepts of space
and time through history, emphasizing the newest developments and ideas.
The presentation will be non-mathematical: the concepts will be introduced
and explained, but no real calculations will be performed. The various
concepts will be introduced in a historical order (whenever possible), this
provides a measure of understanding as to how the ideas on which the mod-
ern theory of space and time is based were developed. In a real sense this
has been an adventure for humanity, very similar to what a child undergoes
from the moment he or she first looks at the world to the point he or she
understands some of its rules. Part of this adventure will be told here.

Every single culture has had a theory of the formation of the universe
and the laws that rule it. Such a system is called a cosmology (from the
Greek kosmos: world, and logia from legem: to speak). The first coherent
non-religious cosmology was developed during ancient Greece, and much
attention will be paid to it after a brief overview of Egyptian and Baby-
onian comologies 1 The system of the world devised by the Greeks described
correctly all phenomena known at the time, and was able to predict most
astronomical phenomena with great accuracy. Its most refined version, the
Ptolemaic system, survived for more than one thousand years.

X A few other comologies will be only summarily described. This is for lack of erudition,
Indian, Chinese and American comologies are equally fascinating.

These promising developments came to a stop during the Middle Ages,
but took off with a vengeance during the Renaissance; the next landmark in
this saga. During this time Copernicus developed his system of the world,
where the center of the Universe was the Sun and not the Earth. In the
same era Galileo defined and developed the science of mechanics with all its
basic postulates; he was also the creator of the idea of relativity, later used
by Einstein to construct his Special and General theories.

The next great player was Isaac Newton, who provided a framework
for understanding all the phenomena known at the time. In fact most of
our daily experience is perfectly well described by Newton's mathematical
formulae.

The cosmology based on the ideas of Galileo and Newton reigned supreme
up until the end of the 19th century: by this time it became clear that New-
ton's laws were unable to describe correctly electric and magnetic phenom-
ena. It is here that Einstein enters the field, he showed that the Newtonian
approach does not describe correctly situations in which bodies move at
speeds close to that of light ( in particular it does not describe light accu-
rately). Einstein also provided the generalization of Newton's equations to
the realm of such high speeds: the Special Theory of Relativity. Perhaps
more importantly, he also demonstrated that certain properties of space and
time taken for granted are, in fact, incorrect. We will see, for example, that
the concept of two events occurring at the same time in different places is
not absolute, but depends on the state of motion of the observer.

Not content with this momentous achievements, Einstein argued that the
Special Theory of Relativity itself was inapplicable under certain conditions,
for example, near very heavy bodies. He then provided the generalization
which encompasses these situations as well: the General Theory of Relativ-
ity. This is perhaps the most amazing development in theoretical physics in
300 years: without any experimental motivation, Einstein single handedly
developed this modern theory of gravitation and used it to predict some of
the most surprising phenomena observed to date. These include the bending
of light near heavy bodies and the existence of black holes, massive objects
whose gravitational force is so strong it traps all objects, including light.

These notes provide an overview of this saga. From the Greeks and their
measuring of the Earth, to Einstein and his description of the universe. But
before plunging into this, it is natural to ask how do scientific theories are
born, and why are they discarded. Why is it that we believe Einstein is
right and Aristotle is wrong? Why is it that we claim that our current
understating of the universe is deeper than the one achieved by the early
Greeks? The answer to these questions lies in the way in which scientists

evaluate the information derived from observations and experiments, and is
the subject of the next section.

1.2 The scientific method

Science is best defined as a careful, disciplined,
logical search for knowledge about any and all as-
pects of the universe, obtained by examination of
the best available evidence and always subject to
correction and improvement upon discovery of bet-
ter evidence. What's left is magic. And it doesn't
work.

James Randi

It took a long while to determine how is the world better investigated.
One way is to just talk about it (for example Aristotle, the Greek philoso-
pher, stated that males and females have different number of teeth, without
bothering to check; he then provided long arguments as to why this is the
way things ought to be). This method is unreliable: arguments cannot
determine whether a statement is correct, this requires proofs.

A better approach is to do experiments and perform careful observations.
The results of this approach are universal in the sense that they can be
reproduced by any skeptic. It is from these ideas that the scientific method
was developed. Most of science is based on this procedure for studying
Nature.

1.2.1 What is the "scientific method"?

The scientific method is the best way yet discovered for winnowing the truth
from lies and delusion. The simple version looks something like this:

1. Observe some aspect of the universe.

2. Invent a tentative description, called a hypothesis, that is consistent
with what you have observed.

3. Use the hypothesis to make predictions.

4. Test those predictions by experiments or further observations and
modify the hypothesis in the light of your results.

5. Repeat steps 3 and 4 until there are no discrepancies between theory
and experiment and/or observation.

Figure 1.1: Flow diagram describing the scientific method.

When consistency is obtained the hypothesis becomes a theory and pro-
vides a coherent set of propositions which explain a class of phenomena.
A theory is then a framework within which observations are explained and
predictions are made.

The great advantage of the scientific method is that it is unprejudiced:
one does not have to believe a given researcher, one can redo the experiment
and determine whether his/her results are true or false. The conclusions
will hold irrespective of the state of mind, or the religious persuasion, or
the state of consciousness of the investigator and/or the subject of the in-
vestigation. Faith, defined as 2 belief that does not rest on logical proof or
material evidence, does not determine whether a scientific theory is adopted
or discarded.

A theory is accepted not based on the prestige or convincing powers of
the proponent, but on the results obtained through observations and/or ex-

2 The American Heritage Dictionary (second college edition)

periments which anyone can reproduce: the results obtained using the scien-
tific method are repeatable. In fact, most experiments and observations are The results obtained using
repeated many times (certain experiments are not repeated independently the scientific method are
but are repeated as parts of other experiments). If the original claims are
not verified the origin of such discrepancies is hunted down and exhaustively
studied.

When studying the cosmos we cannot perform experiments; all informa-
tion is obtained from observations and measurements. Theories are then
devised by extracting some regularity in the observations and coding this
into physical laws.

There is a very important characteristic of a scientific theory or hypoth-
esis which differentiates it from, for example, an act of faith: a theory must

be "falsifiable" . This means that there must be some experiment or possible Every scientific theory must
discovery that could prove the theory untrue. For example, Einstein's the- be falsifiable
ory of Relativity made predictions about the results of experiments. These
experiments could have produced results that contradicted Einstein, so the
theory was (and still is) falsifiable.

In contrast, the theory that "the moon is populated by little green men
who can read our minds and will hide whenever anyone on Earth looks for
them, and will flee into deep space whenever a spacecraft comes near" is not
falsifiable: these green men are designed so that no one can ever see them.
On the other hand, the theory that there are no little green men on the
moon is scientific: you can disprove it by catching one. Similar arguments
apply to abominable snow-persons, UFOs and the Loch Ness Monster (s?).

A frequent criticism made of the scientific method is that it cannot ac-
commodate anything that has not been proved. The argument then points
out that many things thought to be impossible in the past are now every-
day realities. This criticism is based on a misinterpretation of the scientific
method. When a hypothesis passes the test it is adopted as a theory it
correctly explains a range of phenomena it can, at any time, be falsified by
new experimental evidence. When exploring a new set or phenomena scien-
tists do use existing theories but, since this is a new area of investigation,
it is always kept in mind that the old theories might fail to explain the new
experiments and observations. In this case new hypotheses are devised and
tested until a new theory emerges.

There are many types of "pseudo-scientific" theories which wrap them-
selves in a mantle of apparent experimental evidence but that, when exam-
ined closely, are nothing but statements of faith. The argument 3 , cited by

3 From http://puffin.ptialaska.net/~svend/award.html

some creationists, that science is just another kind of faith is a philosophic
stance which ignores the trans-cultural nature of science. Science's theory
of gravity explains why both creationists and scientists don't float off the
earth. All you have to do is jump to verify this theory - no leap of faith
required.

A theory is a conceptual
framework that explains
existing observations and
predicts new ones

1.2.2 What is the difference between a fact, a theory and a
hypothesis?

In popular usage, a theory is just a vague and fuzzy sort of fact and a
hypothesis is often used as a fancy synonym to 'guess'. But to a scientist
a theory is a conceptual framework that explains existing observations and
predicts new ones. For instance, suppose you see the Sun rise. This is an
existing observation which is explained by the theory of gravity proposed
by Newton. This theory, in addition to explaining why we see the Sun
move across the sky, also explains many other phenomena such as the path
followed by the Sun as it moves (as seen from Earth) across the sky, the
phases of the Moon, the phases of Venus, the tides, just to mention a few.
You can today make a calculation and predict the position of the Sun, the
phases of the Moon and Venus, the hour of maximal tide, all 200 years from
now. The same theory is used to guide spacecraft all over the Solar System.
A hypothesis is a working assumption. Typically, a scientist devises a hy-
pothesis and then sees if it "holds water" by testing it against available data
(obtained from previous experiments and observations). If the hypothesis
does hold water, the scientist declares it to be a theory.

1.2.3 Truth and proof in science.

Experiments sometimes produce results which cannot be explained with
existing theories. In this case it is the job of scientists to produce new
theories which replace the old ones. The new theories should explain all
the observations and experiments the old theory did and, in addition, the
new set of facts which lead to their development. One can say that new
theories devour and assimilate old ones (see Fig, 1.2). Scientists continually
test existing theories in order to probe how far can they be applied.

When a new theory cannot explain new observations it will be (eventu-
ally) replaced by a new theory. This does not mean that the old ones are
"wrong" or "untrue" , it only means that the old theory had a limited appli-
cability and could not explain all current data. The only certain thing about
currently accepted theories is that they explain all available data, which, if

Figure 1.2: Saturn devouring his sons (by F. Goya). A paradigm of how
new theories encompass old ones.

course, does not imply that they will explains all future experiments!

In some cases new theories provide not only extensions of old ones, but a
completely new insight into the workings of nature. Thus when going from
Newton's theory of gravitation to Einstein's our understanding of the nature
of space and time was revolutionized. Nonetheless, no matter how beautiful
and simple a new theory might be, it must explain the same phenomena the
old one did. Even the most beautiful theory can be annihilated by a single
ugly fact.

Scientific theories have various degrees of reliability and one can think
of them as being on a scale of certainty. Up near the top end we have our
theory of gravitation based on a staggering amount of evidence; down at the
bottom we have the theory that the Earth is flat. In the middle we have
our theory of the origin of the moons of Uranus. Some scientific theories are
nearer the top than others, but none of them ever actually reach it.

An extraordinary claim is one that contradicts a fact that is close to the
top of the certainty scale and will give rise to a lot of skepticism. So if you
are trying to contradict such a fact, you had better have facts available that
are even higher up the certainty scale: "extraordinary evidence is needed
for an extraordinary claim" .

1.2.4 If scientific theories keep changing, where is the Truth?

In 1666 Isaac Newton proposed his theory of gravitation. This was one of the
greatest intellectual feats of all time. The theory explained all the observed
facts, and made predictions that were later tested and found to be correct
within the accuracy of the instruments being used. As far as anyone could

see, Newton's theory was "the Truth" .

During the nineteenth century, more accurate instruments were used to
test Newton's theory, these observations uncovered some slight discrepan-
cies. Albert Einstein proposed his theories of Relativity, which explained
the newly observed facts and made more predictions. Those predictions
have now been tested and found to be correct within the accuracy of the
instruments being used. As far as anyone can see, Einstein's theory is "the
Truth" .

So how can the Truth change? Well the answer is that it hasn't. The

Universe is still the same as it ever was. When a theory is said to be "true"

A/hen a theory is said to be it means that it agrees with all known experimental evidence. But even the

'true" it means that it ^est Q f theories have, time and again, been shown to be incomplete: though

:xperimental evidence they might explain a lot of phenomena using a few basic principles, and

even predict many new and exciting results, eventually new experiments

(or more precise ones) show a discrepancy between the workings of nature

and the predictions of the theory. In the strict sense this means that the

theory was not "true" after all; but the fact remains that it is a very good

approximation to the truth, at lest where a certain type of phenomena is

concerned.

When an accepted theory cannot explain some new data (which has been
confirmed), the researchers working in that field strive to construct a new
theory. This task gets increasingly more difficult as our knowledge increases,
for the new theory should not only explain the new data, but also all the
old one: a new theory has, as its first duty, to devour and assimilate its
predecessors.

One other note about truth: science does not make moral judgments.
Anyone who tries to draw moral lessons from the laws of nature is on very
dangerous ground. Evolution in particular seems to suffer from this. At one
time or another it seems to have been used to justify Nazism, Communism,
and every other -ism in between. These justifications are all completely
bogus. Similarly, anyone who says "evolution theory is evil because it is
used to support Communism" (or any other -ism) has also strayed from the
path of Logic (and will not live live long nor prosper).

1.2.5 What is Ockham's Razor?

When a new set of facts requires the creation of a new theory the process is
far from the orderly picture often presented in books. Many hypothses are
proposed, studied, rejected. Researchers discuss their validity (sometimes
quite heatedly) proposing experiments which will determine the validity of

one or the other, exposing flaws in their least favorite ones, etc. Yet, even
when the unfit hypotheses are discarded, several options may remain, in
some cases making the exact same predictions, but having very different
underlying assumptions. In order to choose among these possible theories a
very useful tool is what is called Ockham 's razor.

Ockham's Razor is the principle proposed by William of Ockham in the
fourteenth century: "Pluralitas non est ponenda sine neccesitate" , which
translates as "entities should not be multiplied unnecessarily" .

In many cases this is interpreted as "keep it simple" , but in reality the
Razor has a more subtle and interesting meaning. Suppose that you have two
competing theories which describe the same system, if these theories have
different predictions than it is a relatively simple matter to find which one is
better: one does experiments with the required sensitivity and determines
which one give the most accurate predictions. For example, in Copernicus'
theory of the solar system the planets move in circles around the sun, in
Kepler's theory they move in ellipses. By measuring carefully the path of
the planets it was determined that they move on ellipses, and Copernicus'
theory was then replaced by Kepler's.

But there are are theories which have the very same predictions and it
is here that the Razor is useful. Consider form example the following two
theories aimed at describing the motions of the planets around the sun

• The planets move around the sun in ellipses because there is a force
between any of them and the sun which decreases as the square of the
distance.

• The planets move around the sun in ellipses because there is a force
between any of them and the sun which decreases as the square of the
distance. This force is generated by the will of some powerful aliens.

Since the force between the planets and the sun determines the motion of
the former and both theories posit the same type of force, the predicted
motion of the planets will be identical for both theories, the second theory,
however, has additional baggage (the will of the aliens) which is unnecessary
for the description of the system.

If one accepts the second theory solely on the basis that it predicts cor-
rectly the motion of the planets one has also accepted the existence of aliens
whose will affect the behavior of things, despite the fact that the presence
or absence of such beings is irrelevant to planetary motion (the only rel-
evant item is the type of force). In this instance Ockham's Razor would
unequivocally reject the second theory. By rejecting this type of additional

irrelevant hypotheses guards against the use of solid scientific results (such
as the prediction of planetary motion) to justify unrelated statements (such
as the existence of the aliens) which may have dramatic consequences. In
this case the consequence is that the way planets move, the reason we fall to
the ground when we trip, etc. is due to some powerful alien intellect, that
this intellect permeates our whole solar system, it is with us even now. ..and
from here an infinite number of paranoid derivations.

For all we know the solar system is permeated by an alien intellect, but
the motion of the planets, which can be explained by the simple idea that
there is a force between them and the sun, provides no evidence of the aliens'
presence nor proves their absence.

A more straightforward application of the Razor is when we are face
with two theories which have the same predictions and the available data
cannot distinguish between them. In this case the Razor directs us to study
in depth the simplest of the theories. It does not guarantee that the simplest
theory will be correct, it merely establishes priorities.

A related rule, which can be used to slice open conspiracy theories, is
Hanlon's Razor: "Never attribute to malice that which can be adequately
explained by stupidity".

1.2.6 How much fraud is there in science?

The picture of scientists politely discussing theories, prposing new ones in
view of new data, etc. appears to be completely devoid of any emotions. In
fact this is far from the truth, the discussions are very human, even though
the bulk of the scientific community will eventually accept a single theory
based on it explaining the data and making a series of verified predictions.
But before this is achieved, does it happen that researchers fake results
or experiments for prestige and/or money? How frequent is this kind of
scientific fraud?

In its simplest form this question is unanswerable, since undetected fraud
is by definition unmeasurable. Of course there are many known cases of fraud
in science. Some use this to argue that all scientific findings (especially those
they dislike) are worthless.

This ignores the replication of results which is routinely undertaken by
scientists. Any important result will be replicated many times by many
different people. So an assertion that (for instance) scientists are lying
about carbon-14 dating requires that a great many scientists are engaging in
a conspiracy. In fact the existence of known and documented fraud is a good
illustration of the self-correcting nature of science. It does not matter (for

the progress of science) if a proportion of scientists are fraudsters because
any important work they do will not be taken seriously without independent
verification.

Also, most scientists are idealists. They perceive beauty in scientific
truth and see its discovery as their vocation. Without this most would
have gone into something more lucrative. These arguments suggest that
undetected fraud in science is both rare and unimportant.

The above arguments are weaker in medical research, where companies
frequently suppress or distort data in order to support their own products.
Tobacco companies regularly produce reports "proving" that smoking is
harmless, and drug companies have both faked and suppressed data related
to the safety or effectiveness or major products. This type of fraud does
not, of course, reflect on the validity of the scientific method.

1.2.7 Are scientists wearing blinkers?

One of the commonest allegations against mainstream science is that its
practitioners only see what they expect to see. Scientists often refuse to test
fringe ideas because "science" tells them that this will be a waste of time
and effort. Hence they miss ideas which could be very valuable.

This is the "blinkers" argument, by analogy with the leather shields
placed over horses eyes so that they only see the road ahead. It is often put
forward by proponents of new-age beliefs and alternative health.

It is certainly true that ideas from outside the mainstream of science can
have a hard time getting established. But on the other hand the opportunity
to create a scientific revolution is a very tempting one: wealth, fame and
Nobel prizes tend to follow from such work. So there will always be one or
two scientists who are willing to look at anything new.

If you have such an idea, remember that the burden of proof is on you.
The new theory should explain the existing data, provide new predictions
and should be testable; remember that all scientific theories are falsifiable.
Read the articles and improve your theory in the light of your new knowl-
edge. Starting a scientific revolution is a long, hard slog. Don't expect it to
be easy. If it was, we would have them every week. People putting forward
extraordinary claims often refer to Galileo as an example of a great genius
being persecuted by the establishment for heretic theories. They claim that
the scientific establishment is afraid of being proved wrong, and hence is
trying to suppress the truth. This is a classic conspiracy theory. The Con-
spirators are all those scientists who have bothered to point out flaws in the
claims put forward by the researchers. The usual rejoinder to someone who

says "They laughed at Columbus, they laughed at Galileo" is to say "But
they also laughed at Bozo the Clown".

1.2.8 Why should we worry?

I have argued that the scientific method provides an excellent guideline for
studying the world around us. It is, of course, conceivable that there are
other "planes of thought" but their presence and properties, and what may
happen in them is a matter of belief.

Through time "alternative" sciences regularly rise their head and are
debunked. One might be bothered about their presence since it does say
something less than flattering about human psychology. But even if one
defends these beliefs on the basis of free speech, one should be aware that
they sometimes represent more than idle talk. For example, there is this
recent news article

• ALTERNATIVE MEDICINE: REPORT SEEKS TO TAKE NIH INTO A
NEW AGE! What may rank as the most credulous document in medical
history was unveiled yesterday in a Senate conference room. Senator Tom
Harkin (D-IA), who fathered the 1991 legislation that created the NIH Office
of Alternative Medicine, admitted that the program had "gotten off to a slow
start" due to opposition from "traditional" medicine. It should soar now; the
420-page report, "Alternative Medicine: Expa / !edia Hon ons," lays

out an OAM agenda for research into everything from Lakota medicine wheels
to laying on of hands and homeopathic medicines. Homeopathic medicines
employ dilutions far beyond the point at which a single molecule would re-
main, but the water "remembers." Where does physics fit in? Well, when
really weird things happen, like mental healing at a distance, it must be quan-
tum mechanics (Brian Josephson is cited for authority). Medical ethics are
not ignored: the possibility of distant organisms being harmed by non-local
mental influence is raised, and board certification of mental healers is pro-
posed "l,o protect consumers from predatory quacks. " An entire chapter is
devoted to "Bioelectromagnetics." This is tricky stuff: "Weak EMF may,
at the proper frequent'!) and site of application, produce large effects thai, are
either clinically beneficial or harmful. " 4

It truly is amazing that people will even consider this statement. In fact
it is not dismissed because it refers to science, but imagine a similar situation

4 Extracted from "What's New", by Robert L. Park (March 3, 1995) produced by The
l Physical Society.

where "really important matters" are involved, such as money, suppose a
banker were to empty an account and claim that, even though there is no
money left, the owner of the account is just as rich because his bank book
still "remembers" the balance and that this miraculous memory of wealth
past can be used to "cure" the owner's credit-card balance. Without a doubt
this banker would end up in jail or in the loony bin.

Various tests using the scientific method have proven the fallacy of the
"water with deep memory" theory. Yet these items are seriously consid-
ered and sometimes funded by Congress, diverting monies from important
programs such as education. In the OAM has had an interesting and con-
troversial history 5 , despite this it has a budget of $12 million; in 1993-1994
it dispersed about 10% of this in grants.

This is not a unique occurrence. There are many many claims which use
high-sounding scientific jargon; for example J. Randi 6 mentions that the
NIH Office of Alternative Medicine has given credence to such claims as a
cure for multiple sclerosis (despite the fact that the staff must know there is
no such thing). When such startling claims are investigated, they are found
to be merely ridiculous statements. If you are curious about these I provide
a list of WWW sites for your amusement

• A page of links, ranging from free universal energy claims to antigrav-
ity, is found in http://www.padrak.com/ine/SUBJECTS.html

• Free energy http://jabi.com/ucsa/ which is exposed in
http : //www . voicenet . com/~eric/dennis . html

• Perpetual motion machines http://www.overunity.de/finsrud.htm

• Products that miraculously improve your car's performance
http : //widget . ecn. purdue . edu/~f eiereis/magic .html

• Flat Earth Society links (pro and against)

http : //www . town . hanna . ab . ca/hemaruka/hemlinks . htm.

And yes, in case you are wondering, some of these people are serious.

It is important to differentiate between these "pseudo-scientific" cre-
ations and true science-based developments. Pseudo-science is either not

5 See for example, http://www.nas.org/nassnl/2-ll.htm,
http : //cyberwarped . com/~gcahf /ncahf /newslett/nll9-2 .html,
http://washingtonpost.com/wp-srv/WPlate/1997-08/10/0971-081097-idx.html

6 http : //www . mindspring . com/~anson/randi-hotline/1995/0046 . html

falsifiable or its results cannot be reproduced in a laboratory. If anything
like this were to happen to a scientific hypothesis it would be dismissed
forthright independently of the, belief, feelings, etc. of the researchers.

Below I present excerpts from an essay by R. Feynman on this same
issue 7 .

Cargo Cult Science (excerpts)

by Richard Feynman

During the Middle Ages there were all kinds of crazy ideas, such as that a piece
of of rhinoceros horn would increase potency. Then a method was discovered for
separating the ideas-which was to try one to see if it worked, and if it didn't work,
to eliminate it. This method became organized, of course, into science. And it
developed very well, so that we are now in the scientific age. It is such a scientific
age, in fact, that we have difficulty in understanding how witch doctors could ever
have existed, when nothing that they proposed ever really worked-or very little of
it did.

But even today I meet lots of people who sooner or later get me into a
tion about UFO's, or astrology, or some form of mysticism, expanded c
new types of awareness, ESP, and so forth. And I've concluded that it's not a sci-
entific world.

Most people believe so many wonderful things that I decided to investigate
why they did. And what has been referred to as my curiosity for investigation has
landed me in a difficulty where I found so much junk that I'm overwhelmed. First
I started out by investigating various ideas of mysticism and mystic experiences. I
went into isolation tanks and got many hours of hallucinations, so I know something
about that. Then I went to Esalen, which is a hotbed of this kind of thought (it's a
wonderful place; you should go visit there). Then I became overwhelmed. I didn't
realize how MUCH there was.

I also looked into extrasensory perception, and PSI phenomena, and the latest
craze there was Uri Geller, a man who is supposed to be able to bend keys by
rubbing them with his finger. So I went to his hotel room, on his invitation, to see
a demonstration of both mind reading and bending keys. He didn't do any mind
reading that succeeded; nobody can read my mind, I guess. And my boy held a
key and Geller rubbed it, and nothing happened. Then he told us it works better
under water, and so you can picture all of us standing in the bathroom with the
water turned on and the key under it, and him rubbing the key with his finger.
Nothing happened. So I was unable to investigate that phenomenon.

7 The complete version can be found in the World-Wide- Web at
http://www.pd.infn.it/wwwcdf/science.html

But then I began to think, what else is there that we believe? (And I thought
then about the witch doctors, and how easy it would have been to check on them
by noticing that nothing really worked.) So I found things that even more people
believe, such as that we have some knowledge of how to educate. There are big
schools of reading methods and mathematics methods, and so forth, but if you
notice, you'll see the reading scores keep going down-or hardly going up-in spite of
the fact that we continually use these same people to improve the methods. There's
a witch doctor remedy that doesn't work. It ought to be looked into; how do they
know that their method should work? Another example is how to treat criminals.
We obviously have made no progress-lots of theory, but no progress-in decreasing
the amount of crime by the method that we use to handle criminals.

Yet these things are said to be scientific. We study them. And I think ordinary
people with common sense ideas are intimidated by this pseudo-science. A teacher
who has some good idea of how to teach her children to read is forced by the school
system to do it some other way-or is even fooled by the school system into thinking
that her method is not necessarily a good one. Or a parent of bad boys, after
disciplining them in one way or another, feels guilty for the rest of her life because
she didn't do "the right thing," according to the experts.

So we really ought to look into theories that don't work, and science that isn't
science.

I think the educational and psychological studies I mentioned are examples of
what I would like to call cargo cult science. In the South Seas there is a cargo
cult of people. During the war they saw airplanes with lots of good materials, and
they want the same thing to happen now. So they've arranged to make things
like runways, to put fires along the sides of the runways, to make a wooden hut
for a man to sit in, with two wooden pieces on his head to headphones and bars
of bamboo sticking out like antennas-he's the controller-and they wait for the
airplanes to land. They're doing everything right. The form is perfect. It looks
exactly the way it looked before. But it doesn't work. No airplanes land. So I call
these things cargo cult science, because they follow all the apparent precepts and
forms of scientific investigation, but they're missing something essential, because
the planes don't land.

Now it behooves me, of course, to tell you what they're missing. But it would
be just about as difficult to explain to the South Sea islanders how they have to
arrange things so that they get some wealth in their system. It is not something
simple like telling them how to improve the shapes of the earphones. But there
is one feature I notice that is generally missing in cargo cult science. That is the
idea that we all hope you have learned in studying science in school-we never say
explicitly what this is, but just hope that you catch on by all the examples of
scientific investigation. It is interesting, therefore, to bring it out now and speak
of it explicitly. It's a kind of scientific integrity, a principle of scientific thought
that corresponds to a kind of utter honesty-a kind of leaning over backwards. For
example, if you're doing an experiment, you should report everything that you think
might make it invalid-not only what you think is right about it: other causes that
could possibly explain your results; and things you thought of that you've eliminated

by some other experiment, and how they worked-to make sure the other fellow can
tell they have been eliminated.

Details that could throw doubt on your interpretation must be given, if you
know them. You must do the best you can-if you know anything at all wrong, or
possibly wrong-to explain it. If you make a theory, for example, and advertise it,
or put it out, then you must also put down all the facts that disagree with it, as
well as those that agree with it. There is also a more subtle problem. When you
have put a lot of ideas together to make an elaborate theory, you want to make
sure, when explaining what it fits, that those things it fits are not just the things
that gave you the idea for the theory; but that the finished theory makes something
else come out right, in addition.

In summary, the idea is to give all of the information to help others to judge
the value of your contribution; not just the information that leads to judgment in
one particular direction or another.

The easiest way to explain this idea is to contrast it, for example, with adver-
tising. Last night I heard that Wesson oil doesn't soak through food. Well, that's
true. It's not dishonest; but the thing I'm talking about is not just a matter of not
being dishonest; it's a matter of scientific integrity, which is another level. The fact
that should be added to that advertising statement is that no oils soak through
food, if operated at a certain temperature. If operated at another temperature,
they all will-including Wesson oil. So it's the implication which has been conveyed,
not the fact, which is true, and the difference is what we have to deal with.

We've learned from experience that the truth will come out. Other experi-
menters will repeat your experiment and find out whether you were wrong or right.
Nature's phenomena will agree or they'll disagree with your theory. And, although
you may gain some temporary fame and excitement, you will not gain a good rep-
utation as a scientist if you haven't tried to be very careful in this kind of work.
And it's this type of integrity, this kind of care not to fool yourself, that is missing
to a large extent in much of the research in "alternative science" .

I would like to add something that's not essential to the science, but something
I kind of believe, which is that you should not fool the layman when you're talking
as a scientist. I'm talking about a specific, extra type of integrity that is not lying,
but bending over backwards to show how you're maybe wrong, that you ought to
have when acting as a scientist. And this is our responsibility as scientists, certainly
to other scientists, and I think to laymen.

For example, I was a little surprised when I was talking to a friend who was
going to go on the radio. He does work on cosmology and astronomy, and he
wondered how he would explain what the applications of his work were. "Well,"
I said, "there aren't any." He said, "Yes, but then we won't get support for more
research of this kind." I think that's kind of dishonest. If you're representing
yourself as a scientist, then you should explain to the layman what you're doing-
and if they don't support you under those circumstances, then that's their decision.

One example of the principle is this: If you've made up your mind to test a
theory, or you want to explain some idea, you should always decide to publish it
whichever way it comes out. If we only publish results of a certain kind, we can

make the argument look good. We must publish BOTH kinds of results.

So I have just one wish for you-the good luck to be somewhere where you are
free to maintain the kind of integrity I have described, and where you do not feel
forced by a need to maintain your position in the organization, or financial support,
or so on, to lose your integrity. May you have that freedom.

1.3 Large numbers

These notes deal with space and time. The first thing we notice about the
universe around us is how big it is. In order to quantify things in cosmology
very large numbers are required and the endless writing of zeroes quickly
becomes tedious. Thus people invented what is called the scientific notation
which is a way of avoiding writing many zeroes. For example the quantity
'one million' can be written as 1,000,000 which is a one followed by six
zeroes, this is abbreviated as 10 6 (the little number above the zero is called
the exponent and denotes the number of zeroes after the one) . In this way
we have

one million = 1, 000, 000 = 10 6

one billion = 1, 000, 000, 000 = 10 9

one trillion = 1, 000, 000, 000, 000 = 10 12 , etc.

So much for large numbers. There is a similar short-hand for small
numbers, the only difference is that the exponent has a minus sign in front:

one tenth = 0.1 = 10" 1

one thousandth = 0.001 = 10 -3

one millionth = 0.000001 = 10 -6 , etc.

(1.2)

In order to get several times the above quantities one multiplies by or-
dinary numbers, so, for example, 8 x 10 6 =eight millions, 4 x 10 -12 =four
trillionths, etc.

This notation is a vast improvement also on the one devised by the
Romans, and which was used up until the Renaissance. For example, our
galaxy, the Milky Way, has a diameter of about 10 5 light years (a light year
is the distance light travels in one year), in Roman numerals

10 5 = MMMMMMMMMMMMMMMMMMMM

MMMMMMMM M M M M M M MM MM MM
MMMMMMMMMMMMMMMMMMMM
MMMMMMMMMMMMMMMMMMMM
MMMMMMMMMMMMMMMMMMMM

The Andromeda galaxy is about 2 x 10 6 (two million) light years from our
galaxy, in Roman numerals writing this distance requires 40 lines.

Appendix: Examples of large numbers

Very small and very large numbers are not the sole property of cosmology,
there are many cases where such numbers appear. What is hard to do is
visualize the meaning of something like a million or a billion. Below I provide
several examples of large and small numbers.

In the table for temperatures the values are given in degrees Kelvin; a degree
Kelvin equals a degree Celsius, but zero degrees Kelvin corresponds to —273.16
degrees Celsius. In order to change to degrees Fahrenheit you need to do the
following operation:

Deg. Fahrenheit = 1.8 x Deg. Kelvin - 459.

Absolute zero, the temperature at which all systems reach their lowest energy level,
corresponds to zero degrees Kelvin, and —459 degrees Fahrenheit.

Times (in seconds)

Distances (in meters)

Velocities (in meters per second)

Masses (in kilograms)

Tyranno

3 Typical

4 Typical

Temperatures (in deg. Kelvin)

Monies (in 1994 US dollars)

T*hTjSicS 7

Chapter 2

Greek cosmology

The first "cosmologies" were based on creation myths in which one or
more deities made the universe out of sheer will, or out of their bodily
fluids, or of the carcass of some god they defeated, etc. A few examples
of such "theories" of the universe are provided in this chapter. These are
hardly scientific theories in the sense that they have almost no support form
observation and in that they predict very few things outside of the fact that
there is a world (if everything is due to the whims of the Gods then there
is very little one can predict). It is an interesting comment on the workings
of the human mind that quite different cultures produced similar creation
myths.

The first scientific cosmology was created by the Greeks more than 2000
years ago, and this chapter also describes these ideas and their origin. The
Greeks used some of the knowledge accumulated by earlier civilizations,
thus this chapter begins with a brief description of the achievements of
the Egyptians and Babylonians. We then consider the highlights of Greek
cosmology culminating with Ptolemy's system of the world.

2.1 Egypt and Babylon
2.1.1 Babylon

The Babylonians lived in Mesopotamia, a fertile plain between the Tigris
and Euphrates rivers (see Fig. 2.1). They developed an abstract form of
writing based on cuneiform (wedge-shaped) symbols. Their symbols were
written on wet clay tablets which were baked in the sun; many thousands
of these tablets have survived to this day; an example is shown in Fig. 2.1.

1

Figure 2.1: Left: Region dominated by the Babylonian civilization. Right:
example of a cuneiform tablet containing Pythagorean triples.

The Babylonian apparently believed the Earth to be a big circular plane
surrounded by a river beyond which lies an impassable mountain barrier,
with the whole thing resting on a cosmic sea. No human may cross the river
surrounding the Earth. The mountains support the vault of heaven, which
is made of a very strong metal. There is a tunnel in the northern mountains
that opens to the outer space and which also connects two doors, one in the
East and one in the West. The sun comes out through the eastern door,
travels below the metallic heavens and then exits through the western door;
he spends the nights in the tunnel.

The creation myth is more lively than the Egyptian version. It imagines
that the cosmic ocean Apsu mixed with chaos Tiamat and eventually gener-
ated life. For a while life was good for the gods but there came a time when
Tiamat felt her domain was too small and made war against the other gods.
All but Marduk were afraid of her, so Marduk, after getting all the powers
from the frightened gods, fought Tiamat. When Tiamat opened her mouth
to swallow him he thrust a bag full with hurricane winds into her so that
she swelled and, taking advantage of her indisposition, Marduk pierced her
with his lance and killed her. Then he split Tiamat's carcass making the
lower half the earth and the upper the heavens. Finally Marduk mixed his
bloodown blood with the earth to make men for the service of the gods.

Babylonians and Chaldeans observed the motion of the stars and planets
from the earliest antiquity (since the middle of the 23rd century B.C.). They
cataloged the motion of the stars and planets as well as the occurrence of
eclipses and attempted to fit their behavior to some numerical theories.
Many of these observations were used for astrological prophesying and, in
fact, they were the originators of astrology. They believed that the motions

and changes in the stars and planets determine (or so they believed) what
occurs on this planet.

The Babylonians excelled in computational mathematics, they were able
to solve algebraic equations of the first degree, understood the concept of
function and realized the truth of Pythagoras' theorem (without furnishing
an abstract proof). One of the clay tablets dated from between 1900 and
1600 B.C. contains answers to a problem containing Pythagorean triples, i.e.
numbers a, b, c with a 2 + b 2 = c 2 . It is said to be the oldest number theory
document in existence. The Babylonians had an advanced number system
with base 60 rather than the base 10 of common today. The Babylonians
divided the day into 24 hours, each hour into 60 minutes, each minute into
60 seconds. This form of counting has survived for 40 centuries.

2.1.2 Egypt

The anciebloodnt Egyptians conceived the sky as a roof placed over the
world supported by columns placed at the four cardinal points. The Earth
was a flat rectangle, longer from north to south, whose surface bulges slightly
and having (of course) the Nile as its center. On the south there was a river
in the sky supported by mountains and on this river the sun god made his
daily trip (this river was wide enough to allow the sun to vary its path as it
is seen to do). The stars were suspended from the heavens by strong cables,
but no apparent explanation was given for their movements.

There is no unique Egyptian creation myth, yet one of the most colorful
versions states that at the beginning of the world, Nuit, the goddess of the
night, was in a tight embrace with her husband Sibil, the earth god. Then
one day, without an obvious reason, the god Shu grabed her and elevated her
to the sky (to become the sky) despite the protests and painful squirmings
of Sibu. But Shu has no sympathy for him and freezes Sibu even as he is
thrashing about. And so he remains to this day, his twisted pose generating
the irregularities we see on the Earth's surface (see Fig. 2.2). Nuit is sup-
ported by her arms and legs which become the columns holding the sky. The
newly created world was divided into four regions or houses, each dominated
by a god. Since the day of creation Sibu has been clothed in verdure and
generations of animals prospered on his back, but his pain persists.

An extended version of this myth imagines that in the beginning the
god Tumu suddenly cried "Come to me!" across the cosmic ocean, whence a
giant lotus flower appeared which had the god Ra inside, then Ra separates
Nuit and Sibu, and the story proceeds as above. It is noteworthy that
creation did not come through muscular effort, but through Tumu's voiced

Figure 2.2: Nuit the sky above Sibu the Earth after being separated by Shu
in a version of the Egyptian creation myth.

command. This later evolved into the belief that the creator made the world
with a single word, then with a single sound (yet the creation through pure
thought was not considered).

After creation the gods, especially Thot (Fig. 2.3), teach the arts and
sciences to the Egyptians. In particular Thot taught the Egyptians how
to observe the heavens and the manner in which the planets and the sun
move, as well as the names of the (36) constellations (though he apparently
neglected to tell them about eclipses which are never referred to) .

Figure 2.3: The Egyptian god Thot.

The study of the heavens was not made for altruistic purposes but with
very practical aims: a good calendar was necessary in order to prepare
for the regular flooding of the Nile as well as for religious purposes. The
Egyptian calendar had a year of precisely 365 days and was used for many
centuries; curiously they never corrected for the fact that the year is 365

1/4 days in length (this is why every four years we have a leap year and
add a day to February) and so their time reckoning was off one day every
four years. After 730 years this deficiency adds up to 6 months so that the
calendar announced the arrival of summer at the beginning of winter. After
1460 years the Egyptian calendar came back on track and big celebrations
ensued.

Egyptians knew and used the water clock whose origin is lost in the
mists of time, the oldest clock in existence dates from the reign of the pharo
Thutmose III (about 1450 B.C.) and is now in th Berlin Museum.

Most of Egyptian mathematics was aimed at practical calculations such
as measuring the Earth (important as the periodic Nile floods erased prop-
erty boundary marks) and business mathematics. Their number system was
clumsy (addition was not too bad but multiplication is very cumbersome).
To overcome this deficiency the Egyptians devised cunning ways to multiply
numbers, the method, however, was very tedious: to obtain 41 x 59 = 2419,
nine operations had to be performed (all additions and subtractions); yet
they were able to calculate areas and estimate the number n. Examples of
calculations have survived in several papyri (Fig. 2.4).

Unlike the Greeks who thought abstractly about mathematical ideas, the
Egyptians were only concerned with practical arithmetic. In fact the Egyp-
tians probably did not think of numbers as abstract quantities but always
thought of a specific collection of objects when a number was mentioned.

2.2 Other nations

None of the early civilizations lacked a cosmology or creation myths. In this
section a brief summary of some of these myths is presented.

2.2.1 India

The traditional Indian cosmology states that the universe undergoes cyclic
periods of birth, development and decay, lasting 4.32 x 10 9 years, each of
these periods is called a Kalpa or "day of Brahma" . During each Kalpa the
universe develops by natural means and processes, and by natural means
and processes it decays; the destruction of the universe is as certain as the
death of a mouse (and equally important). Each Kalpa is divided into 1000
"great ages" , and each great age into 4 ages; during each age humankind de-
teriorates gradually (the present age will terminate in 426,902 years). These
is no final purpose towards which the universe moves, there is no progress,
only endless repetition. We do not know how the universe began, perhaps

1 i i <■ I irfa of] a bastet [I ispliere] ^ ^ ^ H 1 1" I I I - — -» CT>

fl^ £=> 3 /7 "fr^ tL ™" After subtracting 2/3 +1/16 +1/18. You get 7 +1/9.

111! KM. ! h @ ^c^ _ „ni-o>iiiifl><=»<n>

•,8f»->t

mm /J^fl <

Figure 2.4: An example of Egyptian papyri, the Moscow papyrus and its
translation; the text contains the estimate -k ~ 256/81 = 3.1605.

Brahma laid it as an egg and hatched it; perhaps it is but an error or a joke
of the Maker.

This description of the universe is remarkable for the enormous numbers
it uses. The currently accepted age of the universe is about 10 18 seconds
and this corresponds to about 7 Kalpas+335 great ages. A unique feature
of Indian cosmology is that no other ancient cosmology manipulates such
time periods.

In the Surya Siddanta it is stated that the stars revolved around the
cosmic mountain Meru at whose summit dwell the gods. The Earth is
a sphere divided into four continents, the planets move by the action of
a cosmic wind and, in fact, the Vedic conception of nature attributes all
motion to such a wind. It was noted that the planets do not move in perfect
circles and this was attributed to "weather forms" whose hands were tied to
the planets by "cords of wind"

2.2.2 China

The Chinese have a very long history of astronomical observations reaching
back to the 13th century B.C. They noted solar eclipses as well as supernova
events (exploding stars). The most impressive of these events was the ob-
servation on 1054 A.D. of such a supernova event which lasted for 2 years,
after that the star dimmed and disappeared from view. The astronomical
observations were sufficiently precise for later astronomers to determine that
the location of that exploding star is now occupied by the crab nebula (Fig.
2.5); it was then shown that this nebula is expanding and, extrapolating
backwards, that this expansion started in 1054 A.D.

Figure 2.5: The Crab nebula, the remnant of a supernova.

The first Chinese cosmography imagines a round sky over a square Earth
with the sun and heavens revolving around the Earth. Later this was re-
placed by a round Earth around which all heavenly bodies rotate. These
theories propagated throughout Eastern Asia.

2.3 Early Greeks

The Greeks were apparently the first people to look upon the heavens as a
set of phenomena amenable to human comprehension and separated from
the sometimes fickle whims of the gods. They were able to extract an great
amount of information using nothing but basic reasoning and very elemen-
tary observations. This makes their results all the more amazing.

In the earliest times their view of the world and its origin was firmly
based on creation myths consolidated by Homer in the Iliad and Odyssey,
as the culture evolved this view of the universe evolved and distanced itself
from the purely religious outlook.

2.3.1 Mythology

A simplified version of the Greek creation myth follows.

In the beginning there was only chaos. Then out of the void appeared
Night and Erebus, the unknowable place where death dwells. All else was
empty, silent, endless, darkness. Then somehow Love (Eros) was born bring-
ing a start of order. From Love came Light and Day. Once there was Light
and Day, Gaea, the earth appeared.

Then Erebus slept with Night, who gave birth to Aether, the heavenly
light, and to Day, the earthly light. Then Night alone produced Doom,
Fate, Death, Sleep, Dreams, Nemesis, and others that come to man out of
darkness.

Meanwhile Gaea alone gave birth to Uranus, the heavens. Uranus be-
came Gaea's mate covering her on all sides. Together they produced the
three Cyclops, the three Hecatoncheires, and twelve Titans.

However, Uranus was a bad father and husband. He hated the Heca-
toncheires and imprisoned them by pushing them into the hidden places of
the earth, Gaea's womb. This angered Gaea and she plotted against Uranus.
She made a flint sickle and tried to get her children to attack Uranus. All
were too afraid except, the youngest Titan, Cronus.

Gaea and Cronus set up an ambush of Uranus as he lay with Gaea at
night. Cronus grabbed his father and castrated him, with the stone sickle,
throwing the severed genitals into the ocean. The fate of Uranus is not clear.
He either died, withdrew from the earth, or exiled himself to Italy. As he
departed he promised that Cronus and the Titans would be punished. From
his spilt blood came the Giants, the Ash Tree Nymphs, and the Erinyes.
From the sea foam where his genitals fell came Aphrodite.

Cronus became the next ruler. He imprisoned the Cyclops and the Heca-
toncheires in Tartarus. He married his sister Rhea, under his rule he and
the other Titans had many offspring. He ruled for many ages. However,
Gaea and Uranus both had prophesied that he would be overthrown by a
son. To avoid this Cronus swallowed each of his children as they were born.
Rhea was angry at the treatment of the children and plotted against Cronus.
When it came time to give birth to her sixth child, Rhea hid herself, and
after the birth she secretly left the child to be raised by nymphs. To conceal
her act she wrapped a stone in swaddling clothes and passed it off as the
baby to Cronus, who swallowed it.

This child was Zeus. He grew into a handsome youth on Crete. He
consulted Metis on how to defeat Cronus. She prepared a drink for Cronus
designed to make him vomit up the other children. Rhea convinced Cronus

to accept his son and Zeus was allowed to return to Mount Olympus as
Cronus's cup-bearer. This gave Zeus the opportunity to slip Cronus the
specially prepared drink. This worked as planned and the other five children
were vomited up. Being gods they were unharmed. They were thankful to
Zeus and made him their leader.

Cronus was yet to be defeated. He and the Titans, except Prometheus,
Epimetheus, and Oceanus, fought to retain their power. Atlas became their
leader in battle and it looked for some time as though they would win and
put the young gods down. However, Zeus was cunning. He went down to
Tartarus and freed the Cyclops and the Hecatoncheires. Prometheus joined
Zeus as well who returned to battle with his new allies. The Cyclops pro-
vided Zeus with lighting bolts for weapons. The Hecatoncheires he set in
ambush armed with boulders. With the time right, Zeus retreated drawing
the Titans into the Hecatoncheires's ambush. The Hecatoncheires rained
down hundreds of boulders with such a fury the Titans thought the moun-
tains were falling on them. They broke and ran giving Zeus victory.

Zeus exiled the Titans who had fought against him into Tartarus. Except
for Atlas, who was singled out for the special punishment of holding the
world on his shoulders.

However, even after this victory Zeus was not safe. Gaea angry that
her children had been imprisoned gave birth to a last offspring, Typhoeus.
Typhoeus was so fearsome that most of the gods fled. However, Zeus faced
the monster and flinging his lighting bolts was able to kill it. Typhoeus was
buried under Mount Etna in Sicily.

Much later a final challenge to Zeus rule was made by the Giants. They
went so far as to attempt to invade Mount Olympus, piling mountain upon
mountain in an effort to reach the top. But, the gods had grown strong and
with the help of Heracles the Giants were subdued or killed.

One of the most significant features of the Greek mythology is the pres-
ence of the Fates: these were three goddesses who spend the time weaving
a rug where all the affairs of men and gods appear. There is nothing that
can be done to alter this rug, even the gods are powerless to do so, and it is
this that is interesting. For the first time the idea appears of a force which
rules everything, even the gods.

2.3.2 Early cosmology

In their many travels the early Greeks came into contact with older civi-
lizations and learned their mathematics and cosmologies. Early sailors re-

lied heavily on the celestial bodies for guidance and the observation that
the heavens presented very clear regularities gave birth to the concept that
these regularities resulted, not from the whims of the gods, but from physical
laws. Similar conclusions must have been drawn from the regular change of
the seasons. This realization was not sudden, but required a lapse of many
centuries, yet its importance cannot be underestimated for it is the birth of
modern science.

The earliest of the Greek cosmologies were intimately related to mythol-
ogy: earth was surrounded by air above, water around and Hades below;
ether surrounded the earth- water-Hades set (Fig. 2.6),

The MythofogicaCWorfdView

Ocean
J Tartaros CHadix)

Figure 2.6: The universe according to Greek mythology.

This system was soon replaced by more sophisticated views on the nature
of the cosmos. Two interesting examples were first the claim of Anaxagoras
of Clazomenae that the Moon shines only through the light it reflects from
the sun, and that that lunar eclipses are a result of the earth blocking the
sunlight in its path to the moon; he also believed the Sun to be a ball of
molten iron larger than the Peloponesus.

Another remarkable feat was the prediction of a solar eclipse by Thales in
585 B.C. (for which he used the data obtained by Babylonian astronomers).
During this period other ideas were suggested, such as the possibility of
an infinite, eternal universe (Democritus) and a spherical immovable Earth
(Parmenides).

Thales of Miletus (624 B.C. - 546 B.C.). Born and died
in Miletus, Turkey. Thales of Miletus was the first known
Greek philosopher, scientist and mathematician. None of
his writing survives so it is difficult to determine his views
and to be certain about his mathematical discoveries. Ik-
is credited with five theorems of elementary geometry: (i)
A circle is bisected by any diameter, (ii) The base angles
of an isosceles triangle are equal. (Hi) The angles between
two intersecting straight lines are equal, (iv) Two triangles
are congruent if they have two angles and one side equal.
(v) An angle in a semi-circle is a right angle. Thales is
believed to have been the teacher of Anaximander and he
is the first natural philosopher in the Milesian School. 1 .

Despite these strikingly "modern" views about the sun and moon, the
accepted cosmologies of the time were not so advanced. For example, Thales
believed that the Earth floats on water (and earthquakes were the result of
waves in this cosmic ocean), and all things come to be from this cosmic
ocean. In particular the stars float in the upper waters which feed these
celestial fires with their "exhalations". The motion of the stars were as-
sumed to be governed by (then unknown) laws which are responsible for the
observed regularities.

A good example of the manner in which the Greeks drew logical con-
clusions from existing data is provided by the argument of Anaxagoras who
pointed out that meteors, which are seen to fall from the heavens, are made
of the same materials as found on Earth, and then hypothesized that the
heavenly bodies were originally part of the Earth and were thrown out by
the rapid rotation of the Earth; as the rapid rotation of these bodies de-
creases they are pulled back and fall as meteors. This conclusion is, of
course, wrong, but the hypothesis proposed does demonstrate imagination
as well as close adherence to the observed facts.

The early Greek cosmological theories did explain all the data avail-
able at the time (though they made no predictions). And, even with these
deficiencies, this period is notable for the efforts made to understand the
workings of Nature using a rational basis. This idea was later adopted by
Plato and is the basis of all modern science.

There are many other early cosmologies, for example, Anaximander believed
the Earth to be surrounded by a series of spheres made of mist and surrounded by

a big fire; the Sun, Moon and stars are glimpses of this fire through the mist. In a
different version of his cosmology he imagined the Earth to be a cylinder floating in
space. In a more poetical vein, Empedocles believed the cosmos to be egg-shaped
and governed by alternating reigns of love and hate.

Parmenides of Elea (515 B.C. - 445 B.C.). Born in Elea,
a Greek city in southern Italy (today called Velia); almost
certainly studied in Athens and there is ample evidence
that he was a student of Anaximander and deeply influ-
enced by the teachings of the Pythagoreans, whose religious
and philosophical brotherhood he joined at their school in
Crotona. All we have left of his writings are about 160
lines of a poem called Nature, written for his illustrious
student Zeno and preserved in the writings of later philoso-
phers such as Sextus Empiricus. His style influenced by
Pythagorean mysticism.

Anaxagoras of Clazomenae (499 B. C. - 4^8 B. C). Greek,
born in Ionia, lived in Athens He was imprisoned for claim-
ing that the sun was not a god and that the moon reflected
the sun's light. While in prison he tried to solve the prob-
lem of squaring the circle, that is constructing with ruler
and compasses a square with area equal to that of a given
circle (this is the first record of this problem being stud-
ied). He was saved from prison by Pericles but had to leave
.\i hcu.v

The early Greeks also considered the composition of things. It was during
these times that it was first proposed (by Anaximines of Miletus, c. 525
B.C.) that everything was supposed to be made out of four "elements":

-ere supposed to earth, water, air and fire. This idea prevailed for many centuries. It was
t of four believed that earth was some sort of condensation of air, while fire was some

sort of emission form air. When earth condenses out of air, fire is created
in the process. Thus we have the first table of the elements (see Fig. 2.7)

A

FIRE

AIR

WATER

EARTH

Figure 2.7: The earliest table of the elements.

This, however was not universally accepted. The most notable detractor
was Democritus who postulated the existence of indestructible atoms (from
the Greek a-tome: that which cannot be cut) of an infinite variety of shapes
and sizes. He imagined an infinite universe containing an infinite number of
such atoms, in between the atoms there is an absolute void.

Democritus of Abdera ( 460 B.C. - 310 B.C.). Democri-
tus is best known for his atomic theory but he was also
an excellent geometer. Very little is known of his life but
we know that Leucippus was his teacher. He's believed to
have traveled widely, perhaps spent a considerable time in
Egypt, and he certainly visited Persia. Democritus wrote
many mathematical works but none survive. He claimed
that the universe was a purely mechanical system obeying
fixed laws. He explained the origin of the universe through
atoms moving randomly and colliding lo form larger bodies
and worlds. He also believed that space is infinite and eter-
nal, and that the number of atoms are infinite. Democri-
tus's philosophy contains an early form of the conservation
of energy.

The Pythagoreans
originated the idea that the
world could be understood
through mathematics was

2.3.3 The Pythagoreans

About five centuries B.C. the school founded by the Greek philosopher,
mathematician and astronomer Pythagoras flourished in Samos, Greece.
The Pythagoreans believed (but failed to prove) that the universe could
be understood in terms of whole numbers. This belief stemmed from obser-
vations in music, mathematics and astronomy. For example, they noticed
that vibrating strings produce harmonious tones when the ratios of the their
lengths are whole numbers. From this first attempt to express the universe
in terms of numbers the idea that the world could be understood through
mathematics was born, a central concept in the development of mathematics
and science.

The importance of pure numbers is central to the Pythagorean view of
the world. A point was associated with 1, a line with 2 a surface with 3 and
a solid with 4. Their sum, 10, was sacred and omnipotent 2 .

Pythagoras of Samos (580-500 B.C.). Born Samos,

Greece, died in Italy. Pythagoras was a Greek philoso-
pher responsible for important developments in mathemat-
ics, astronomy, and the theory of music. He founded a
philosophical and religious school in i 'rot on that had many
followers. Of his act ual work not hiug is directly known. His
school practiced secrecy and communalism making it hard
to distinguish between the work of Pythagoras and that of
his followers.

Pythagoras also developed a rather sophisticated cosmology. He and
his followers believed the earth to be perfectly spherical and that heavenly
bodies, likewise perfect spheres, moved as the Earth around a central fire
invisible to human eyes (this was not the sun for it also circled this central
fire) as shown in Fig. 2.8. There were 10 objects circling the central fire
which included a counter-earth assumed to be there to account from some
eclipses but also because they believed the number 10 to be particularly

2 Some relate this to the origin of the decimal system, but it seems to me more reason-
able to associate the decimal system to our having ten fingers.

sacred. This is the first coherent system in which celestial bodies move in
circles, an idea that was to survive for two thousand years.

Figure 2.8: The universe according to the Pythagoreans.

It was also stated that heavenly bodies give forth musical sounds "the
harmony of the spheres" as they move in the cosmos, a music which we
cannot discern, being used to it from childhood (a sort of background noise);
though we would certainly notice if anything went wrong! The Pythagoreans
did not believe that music, numbers and cosmos were just related, they
believed that music was number and that the cosmos was music

Pythagoras is best known for the mathematical result (Pythagoras' the-
orem) that states that the sum of the squares of the sides of a right triangle
equals the square of the diagonal; see Fig. 2.9. This result, although known
to the Babylonians 1000 years earlier, was first proved by Pythagoras (al-
legedly: no manuscript remains). Pythagoras' theorem will be particularly
important when we study relativity for, as it turns out, it is not valid in
the vicinity of very massive bodies! Similar statements hold for Euclid's
postulate that parallel lines never meet, see Sect. ??.

A

*

Figure 2.9: Pythagoras' theorem (the areas of the squares attached to the
smaller sides of the triangle equal the area of the largest square) .

2.4 Early heliocentric systems

By the IV century B.C. observations had shown that there are two types
of stars: fixed stars whose relative position remained constant, and "wan-
dering stars" , or planets, whose position relative to the fixed stars changed
regularly. Fixed stars moved as if fixed to a sphere that turned with the
earth at its center, the planets moved about these fixed stars driven by an
unknown agency. In fact, Plato regarded the investigation of the rules that
determined the motion of the planets as a very pressing research problem.

A remarkable answer was provided by the heliocentric (!!) system of
Aristarchus of Samos. Using a clever geometric argument Aristarchus esti-
mated the size of the Sun and concluded it must be enormously larger than
the Earth; he then argued that it was inconceivable that such a behemoth
would slavishly circle a puny object like the Earth. Once he concluded this,
he concluded that the Earth must rotate on its axis in order to explain the
(apparent) motion of the stars. Thus Aristarchus conceived the main ingre-
dients of the Copernican system 17 centuries before the birth of Copernicus!
Unfortunately these views were soundly rejected by Aristotle: if the Earth
is rotating, how is it that an object thrown upwards falls on the same place?
How come this rotation does not generate a very strong wind? Due to ar-
guments such as this the heliocentric theory was almost universally rejected
until Copernicus' answered these criticisms.

;-,:''."

Aristarchus of Samos (310 B.C. - 230 B.C.). Born
and died in Greece. Aristarchus was a mathematician
and astronomer who is celebrated as the exponent of
a Sun-centered universe and for his pioneering attempt
to determine the sizes and distances of the Sun and
Moon. Aristarchus was a student of Strato of Lampsacus,
head of Aristotle's Lyceum, coming between Euclid and
Archimedes. Little evidence exists concerning the origin
of his belief in a heliocentric system; the theory was not
accepted by the Greeks and is known only because of a
summary statement in Archimedes' The Sand-Reckoner

and a reference by Plutarch. The only surviving work of
Aristarchus, On the Sizes and Distances of the Sun and
Moon, pro\ id< 1 U < t ail of 1 rouiar ka 1 >1 2 1 n 1 1 1 1 n
gmiiout . based on observation, whereby he dole] mined that
the Sun was about 20 times as distant from the Earth as the
Moon, and 20 times the Moon's size. Both these estimates
were an order of magnitude too small, but the fault was in
Aristarchus' lack of accurate instruments rather than in his
correct method of reasoning. Aristarchus also found an im-
proved value for the length of the length of the solar year.

Astronomy also progressed, with the most striking result, due to Eratos-
thenes, was accurate measurement of the Earth's circumference 3 (the fact

that the Earth is round was common knowledge) He noted that the distance The fact that the Earth is
from Alexandria to Aswan is 5,000 stadia and that when the sun casts no round was common
n i • a i i • ii i- i c r, nr> knowledge

shadow in Alexandria it casts a shadow corresponding to an angle oi 7.2"

(see Fig. 2.10). From this he determined the circumference of the Earth less
than 2% accuracy!

It is important to remember that the realization that the Earth was
round was not lost to the following centuries, so that neither Columbus nor
any of his (cultivated) contemporaries had any fear of falling off the edge of
the world when traveling West trying to reach the Indies. The controversy
surrounding Columbus' trip was due to a disagreement on the size of the
Earth. Columbus had, in fact, seriously underestimated the radius of the
Earth and so believed that the tiny ships he would command had a fair
chance of getting to their destination. He was, of course, unaware of the
interloping piece of land we now call America, had this continent not existed,
Columbus and his crew would have perished miserably in the middle of the

3 Aristotle had previously estimated i
which is about 1.6 times its actual size.

value of 400,000 stadia (1 stadiu

Figure 2.10: Description of Eratosthenes' procedure for measuring the
Earth. He reasoned that the change in angle of the shadow was caused by
moving about the Earth. By measuring the angle of the shadow at Seyene,
and then in a city that was directly north of Seyene (Alexandria) , he deter-
mined that the two cities were 7 degrees apart. That is to say, out of the 360
degrees needed to travel all the way around the world, the two cities were
360/7 of that distance. Since he knew that the two cities were about 500
miles apart, he concluded that the the Earth must be (360/7) x 500 miles
in circumference, or roughly 25,000 miles.

Eratosthenes of Cyrene (276 B.C. - 197 B.C.). Greek,
lived in Alexandria He was born in Cyrene which is now
in Libya. He worked on geometry and prime numbers. He
is best remembered for his prime number sieve which, in
modified form, is still an important tool in number theory
research. Eratosthenes measured the tilt of the earth's axis
with great accuracy and compiled a star catalogue contain-
ing 675 stars; he suggested that a leap day be added every
fourth year and tried to construct an accurately-dated his-
tory. He became blind in his old age and is said to have
committed suicide by starvation.

2.5 Aristotle and Ptolemy

There are few instances of philosophers that have had such a deep influ-
ence as Aristotle, or of cosmologists whose theories have endured as long
as Ptolemy's. Aristotle's influence is enormous ranging form the sciences to
logic. Many of his ideas have endured the test of the centuries. His cosmol-
ogy, based on a geocentric system, is not one of them. In the words of W.
Durant

His curious mind is interested, to begin with, in the process and
techniques of reasoning; and so acutely does he analyze these
that his Organon, or Instrument-the name given after his
death to his logical treatises-became the textbook of logic
for two thousand years. He longs to think clearly, though he
seldom, in extant works, succeeds; he spends half his time
defining his terms, and then he feels that he has solved the
problem.

It must me noted, however, that he forcefully argued for the sphericity of
the Earth based on data: he noted that only a spherical Earth can account
for the shadow seen on the Moon during a lunar eclipse

Ptolemy enlarged Aristotle's ideas creating a very involved model of the
solar system which endured until the Copernican revolution of the middle
16th century. When comparing the Ptolemaic system with the Copernican
heliocentric system Occam's razor (Sect. ??) instantly tells us to consider
the latter first: it provide a much simpler explanation (and, as it turns out,
a much better one) that the former.

2.5.1 Aristotelian Cosmology

Aristotle's cosmological work On The Heavens is the most influential treatise

of its kind in the history of humanity. It was accepted for more that 18

centuries from its inception (around 350 B.C.) until the works of Copernicus

in the early 1500s. In this work Aristotle discussed the general nature of the

cosmos and certain properties of individual bodies.

Aristotle believed that all bodies are made up of four elements: earth,

water, air and fire (see Fig. 2.7). These elements naturally move up or Aristotle believed that all

down, fire being the lightest and earth the heaviest. A composite object will bodies are made up of four

elements: earth, water, air
have the features ol the element which dominates; most things are of this and fire

sort. But since the elements in, for example, a worm, are not where they The elements naturally

belong (the fiery part is too low being bound by the earth part, which is move up or down < fire being

According to Aristotle all
bodies, by their very
nature, have a natural way

a bit too high), then the worm is imperfect. All things on earth are thus
imperfect.

The idea that all bodies, by their very nature, have a natural way of
moving is central to Aristotelian cosmology. Movement is not, he states, the
result of the influence of one body on another

Aristotle (384 B.C. - 322 B.C.). Born Stagirus, Greece,
died Chalcis, Greece. In 367 Aristotle became a student
at Plato's Academy in Athens. Soon he became a teacher
at the Academy. After Plato's death in 347 B.C., Aristotle
joined the court of Hermias of Atarneus. In 343 B.C. he be-
came tutor to the young Alexander the Great at the court
of Philip II of Macedonia. In 335 B.C. Aristotle founded
his own school the Lyceum in Athens. The Academy had
become narrow in its interests after Plato's death but the
Lyceum under Aristotle pursued a broader range of sub-
jects. Prominence was given to the detailed study of na-
ture. After the death of Alexander the Great in 323 B.C.,
anti-Macedonian feeling in Athens made Aristotle retire to
Chalcis where he died l be billowing year. Aristotle was not
primarily a mathematician but made important contribu-
tions by systematizing deductive logic. He wrote on phys-
ical subjects; some parts of his Analytica Posteriora show
an unusual grasp of mathematics. He also had a strong
interest in anatomy and the structure of living things in
general which helped him to develop a remarkable talent
lor observation.

Some bodies naturally move in straight lines, others naturally stay put.
But there is yet another natural movement: the circular motion. Since to
each motion there must correspond a substance, there ought to be some
things that naturally move in circles. Aristotle then states that such things
are the heavenly bodies which are made of a more exalted and perfect sub-
stance than all earthly objects.

Since the stars and planets are made of this exalted substance and then
move in circles, it is also natural, according to Aristotle, for these objects
to be spheres also. The cosmos is then made of a central earth (which he
accepted as spherical) surrounded by the moon, sun and stars all moving in
circles around it. This conglomerate he called "the world". Note the strange
idea that all celestial bodies are perfect, yet they must circle the imperfect
Earth. The initial motion of these spheres was caused by the action of a

21

"prime mover" which (who?) acts on the outermost sphere of the fixed stars;
the motion then trickles down to the other spheres through a dragging force.
Aristotle also addresses the question whether this world is unique or
not; he argues that it is unique. The argument goes as follows: earth (the
substance) moves naturally to the center, if the world is not unique there
ought to be at least two centers, but then, how can earth know to which of
the two centers to go? But since "earthy" objects have no trouble deciding
how to move he concludes that there can only be one center (the Earth)
circled endlessly by all heavenly bodies. The clearest counterexample was
found by Galileo when he saw Jupiter and its miniature satellite system (see
Fig. 2.11), which looks like a copy of our "world". Aristotle was wrong not
in the logic, but in the initial assumptions: things do not have a natural
motion.

Figure 2.11: Montage of Jupiter and the Galilean satellites, Io, Europa,
Ganymede, and Callisto.

It is interesting to note that Aristotle asserts that the world did not come
into being at one point, but that it has existed, unchanged, for all eternity Aristotle asserts that the

(it had to be that way since it was "perfect" ): the universe is in a kind world dld not come mt0
\« , . „ ,-, .„ • T , t -, i , , , being at one point, but that

of steady state scenario . Still, since he believed that the sphere was the it nas ex j Ste d unchanged
most perfect of the geometrical shapes, the universe did have a center (the for all eternity
Earth) and its "material" part had an edge, which was "gradual" starting
in the lunar and ending in the fixed star sphere. Beyond the sphere of the
stars the universe continued into the spiritual realm where material things

cannot be (Fig. 2.12). This is in direct conflict with the Biblical description
of creation, and an enormous amount of effort was spent by the medieval
philosophers in trying to reconcile these views.

ijk;

Figure 2.12: A pictorial view of the Aristotelian model of the cosmos.

On the specific description of the heavens, Aristotle created a complex
system containing 55 spheres(!) which, despite it complexity, had the virtue
of explaining and predicting most of the observed motions of the stars and
planets. Thus, despite all the bad publicity it has received, this model had
all the characteristics of a scientific theory (see Sect. ??): starting from
the hypothesis that heavenly bodies move in spheres around the Earth,
Aristotle painstakingly modified this idea, matching it to the observations,
until all data could be accurately explained. He then used this theory to
make predictions (such as where will Mars be a year from now) which were
confirmed by subsequent observations.

One of the fundamental
propositions of Aristoteliar
philosophy is that there is
no effect without a cause

2.5.2 The motion according to Aristotle

One of the fundamental propositions of Aristotelian philosophy is that there
is no effect without a cause. Applied to moving bodies, this proposition
dictates that there is no motion without a force. Speed, then is proportional
to force and inversely proportional to resistance

force = (resistance) x (speed)

(though none of these quantities were unambiguously defined) . This notion
is not at all unreasonable if one takes as one's defining case of motion, say,
an ox pulling a cart: the cart only moves if the ox pulls, and when the ox
stops pulling the cart stops.

Aristotle's law of motion, (from Physics, book VII, chapter
5). Then, A the movement have moved B a distance T in a
time A, then in the same time the same force A will move
|.B twice the distance T, and in |A it will move \B the
whole distance for Y: thus the rules of proportion will be
observed.

The translation into modern concepts is A —> F =force,
B — ► m = mass, F — > d =distance, and A — » t =time.

• Given a force F which moves a mass m a distance d
in a time i, it will also move half the mass by twice
the distance in the same time.

• Given a force F which moves a mass m a distance d
in a time t, it will also move half the mass the same
distance in half the time.

These three rules imply that the product of the mass and
the average speed depends only on the force. For example,
a body of constant mass under the action of a constant
force will have a constant speed. This is wrong: the speed

nth time.

Qualitatively this implies that a body will traverse a thinner medium
in a shorter time than a thicker medium (of the same length): things will
go faster through air than through water. A natural (though erroneous)

conclusion is that there could be no vacuum in Nature, for if the resistance Aristotle argued that there
became vanishingly small, a tiny force would produce a very large "motion"; could be no vacuum In
in the limit where there is no resistance any force on any body would produce
an infinite speed. This conclusion put him in direct contradiction with the
ideas of the atomists such as Democritus (see Sect. 2.3.2). Aristotle (of
course) concluded the atomists were wrong, stating that matter is in fact
continuous and infinitely divisible.

For falling bodies, the force is the weight pulling down a body and the
resistance is that of the medium (air, water, etc.). Aristotle noted that a
falling object gains speed, which he then attributed to a gain in weight.
If weight determines the speed of fall, then when two different weights are

dropped from a high place the heavier will fall faster and the lighter slower, Aristotle asserted that when
in proportion to the two weights. A ten pound weight would reach the Earth two dlfferent weights are
by the time a one-pound weight had fallen one-tenth as far. the heavier will fall faster

and the lighter slower

2.5.3 Ptolemy

The Aristotelian system was modified by Hipparchus whose ideas were popu-
larized and perfected by Ptolemy In his treatise the Almagest ( "The Great
System") Ptolemy provided a mathematical theory of the motions of the
Sun, Moon, and planets. Ptolemy vision (based on previous work by Hip-
parchus) was to envision the Earth surrounded by circles, on these circles he
imagined other (smaller) circles moving, and the planets, Sun, etc. moving
on these smaller circles. This model remained unchallenged for 14 centuries.
The system of circles upon circles was called a system of epicycles (see
Fig. 2.14). It was extremely complicated (requiring several correction fac-
tors) but it did account for all the observations of the time, including the
peculiar behavior of the planets as illustrated in Fig. 2.15. The Almagest
was not superseded until a century after Copernicus presented his heliocen-
tric theory in Copernicus' De Revolutionibus of 1543.

Ptolemy (100 - 170). Born in Ptolemais Hermii,

Egypt, died Alexandria, Egypt. One of the most in-
fluential Greek astronomers and geographers of his time,
Ptolemy propounded the geocentric theory that prevailed
for 1400 years. Ptolemy made astronomical observations
from Alexandria Egypt during the years A.D. 127-41. He
probably spent most of his life in Alexandria. He used his
observations to construct a geometric model of the universe
which accurately predicted the positions of all significant
planets and stars. This model employed combinations of
circles known as epicycles, within the framework of the ba-
sic Earth-centered system supplied by Aristotle. His model
is presented in his treatise Almagest. In a book entitled
Analemma he discussed the projection of points on the
celestial sphere. In Piaiii.spii;irrnin.i lie is concerned with
stereographic project ion. He also devised a calendar that
was followed for many centuries. There where problems
with it, however, and this required corrections of about 1
month every 6 years. This generated a lot of problems in
particular in agriculture and religion!

This model was devised in order to explain the motion of certain planets.
Imagine that the stars are a fixed background in which the planets move,
then you can imagine tracing a curve which joins the positions of a given
planet everyday at midnight (a "join the dots" game); see, for example
Fig 2.13. Most of the planets move in one direction, but Mars does not,

its motion over several months is seen sometimes to backtrack (the same
behavior would have been observed for other celestial objects had Ptolemy
had the necessary precision instruments).

Figure 2.13: This computer simulations shows the retrograde motion of
Mars (left) and the asteroid Vesta (right). Vesta's trajectory is followed
over several years; it moves from right to left (west to east), and each loop
occurs once per year. The shape of the retrograde loop depends on where
Vesta is with respect to Earth.

P

%

Figure 2.14: The simplest form of an epicycle (left) and the actual form
required to explain the details of the motion of the planets (right).

Figure 2.15: Example of how a system of epicycles can account for the
backtracking in the motion of a planet. The solid line corresponds to the
motion of Mars as it goes around the epicycle, while the epicycle itself goes
around the Earth. As seen from Earth, Mars would move back and forth
with respect to the background stars.

T^hiiSics 7

Chapter 3

From the Middle Ages to
Heliocentrism

3.1 Preamble

The Roman empire produced no scientific progress in the area of cosmol-
ogy, and the Church tainted it during most of the Middle Ages. Europe
forgot most of the discoveries of the Greeks until they were reintroduced
by Arab astronomers in the XH-th century through the Crusades and other
less distressing contacts. The Renaissance brought a breath of fresh air to
this situation, and allowed for the heretofore untouchable dogmas to be re-
examined, yet, even in this progressive climate, the influence of the Church
was still enormous and this hampered progress.

In the XVI-th century the Copernican view of the solar system saw the
light. In this same era the quality of astronomical observations improved
significantly and Kepler used these data to determine his famous three laws
describing the motion of the planets. These discoveries laid the foundation
for the enormous progress to be achieved by Galileo and then Newton.

3.2 The Middle Ages.

The development of new scientific theories came almost to a stop during the
centuries covering the Roman Empire and the Middle Ages. During this
long period there was a gradual emergence of irrational theories that threat-
ened to engulf the whole of science: astrology challenged astronomy, magic
insinuated itself into medicine and alchemy infiltrated natural science. The

1

beginning of the Christian era, when Oriental mysticism became the rage
in Greece and Rome, witnessed the appearance of exotic sects such as the
Gnostics and the Hermetics who propagated distorted and over-simplified
cosmologies ostensibly given to them by God 1 .

During the Middle Ages European mental efforts were directed towards
non-scientific pursuits. This attitude was perpetuated by the absence of
libraries and the scarcity of books (both a consequence of the economic
depression suffered by Europe at that time), and by the constraints imposed
by the Church which forbade various ares of investigations as they were felt
to be against the teachings of the Bible.

These problems did not permeate the whole world, however, and, in
fact, Arab science flourished during this time devising the now-common
Arab numerals, increasingly accurate time-keeping devices and astronomical
instruments, and providing corrections to Ptolemy's observations. Later,
through the close contacts generated by the Crusades, Arab knowledge was
carried to Europe.

The scientific climate in Europe improved by the XIII century with the
creation of the first universities. It was during this last part of the Middle
Ages that the 3 dimensional nature of space was determined and the con-
cept of force was made precise. The experimental basis of scientific inquiry
was recognized as well as the need for internal logical consistency. With
these developments the field was ready for the scientific developments of the
Renaissance.

Through all these medieval tribulations Ptolemy's magnum opus, the
Alamgest, together with Aristotle's On The Heavens survived as the cos-
mological treatises. Their influence became widespread after translations
into Latin became readily available (at least at the universities). There was
much discussion on the reconciliation of Aristotle's view of the world and
the descriptions found in the Bible. Issues such as whether the universe is
infinite and whether God can create an infinite object were the subject of
heated discussions.

Sometimes the conclusions reached by the philosophers were not sat-
isfactory to the theologians of the era and, in fact, in 1277 the bishop of
Paris collected a list of 219 propositions connected with Aristotle's doctrine
which no-one could teach, discuss or consider in any light under penalty of
excommunication. For example,

• Aristotle argued against the possibility of there being other worlds,
that is, copies of his set of spheres which are supposed to describe our

1 From, A History of Science by H. Smith Williams.

world; these arguments can be interpreted as stating that God does
not have the power to create such other worlds, an idea unacceptable
to the Church.

• Aristotle assumed matter to be eternal and this contradicted the cre-
ation of matter, and in fact, of the whole universe by the will of God.

• Aristotelian advocates believed in the eternal pre-ordained motion of
heavenly bodies which nothing could alter, this again implied limits
on the powers of God.

In my opinion there is an interesting issue connected with the conflict
between the Bible and Aristotle. It was Aristotle's belief that there are rules
which objects are, by their very nature, forced to obey without the need for
divine intervention. It is this idea that is prevalent in science today: there
are natural laws that determine the behavior of inanimate objects without
the intervention of higher authority. It is always possible to argue who or It was Aristotle's belief that

what determines these natural laws, whether there is some underlying will there are natural laws that

. determine the behavior of

behind all of this. But that lies beyond the reach of science (at least in inanimate objects without

its present form), not because the question is of no interest, but because the intervention of higher

it cannot be probed using the reliable framework provided by the scientific autnont y
method (Sect. ??).

The problems with the theory of the universe perfected by Ptolemy were
not apparent due to deficiencies in the instruments of the time. First was the
problem of keeping time accurately: there were no precise clocks (a problem
solved only when Galileo discovered the pendulum clock); a state of the art
time-keeping mechanism of that time, the water-clock, is illustrated in Fig.
3.1; such mechanisms were not significant better than the water clocks used
in Egypt starting form 1600 B.C. Secondly there was a notational problem:
large numbers were very cumbersome to write since only Roman numerals
were known (this notation has no notion of zero and of positional value; see
Sect. ?? for a comparison between modern and Roman numerals).

These problems were recognized and (eventually) solved. The Arabic
number system was slowly accepted in the Western world after its first in-
troduction around 1100 A.D. during the Crusades. The discoveries of the
other Greek scientists (not belonging to the Ptolemaic school) were also
introduced in the West during this period in the same way. The first me-
chanical clocks waere developed in Europe in the XHI-th century. They
worked using pulleys and weights but were still very inaccurate: the best
ones were able only to give the nearest hour!

<EE

W

^

.A

Figure 3.1: Illustrations of a water clock (left) and its use (right).

Despite the bad connotation the Middle Ages have, not all aspects of life
during that time were horrible. In fact the basic ideas behind the universe
in this time were very comforting to Jews, Christians and Muslims. These
ideas provided a stable framework where most people had a (reasonably)
clear view of their place in society, their duties and expectations.

The universe had the Earth at its center with all heavenly bodies circling
it. Beyond the last sphere (that of the fixed stars) lay paradise, hell was in
the bowels of the Earth (a sort of "under-Earth" ) , and purgatory was in the
regions between Earth and the Moon (Fig. 3.2). One of the main architects
of this vision was Thomas Aquinas whose view was adopted by Dante in his
Divine Comedy.

The Middle Ages provided the gestation period during which the neces-
sary conditions for the Renaissance were created. This is witnessed by the
writings of various visionaries, with Roger Bacon as the best example. Ba-
con believed that Nature can be described using mathematics and required
that all accepted theories be based on experimental evidence, not merely as
conclusions drawn from ancient treatises (which themselves have not been
tested). Many of these ideas were, of course, of Greek ancestry.

Figure 3.2: Illustration of a typical Medieval cosmological model.

Roger Bacon (1220-1292). He is remembered for his work
in mathematics, and as a early advocate of the scientific
method. He was a student at the university in Paris and
later at Oxford in England. He became a Franciscan friar
during the 1250s. His work.-, include wril ings in mathemat-
ics, alchemy, and optics. He is known to have authored
Compendium of the Si tnly <>!' I'hilo.^nphy (1272) and Com-
pendium of the Study of Theology (1292). During his life
time he experimented with ideas about the development of
gunpowder, flying machines, motorized vehicles, and tele-
scopes.

Also worth of mentioning is William of Ockham, who parted from Plato's

Entities must not be
needlessly multiplied

claim that ideas are the true and eternal reality (we only see imperfect shad-
ows cast by these ideas, and this taints our perception of Nature). Ockham
argued in his famous "razor" statement that this is an unnecessary complica-
tion in the description of Nature: pluralitas non est ponenda sine neccesitate,
entities must not be needlessly multiplied, which was discussed extensively
in Sect. ??.

William of Ockham (1285-1349). Born in Ockham - near
Ripley, Surrey - England, died in Munich, Bavaria - now
Germany. Ockham's early Franciscan education concen-
trated on logic. He studied theology at Oxford and be-
tween 1317 and 1319 he lectured on the Sentences , the
standard theology text used in imiversh ies up to the 1600's.
His opinions aroused strong opposition and he left Oxford
without his Master's Degree. He continued studying math-
ematical logic and made important contributions to it. He
considered a three valued logic where propositions can take
one of three truth values. This became i mportant for math-
ematics in the 20th Century but it is remarkable that it was
first studied by Oekbam (>00 years earlier. Ockham went to
France and was denounced by the Pope. He was excommu-
nicated and in 1328 he fled seeking the protection of Louis
IV in Bavaria (Louis had also been excommunicated). He
continued to attack papal power always employing logical
reasoning in his arguments until his death.

Yet the great majority of intellectuals accepted Ptolemy's model of the
world. But, was this acceptance based on a belief that this was an accurate
description of nature, or just on the fact that there no superior models to re-
place Ptolemy's? Some astronomers were of the second opinion, for example,
the Arab astronomer Averroes declared (in his commentary on Aristotle's
works) "we find nothing in the mathematical sciences that would lead us to
believe that eccentrics and epicycles exist" and "actually in our time astron-
omy is nonexistent; what we have is something that fits calculation but does
not agree with what is". Similarly, Bacon believed that epicycles were a
convenient mathematical description of the universe, but had no physical
reality. Another notable exception to the general acceptance of Ptolemy's
model was, perhaps not surprisingly, Leonardo da Vinci who at the dawn
of the Renaissance concluded that the Earth moves (which implies that the
Sun does not).

fc-

: '•■*''

Leonardo da Vinci (1452-1519). Born in Vinci - near Em-
polia, Italy - died in Cloux - Amboise, France. Leonardo
da Vinci had many talents in addition to his painting. He
worked in mechanics but geometry was his main love. Re-
ceived the usual elementary education of reading, writing
and arithmetic at his father's house. From 1467 to 1477
he was an apprentice learning painting, sculpture and ac-
quiring techmc il nieal skill uccpted into the
painters' guild in Florence in 1472. From that time he
worked for himself in Florence as a painter. During this
time he sketched pumps, military weapons and other ma-
Was in the service of the Duke of Milan (1482-1499) as a
painter and engine!!'. Completed six paintings and advised
on architecture, fortifications and military matters. Also
considered a hydraulic and mechanical engineer. During
this time he became interested in geometry to the point of
being neglectful of his paintings.

Leonardo studied Undid and I'acioli's Smna and began his
own geometry research, sometimes giving mechanical solu-
tions, for example gave several such methods of squaring
the circle. Wrote a book on the elementary theory of me-
chanics which appeared in Milan around 1498.
Leonardo certain]; realized the possibility of constructing
a telescope (as verified by sections of Codex Atlanticus and
Codex Arundul). He understood the fact that the Moon
shone with reflected light from the Sun. He believed the
Moon to be similar to the Earth with seas and areas of solid
grouud

In 1499 the French armies entered Milan and the Duke was
defeated. Leonardo then left Milan, traveled to Mantua,
Venice and finally Florence. Although he was under con-
stant pressure to paint, but kept his mathematical studies;
for a time was employed by Cesare Borgia as a senior mil-
itary architect and general engineer.

By 1503 he was back in Florence advising on the project
to divert the River Arno behind Pisa to help with the siege
then suffered by the city. He then produced plans for a
canal to allow Florence access to I he sea (neither was ear-

In 1506 Leonardo returned for a second period in Milan,
again his scientific work took precedence over his painting
and he was involved in hydrodynamics, anatomy, mechan-
ics, mathematics and optics.

In 1513 the French were removed from Milan and Leonardo
moved again, this time to Rome. Appears to have led
there a lonely life more devoted to mathematical studies
and technical experiments in his studio than to painting.
After three years of uuhnppiuess Leonardo accepted an in-
vitation from King Francis I to enter his service in France.
The French King gave Leonardo the title of first painter,
architect, and mechanic of the King but seems to have left
him to do as he pleased. This means that Leonardo did no
painting except to finish off some works he had with him,
St. John the Baptist, Alona Lisa and the Virgin and Child
with St Anne. Leonardo spent most of his time arranging
and editing his seicntilie studies. He died in 1519.

Finally I'd like to mention a peculiar alternative to the Aristotle+Ptolemy
view of the world: the "Dairy cosmology" , due to an Italian miller called Domenico
Scandella (1532-1599/1600?), called Menoccio. Scandella believed that God and
the angels were spontaneously generated by nature from the original chaos "just as
worms are produced from a cheese". The chaos was made of the four elements air,
water, earth and fire, and out of them a mass formed "just as cheese forms from
milk". Within this mass of cheese, worms appeared, and "the most holy majesty
declared thai, Ifnsc should be God and Ihc aju/ds." Menoccio was tried by the
Inquisition, found guilty and executed in 1599 or 1600.

3.3 The Copernican Revolution

Copernicus' model
consisted of a central ;
around which all planei
rotated, with the s

The 16th century finally saw what came to be a watershed in the develop-
ment of Cosmology. In 1543 Nicolas Copernicus published his treatise De
Revolutionibus Orbium Coelestium (The Revolution of Celestial Spheres)
where a new view of the world is presented: the heliocentric model.

It is hard to underestimate the importance of this work: it challenged
the age long views of the way the universe worked and the preponderance of
the Earth and, by extension, of human beings. The realization that we, our
planet, and indeed our solar system (and even our galaxy) are quite common
in the heavens and reproduced by myriads of planetary systems provided a
sobering (though unsettling) view of the universe. All the reassurances of
the cosmology of the Middle Ages were gone, and a new view of the world,
less secure and comfortable, came into being. Despite these "problems" and
the many critics the model attracted, the system was soon accepted by the
best minds of the time such as Galileo

Copenicus' model, a rediscovery of the one proposed by Aristarchus cen-
turies before (see Sect. ??), explained the observed motions of the planets
(eg. the peculiar motions of Mars; see Fig. ??) more simply than Ptolemy's
by assuming a central sun around which all planets rotated, with the slower
planets having orbits farther from the sun. Superimposed on this motion,
the planets rotate around their axes. Note that Copernicus was not com-
pletely divorced from the old Aristotelian views: the planets are assumed to

from the s

ving orbits farther move in circles around the sun (Fig. 3.3).

Nicholas Copernicus (Feb 19 1473 - May 24 1543 ). Born
Torun, Poland, died Frauenburg, Poland. Copernicus stud-
ied first at the University of Krakow which was famous for
mathematics, philosophy, and astronomy. Copernicus then
studied liberal arts at Bologna from 1496 to 1501, medicine
at Padua, and law at the University of Ferrara. He grad-
uated in 1503 with a doctorate in canon law. He then
took up duties at the cathedral in Frauenberg. during this
period Copernicus performed his ecclesiastical duties, prac-
ticed medicine, wrote a treatise on monetary reform, and
became interested in astronomy. In May 1514 Copernicus
circulated in manuscript Commentariolus, the first outline
of his heliocentric model; a complete description of which
was provided in 1 > I n nm Coelestium in

1543. Copernicus suffered a stroke in 1542 and was bedrid-
den by the time his magnum opus was published, legend
has it that he saw the first copies (with an unauthorized
3 placate the Church's

Figure 3.3: The page in Copernicus' book De Revolutionibus Orbium
Coelestium outlining the heliocentric model.

It must be noted that Copernicus not only put forth the heliocentric idea,
but also calculated various effects that his model predicted (thus following

the steps outlined in Sect. ??). The presentation of the results was made
to follow Ptolemy's Almagest step by step, chapter by chapter. Copernicus'
results were quite as good as Ptolemy's and his model was simpler; but its
predictions were not superior (since the planets do not actually move in
circles but follow another - though closely related - curve, the ellipse); in
order to achieve the same accuracy as Ptolemy, Copernicus also used epicy-
cles, but now in the motion of the planets around the Sun. The traditional
criticisms to the heliocentric model he answered thusly,

• To the objection that a moving Earth would experience an enormous
centrifugal force which would tear it to pieces, Copernicus answered
that the same would be true of, say, Mars in the Ptolemaic system,
and worse for Saturn since the velocity is much larger.

• To the question of how can one explain that things fall downwards
without using the Aristotelian idea that all things move towards the
center, Copernicus stated that that gravity is just the tendency of
things to the place from which they have been separated; hence a rock
on Earth falls towards the Earth, but one near the Moon would fall
there. Thus he flatly contradicted one of the basic claims of Aristotle
regarding motion.

• To the objection that any object thrown upward would be "left be-
hind" if the Earth moves, and would never fall in the same place,
( lopernicus argued that this will not occur as all objects in the Earth's
vicinity participate in its motion and are being carried by it.

Copernicus was aware that these ideas would inevitably create conflicts
with the Church, and they did. Though he informally discussed his ideas
he waited until he was about to die to publish his magnum opus, of which
he only printed a few hundred copies. Nonetheless this work was far from
ignored and in fact was the first (and perhaps the strongest) blow to the
Medieval cosmology. His caution did not save him from pointed criticisms,
for example, Luther pointed out (from his Tabletalk)

There was mention of a certain new astrologer who wanted to
prove that the Earth moves and not the sky, the Sun, and the
Moon. This would be as if somebody were riding on a cart or in
a ship and imagined that he was standing still while the Earth
and the trees were moving 2 . So it goes now. Whoever wants to

s a prescient remark, see Sect. ?? and Chap. 6.

be clever must agree with nothing that others esteem. He must do
something of his own. This is what that fellow does who wishes
to turn the whole astronomy upside down. Even in these things
that are thrown into disorder I believe the Holy Scriptures, for
Joshua commanded the Sun to stand still and not the Earth. 3

The Pope Paul III was not very critical, but his bishops and cardinals agreed
with Luther and the model was condemned by the Church.

The heliocentric model was eventually universally accepted by the sci-
entific community, but it spread quite slowly There were several reasons
for this, on the one hand there certainly was a reticence to oppose the au-
thority of the Church and of Aristotle, but there was also the fact that
the heliocentric model apparently contradicted the evidence of the senses.
Nonetheless the model became better known and was even improved. For
example, Copernicus' version had the fixed stars attached to an immovable
sphere surrounding the Sun, but its generalizations did and assumed them
to be dispersed throughout the universe (Fig. 3.4); Giordano Bruno even
proposed that the universe is infinite containing many worlds like ours where
intelligent beings live.

In fact it was Bruno's advocacy of the Copernican system that produced
one of the strongest reactions by the Church: Bruno advocated not only the
heliocentric model, but denied that objects posses a natural motion, denied
the existence of a center of the universe, denying even the Sun of a privileged
place in the cosmos. Bruno was executed by the Inquisition in 1600.

3 This statement was produced during an informal after-dinner conversation and \>
published after Luther's death; it should therefore be taken with caution.

Giordano Bruno (1548-1600). Born in Nola, near Naples.
He became a Dominican monk and 1< avned \rist ol.clian phi-
losophy and he was attracted to "unorthodox" streams of
thought (eg. Plato). Left Naples (1576) and later Rome
(1577) to escape the Inquisition. Lived in France until 1583
and in London until 1585. In 1584 he published Cena de
le Ceneri (The Ash Wednesday Supper) and De 1'InGnito,
Universo e Mondi (On the Infinite Universe and Worlds).
In the first he defended I he heliocentric theory (though he
was clearly confused on several points) ; in the second he ar-
gued that the universe was infinite, containing an infinite
number of worlds inhabited by intelligent beings. Wherever
he went, Bruno's passionate utterings led to opposition; he
lived off the munificence of patrons, whom he finally out-
raged. In 1591 he moved to Venice where he was arrested
by the Inquisition and tried; he recanted but was sent to
Rome for another trial, he did not recant a second time. He
was kept imprisoned and repeatedly iutc rrogated until 1600
when he was declared a heretic and burned at the stake.
It is often maintained that Bruno was executed because of
his Copernicanism and his belief in the infinity of inhab-
ited worlds. In fact, we do not know the exact grounds on
which he was declared a heretic because his file is missing
from the records. Scientists such as Galileo and Johannes
Kepler were not sympathetic to Bruno in their writings.

The slow progress of the heliocentric model was also apparent among
part of the scientific community of the time; in particular Tycho Brahe, the
best astronomer of the late 16th century, was opposed to it. He proposed
instead a "compromise": the earth moves around the sun, but the rest of
the planets move around the Earth (Fig. 3.5). Brahe's argument against the
Copernican system was roughly the following: if the Earth moves in circles
around the Sun, nearby stars will appear in different positions at different
times of the year. Since the stars are fixed they must be very far away but
then they should be enormous and this is "unreasonable" (of course they
only need to be enormously bright!)

Figure 3.4: The heliocentric model of Thomas Digges (1546-1595) who en-
larged the Copernican system by asserting that the stars are not fixed in a
celestial orb, but dispersed throughout the universe.

Tycho Brake (14 Dec 1546 - 24 Oct 1601). Born in Den-
mark In was fascinated by astronomy and, being a wealthy
man (and being lirlpc.1 l.y i Ik Danish monarchy), was able
to devote a lot of time to the meticulous recording of the
observed trajectories of the planets. He rented the island
nf 1 1 ven ii'iuii I In- kiim ill' Daumark and set up a state of the
art observatory there (without telescopes, they did not ap-
pear for 100 years). He later had to leave this island having
has a disagreement with the king over religious matters. He
then went to Prague as Imperial Mathematician and it was
there that he interacted with Kepler. He did not adopt the
Copernican system

Figure 3.5: Brahe's model of the universe: a central Earth around which
the sun moves surrounded by the other planets [From the Compendio di un
trattato del Padre Christoforo Borro Giesuita della nuova costitution del mondo
secondo Tichone Brahe e gli altri astologi moderni (Compendium of a treatise of
Father Christoforo Borri, S.J. on the new model of the universe according to Tycho
Brahe and the other modern astronomers) by Pietro della Valle, Risalah- i Padri
Khristafarus Burris Isavi dar tufiq-i jadid dunya.

3.3.1 Aristotle in the 16th century

In 1572 Tycho observed a star which suddenly appeared in the heavens
(we now recognize this as an exploding star: a supernova). He noted that
this "new star" did not change in position with respect to the other stars
and should therefore be in the outer sphere of Aristotle's universe. But
this was supposed to be an eternal, unchanging sphere! He published these
observations in The Nova Stella in 1574. The same type of problems arose
due to his observations of a comet which appeared in 1577, for he could
determine that this object was farther than Venus again contradicting the
Aristotelian idea that the universe beyond the Moon was perfect, eternal and
unchanging. This is a case where better observations when pitted against
the best theory of the time produced discrepancies which, in time, proved
to be fatal to the current model and would eventually give rise to a better,
more precise theory of the universe (see Sect. ??).

By this time also most of the Medieval approach to physics had been
shed, though not completely. For example, the motion of a projectile was
thought to be composed of an initial violent part (when thrown) and a
subsequent natural part (which returns it to the ground) . Still it was during

this time that the importance of velocity and force in determining the motion
of objects was realized.

The birth of new theories is not easy, however. In this case it was not
until the late 17th century that a complete new view of the universe was
polished and could be used as a tool for investigating Nature. By this
time the Aristotelian doctrine was, finally, set aside. The first step in this
long road was taken by Copernicus, the next by Johaness Kepler in his
investigations of the motion of the planets and then by Galielo through
his investigations on the nature of motion and his description of the solar
system.

3.4 Kepler

Johaness Kepler readily accepted the Copernican model, but his first at-
tempts to understand the motion of the planets were still tied to the Aris-
totelian idea that planets "must" move on spheres. Thus his first model of
the solar system was based on the following reasoning: there are, he argued,
six planets (Uranus, Neptune and Pluto would not be discovered for almost
300 years) which move on the surfaces of spheres. There are also five per-
fect geometric figures, the Platonic solids: cube, tetrahedron, octahedron,
icosahedron and dodecahedron. Then, he argued that the relative sizes of
the spheres on which planets move can be obtained as follows (see Fig. 3.6)

• Take the Earth's sphere and put a dodecahedron around it.

• Put a sphere around this dodecahedron, Mars will move on it.

• Put a tetrahedron around Mars' sphere and surround it by a sphere,
Jupiter will move on it.

• Put a cube around Jupiter's sphere and surround it by a sphere, Saturn
will move on it.

• Put an icosahedron inside the Earth's sphere, then Venus will move
on a sphere contained in it.

• Put a octahedron inside Venus sphere, then Mercury will move on a
sphere just contained in it.

Therefore the ordering is octahedron, icosahedron, dodecahedron, tetra-
hedron, cube (8-faces, 20-faces, 12-faces, 4-faces, 6- faces). He spent 20 years
trying to make this model work. ..and failed: the data would just not agree

Figure 3.6: Illustration of Kepler's geometrical model of the solar system

with the model. Hard as this was, he dropped this line of investigation. This
work, however, was of some use: he was recognized as "someone" and, in
1600, was hired by Tycho Brahe (then in Prague) as an assistant (at miserly
wages). Tycho was very reluctant to share his data with Kepler (who was
also made fun for being provincial); Tycho died in 1601 and the king ap-
pointed Kepler as successor (at a much smaller salary which was irregularly
paid).

For many years thereafter Kepler studied Tycho's data using the helio-
centric model as a hypothesis. In 1609 he determined that Mars does not
move in a circle but in an ellipse with the sun in one of the foci and that
in so moving it sweeps equal areas in equal times. This later blossomed
into his first and second laws of planetary motion. Ten years after he dis-
covered his third law: the cube of the average distance of a planet to the
sun is proportional to the square of its period. All this was very important:
Tycho's data, thanks to Kepler's persistence and genius, finally disproved
the epicyclic theory and, on top of this, the idea that planets must move in
circles.

This is a good example of the evolution of a scientific theory (see Sect.
??). The data required Kepler to modify the original hypothesis (planets
move in circles with the sun at the center) to a new hypothesis (planets
move in ellipses with a sun at one focus). He showed that this was the case
for Mars, and then checked whether it was also true for the other planets
(it was).

Johannes Kepler. (Dec 27 1571 - Nov 15 1630). Born
Weil der Stadt, Germany. Died .He^euslmrg, Germany. Ed-
ucated in Tubingen where he became acquainted with the
Copernican system, which he embraced and sought to per-
fect; in 1596 he pn >Ji bed \1\ l< iu.i ( u.smographicum in
which he defended the Copernican theory and described
his ideas on the structure of the planetary system. He
was a devout Lutheran but inclined towards Pythagorean
mysticism. He was intoxicated by numbers and searched
for simple mathen I i! i i mies in the physical world;
in particular he believed that the planets emit music as
they travel and he even gave the various tunes. In 1609
he published Astranomia Nova (" New Astronomy" ) which
contained his first two laws. In 1619 he published Harmon-
ices Mundi (Harmonies of the World) in which he stated
his third law.

The three laws obtained by Kepler are

1. Planets move in ellipses with the sun at one focus; see Fig. 3.7. Kepler's 1st law: Planet

move in ellipses with the

2. Planets sweep equal areas in equal times in their motion around the sun at one focus

sun; see Fig. 3.8. Kepler's 2nd law: Plane-

sweep equal areas in equ;

3. The average distance to the sun cubed is proportional to the period times in their motion
squared; see Table 3.1 for the data which led Kepler to this conclusion,

around the s
Kepler's 3rd law: The
average distance to the sun
proportional to the

The first two laws describe the motion of single planets, the third law
relates the properties of the orbits of different planets. period squared

Kepler did not know why planets behaved in this way. It was only about
50 years later that Newton explained these laws in terms of his universal
law of gravitation. In modern language these results imply the following
(discovered by Newton): the planets move the way they do because they
experience a force from the sun, this force is directed along the line from the
planet to the sun, it is attractive and decreases as the square of the distance.

Figure 3.7: How to draw and ellipse (left) and the elliptical orbit of planets
(right)

|| Planet | Period (years) | Avg. dist. (AU) | Period 2 | Dist 3 ||

Mercury

0.24

0.39

0.06

0.06

Venus

0.62

0.72

0.39

0.37

Earth

1.00

1.00

1.00

1.00

Mars

1.88

1.52

3.53

3.51

Jupiter

11.9

5.20

142

141

Saturn

29.5

9.54

870

868

Table 3.1: Period and average distancs for the innermost five planets, a plot
of the last two columns gives a straight line as claimed by Kepler's third

law.

Figure 3.8: The planet in a given time moves from a to b sometime later
it reaches c and it takes the same time to go from c to d. Kepler's second
law states that the shaded areas are equal

T^hiiSics 7

Chapter 4

Galileo and Newton

4.1 Introduction

The discoveries of Kepler, and the paradigm of the solar system of Coper-
nicus provided a very solid framework for the works of Newton and Galileo.
The resulting theories changed the way we do science to this day and some
of their ideas have withstood the passing of time with little change.

4.2 Galileo Galilei

Only rarely humankind is fortunate to witness the birth and flourishing of
a mind as keen and fertile as Galileo's. To him we owe our current notions
about motion and the concepts of velocity and acceleration. He was the first
to use the telescope as an astronomical tool. Galileo was also creative in
devising practical machines: he invented the first accurate clock, an efficient
water pump, a precision compass and a thermometer. These achievements
distinguish him as the preeminent scietist of his time.

Galileo's research in the exact sciences banished the last vestiges of Aris-
totelian "science" and replaced it with a framework within which the whole
of physics would be constructed. These changes were not achieved without
pain: Galileo was judged and condemned by the Inquisition and died while
under house arrest after being forced to recant his Copernican beliefs.

Underlying all the discoveries made by Galileo there was a modern phi-
losophy of science. He strongly believed, along the Pythagorean tradition,
that the universe should be described by mathematics. He also adopted the
view, following Ockham's razor (Sect. ??), that given various explanations

1

of a phenomenon, the most succinct and economic one was more likely to
be the correct one. Still any model must be tested again and again against
experiment: no matter how beautifuland economical a theory is, should it
fail to describe the data, it is useless except, perhaps, as a lesson.

4.2.1 Galilean relativity

Imagine a person inside a ship which is sailing on a perfectly smooth lake at
constant speed. This passeneger is in the ship's windowless hull and, despite
it being a fine day, is engaged in doing mechanical experiments (such as
studying the behavior of pendula and the trajectories of falling bodies). A
simple question one can ask of this researcher is whether she can determine
that the ship is moving (with respect to the lake shore) without going on
deck or looking out a porthole.

Since the ship is moving at constant speed and direction she will not
feel the motion of the ship. This is the same situation as when flying on
a plane: one cannot tell, without looking out one of the windows, that the
plane is moving once it reaches cruising altitutde (at which point the plane
is flying at constant speed and direction). Still one might wonder whether
the experiments being done in the ship's hull will give some indication of
the its motion. Based on his experiments Galileo concluded that this is in
fact impossible: all mechanical experiments done inside a ship moving at
constant speed in a constant direction would give precisely the same results
as similar experiments done on shore.

The conclusion is that one observer in a house by the shore and another in
the ship will not be able to determine that the ship is moving by comparing
the results of experiments done inside the house and ship. In order to
determine motion these observers must look at each other. It is important
important to note that this is true only if the ship is sailing at constant
speed and direction, should it speed up, slow down or turn the researcher
inside can tell that the ship is moving. For example, if the ship turns you
can see all things hanging from the roof (such as a lamp) tilting with respect
to the floor

Generalizing these observations Galileo postulated his relativity hy-
pothesis:

any two observers moving at constant speed and direction with
respect to one another will obtain the same results for all
mechanical experiments

(it is understood that the apparatuses they use for these experiments move
with them).

In pursuing these ideas Galileo used the scientific method (Sec. ??): he
derived consequences of this hypothesis and determined whether they agree
with the predictions.

This idea has a very important consequence: velocity is not absolute. Velocity is not absolute
This means that velocity can only be measured in reference to some ob-
ject (s), and that the result of this measurment changes if we decide to mea-
sure the velocity with respect to a diferent refernce point(s). Imagine an
observer traveling inside a windowless spaceship moving away from the sun
at constant velocity. Galileo asserted that there are no mechanical experi-
ments that can be made inside the rocket that will tell the occupants that
the rocket is moving . In fact, the question "are we moving" has no mean-
ing unless we specify a reference point ( "are we moving with respect to that
star" is meaningful). This fact, formulated in the 1600's remains very true
today and is one of the cornerstones of Einstein's theories of relativity.

Turbulence, (from Relativity and Its Roots, by B. Hoff-
mann,). Although this question will seem silly, consider it
anyway: Why do the flight attendants on an airplane not
serve meals when the air is turbulent but wait until the
turbulence has passed?

The reason is obvious. If you tried to drink a cup of coffee
during a turbulent lli«ht. you would probably spill it all
over the place.

The question may seem utterly inane. But even so, let us
not be satisfied with only a partial answer. The question
has a second part: Why is it all right for the flight atten-
dants to serve meals when the turbulence lias passed?
Again the reason is obvious. When the plane is in smooth
flight, we can eat and drink in it as easily as we could if it
were at rest on the ground.

Yes indeed! And that is a most remarkable fact of experi-
ence. Think of it.

A concept associated with these ideas is the one of a "frame of reference" .
We intuitively know that the position of a small body relative to a reference
point is determined by three numbers. Indeed consider three long rods at
90° from one another, the position of an object is uniquely determined by
the distance along each of the corresponding three directions one must travel
in order to get from the point where the rods join to the object (Fig. 4.1)

Figure 4.1: A frame of reference.

A reference frame
determines the where
when of anything with

Thus anyone can determine positions and, if he/she carries clocks, mo-
tion of particles accurately by using these rods and good clocks. This set
of rods and clock is called a reference frame. In short: a reference frame
determines the where and when of anything with respect to a reference point.

A prediction of Galileo's principle of relativity is that free objects will
respect to a reference point, move in straight lines at constant speed. A free object does not suffer
form interactions from other bodies or agencies, so if it is at one time at
rest in some reference frame, it will remain at rest forever in this frame.
Now, imagine observing the body form another reference frame moving at
constant speed and direction with respect to the first. In this second frame
the free body is seen to move at constant speed and (opposite) direction.
Still nothing has been done to the body itself, we are merely looking at it
from another reference frame. So, in one frame the body is stationary, in
another frame it moves at constant speed and direction. On the other hand
if the body is influenced by something or other it will change its motion by
speeding up, slowing down or turning. In this case either speed or direction
are not constant as observed in jEM^anyj/EM^ reference frame. From these
arguments Galileo concluded that free bodies are uniquely characterized by
moving at constant speed (which might be zero) and direction.

An interesting sideline about Galilean relativity is the following. Up to
that time the perennial question was, what kept a body moving? Galileo
realized that this was the wrong question, since uniform motion in a straight
line is not an absolute concept. The right question is, what keeps a body
from moving uniformly in a straight line? The answer to that is "forces"
(which are defined by these statements). This illustrates a big problem in

physics, we have at our disposal all the answers (Nature is before us), but
only when the right questions are asked the regularity of the answers before
us becomes apparent. Einstein was able to ask a different set of questions
and this lead to perhaps the most beautiful insights into the workings of
Nature that have been obtained.

Galilean relativity predicts that free motion is in a straight line at con-
stant speed. This important conclusion cannot be accepted without ex-
perimental evidence. Though everyday experience seems to contradict this
conclusion (for example, if we kick a ball, it will eventually stop), Galileo
realized that this is due to the fact that in such motions the objects are
not left alone: they are affected by friction. He then performed a series of
experiments in which he determined that frictionless motion would indeed
be in a straight line at constant speed. Consider a ball rolling in a smooth
bowl (Fig. 4.2).

*; J

Figure 4.2: Illustration of Galileo's experiments with friction

The ball rolls from it's release point to the opposite end and back to a
certain place slightly below the initial point. As the surfaces of the bowl
and ball are made smoother and smoother the ball returns to a point closer
and closer to the initial one. In the limit of zero friction, he concluded, the
ball would endlessly go back and forth in this bowl.

Following this reasoning and "abstracting away" frictional effects he con-
cluded that

Free horizontal motion is constant in speed and direction.

This directly contradicts the Aristotelian philosophy which claimed that

• all objects on Earth, being imperfect, will naturally slow down,

• that in a vacuum infinite speeds would ensue,

• and that perfect celestial bodies must move in circles.

In fact objects on Earth slow down due to friction, an object at rest would
stay at rest even if in vacuum, and celestial bodies, as anything else, move
in a straight line at constant speed or remain at rest unless acted by forces.

Velocity tells how positio
changes with time
Acceleration tells how
velocity changes with tirr

4.2.2 Mechanics

Most of Galileo's investigations in physics had to do with the motion of
bodies; these investigations lead him to the modern description of motion
in terms of position and time. He realized that two important quantities
that describe the motion of all bodies are velocity (which determines how
position changes with time) and acceleration (which determines how velocity
changes with time)

Two important definitions:

Velocity: the rate of change
of position, (how position
changes with time).

Acceleration: the rate of
change of velocity, (how veloc-
ity changes with time).

The motion of falling bodies

Galileo realized, even during his earliest studies (published in his book On
motion) that the speed of a falling body is independent of its weight l . He
argued as follows: suppose, as Aristotle did, that the manner in which a body
falls does depend on it weight (or on some other quality, such as its "fiery" or
"earthy" character), then, for example, a two pound rock should fall faster
than a one pound rock. But if we take a two pound rock, split it in half
and join the halves by a light string then one the one hand this contraption
should fall as fast as a two pound rock, but on the other hand it should
fall as fast as a one-pound rock (see Fig. 4.3). Since any object should
have a definite speed as it falls, this argument shows that the Aristotle's
assumption that the speed of falling bodies is determined by their weight is
inconsistent; it is simply wrong. Two bodies released from a given height
will reach the ground (in general) at different times not because they have
different "earthliness" and "fiery" characteristics, but merely because they
are affected by air friction differently. If the experiment is tried in vacuum
any two objects when released from a given height, will reach the ground
simultaneously (this was verified by the Apollo astronauts on the Moon
using a feather and a wrench).

'Galileo allegedly demonstrated bis conclusions by dropping weights from the leaning
tower of Pisa though this has been doubted by historians.

This result is peculiar to gravity, other forces do not beahve like this at
all. For example, if you kick two objects (thus applying a force to them) the
heavier one will move more slowly than the lighter one. In contrast, objects
being affected by gravity (and starting with the same speed) will have the
same speed at all times. This unique property of gravity was one of the
motivations for Einstein's general theory of relativity (Chap. ??).

Figure 4.3: Illustration of Galileo's experiments with falling bodies.

Also in his investigations of falling bodies Galileo determined that the acceler-
ation of these bodies is constant. He demonstrated that an object released from
a height starts with zero velocity and increases its speed with time (before him it
was thought that bodies when released acquire instantaneously a velocity which
remained constant but was larger the heavier the object was). Experimenting with
inclined planes, and measuring a ball's positions after equal time intervals Galileo
discovered the mathematical expression of the law of falling bodies: the distance
traveled increases as the square of the time.

The motion of projectiles

Galileo also considered the motion of projectiles. He showed that their
motion can be decomposed in a motion along a vertical and horizontal di-
rections. Thus if a ball is thrown horizontally (and air friction is ignored)
it will move in the horizontal direction with constant speed; in the vertical

direction it will experience the pull of gravity and will undergo free fall. The
use of this can be illustrated by the following situation. Suppose a ball is
let fall from a height h and is found to take t seconds to reach the ground.
Now suppose that the ball is instead thrown horizontally with speed v, what
distance will it cover? The answer is vt because the ball, even though it is
moving horizontally, in the vertical direction is still freely falling: the two
motions are completely independent! (see Fig. 4.4). This, of course, was of
great use in warfare.

ARRIVAL TIME

,-Jt ^>

ARRIVAL TIME

Figure 4.4: Horizontal and vertical motion are independent: the cannon
shoots the ball horizontally at the same time the hand drops its ball; they
both hit the ground at the same time.

As another experiment consider the "shoot the monkey" demonstration
(Fig. 4.5). The setup is the following: a hunter wants to shoot a monkey
who is hanging from a branch. As soon as he shoots the monkey lets go
of the branch (thinking that the hunter aimed at the branch, he believes
that the bullet will miss him). But the bullet, to the monkey's surprise (and
distress), does hit him! 2

The reason is the following: if there were no forces the bullet would go
in a straight line (as indicated by the dotted line in the figure) and the
monkey would not fall. So the bullet would hit the monkey. Now, since

No real animals w

i 1 Ids dcmonstrat ion.

-#-*

monkey^

/ f

f

Figure 4.5: Shoot the "monkey" : an illustration of motion in two dimensions.

we have a force acting on the system (gravity) the monkey will not stay at
rest but will accelerate downward. But precisely the same force acts on the
bullet in precisely the same way, hence the bullet will not go in a straight
line but will follow the curve indicated in the figure. The deviation from
their force- free motions (rest for the monkey, straight line for the bullet) are
produced by a force which generates the same acceleration in both objects,
hence these deviations are precisely matched in such a way that the bullet
hits the monkey.

Now, given a force of constant strength, it will affect bodies in varying
degrees; the more massive the object the smaller the effect: a blow from
a hammer will send a small ball flying, the same blow will hardly affect a
planet. On the other hand gravity produces the same acceleration on the
monkey and the bullet; that is why the monkey is hit. Since the mass of the
monkey is very different from that of the bullet we conclude that gravity's

force is very different for each of them. The fact that the accelerations are
independent of the mass but the force is not is actually a very profound fact:
the whole of general relativity is based on it (Chap. ??).

4.2.3 Astronomy

Throughout his life Galileo would provide some of the most compelling ar-
guments in favor of the heliocentric model; though this brought him endless
trouble in his lifetime, he was vindicated by all subsequent investigators.
The beginnings of Galileo's astronomical studies were quite dramatic: in
1604 a "new star" (a supernova — an exploding star) was observed,. Galileo
demonstrated that this object must lie beyond the Moon, contradicting the
Aristotelian doctrine which claimed that the region beyond the Moon was
perfect and unchanging. Yet here was a star that was not there before and
would soon disappear!

A few years later he learned about the discovery of the telescope. He
quickly realized its potential as a tool in astronomical research, and con-
structed several of them (Fig. 4.6), which he used to investigate the heavens.

Figure 4.6: One of Galileo's telescopes

The first object which he studied with his telescope was the Moon of
which he made many drawings (Fig. 4.7) some of which are quite accurate.
He found that the surface of the Moon was heavily scarred, and identified

some of the dark features he observed as shadows. The Moon was not exactly
spherical and hardly perfect.

Figure 4.7: Galileo's drawings of the Moon.

Galileo was the first person to discover that Venus, like the Moon, shows
periodic phases (Fig. 4.8). The simplest explanation is that this planet
goes around the sun in accordance with the Copernican system. Galileo's
astronomical observations were later verified by the Jesuit mathematicians of
the Collegio Romano (although they did not necessarily agree with Galileo's
interpretation!).

But the most dramatic of Galileo's astronomical discoveries was that
of Jupiter's satellites (1610) 3 . He found that Jupiter was surrounded by a
swarm of bodies that circled it and not Earth! These satellites, together with
Jupiter, formed a mini- version of the Copernican model of the solar system
with Jupiter taking the place of the Sun and it's satellites the places of the
planets. All this was in blatant contradiction of the Aristotelian model; any Jupiter and its si
remaining doubts which he might have had in his belief of the heliocentric formcd 3 mini-vc
model vanished.

i of the
Copernican model of the
r system with Jupiter

In 1613, in a book on sunspots, Galileo openly declared the Earth to taking the place of the Sun
circle the Sun. But by then the Church was getting worried about these and it,s satellites the P |aces

of the planets

3 Thi

landed him a permanent position as "Chief Mathematician of the University of
and Philosopher and Mathematician to the Grand Duke of Tuscany"

F^

Figure 4.8: Galileo's drawings of the phases of Venus.

ideas: in 1616 Pope Pius V declared the Earth to be at rest and labeled
the heliocentric model heretical, Copernicus' magnum opus was black-listed
(where it remained until 1822!), and Galileo was called to Rome and told
not to defend Copernicus' ideas.

In 1632 Galileo published his book on the Copernican and Ptolemaic
systems Dialogue Concerning the Two Chief Systems of the World (in Ital-
ian so everyone could understand it). This was originally condoned by the
Church, but the Pope Urban VIII had a change of heart and forbade the
distribution of the book. Galileo was summoned to appear before the Ro-
man Inquisition where, in a penitential garb and on one knee, he was made
to swear on the Bible that he

..abjured, cursed, and detested the error and heresy that the
Sun is fixed and the Earth moves"

and that he would no longer support this idea in any manner. He was put
under house arrest and was made to recite the seven penitential psalms
weekly for three years. This, of course, did not change the fact that the
planets do move around the sun, but it embittered Galielo's last years.

4.2.4 Galileo and the Inquisition

Being one of the most renowned scientist of his time Galileo's opinions were
scrutinized not only be his peers, but by also by Church officials and the
public in general. This made Galileo the lightning-rod of many complaints
against the Copernican doctrine (and also some against Galileo himself).
He did not come out unscathed out of these encounters.

In 1611 Galileo came to the attention of the Inquisition for the first
time for his Copernican views. Four years later a Dominican friar, Niccolo
Lorini, who had earlier criticized Galileo's view in private conversations, files
a written complaint with the Inquisition against Galileo's Copernican views.
Galileo subsequently writes a long letter defending his views to Monsignor
Piero Dini, a well connected official in the Vatican, he then writes his Letter
to the Grand Duchess Christina arguing for freedom of inquiry and travels
to Rome to defend his ideas

In 1616 a committee of consultants declares to the Inquisition that the
propositions that the Sun is the center of the universe and that the Earth has
an annual motion are absurd in philosophy, at least erroneous in theology,
and formally a heresy. On orders of the Pope Paul V, Cardinal Bellarmine
calls Galileo to his residence and administers a warning not to hold or defend
the Copernican theory; Galileo is also forbidden to discuss the theory orally
or in writing. Yet he is reassured by Pope Paul V and by Cardinal Bellarmine
that he has not been on trial nor being condemned by the Inquisition.

In 1624 Galileo meets repeatedly with his (at that time) friend and pa-
tron Pope Urban VIII, he is allowed to write about the Copernican theory
as long as he treated it as a mathematical hypothesis.

In 1625 a complaint against Galileo's publication The Assayer is lodged
at the Inquisition by a person unknown. The complaint charges that the
atomistic theory embraced in this book cannot be reconciled with the offi-
cial church doctrine regarding the Eucharist, in which bread and wine are
"transubstantiated" into Christ's flesh and blood. After an investigation by
the Inquisition, Galileo is cleared.

In 1630 he completed his book Dialogue Concerning the Two Chief
World Systems in which the Ptolemaic and Copernican models are discussed
and compared and was cleared (conditionally) to publish it by the Vatican.
The book was printed in 1632 but Pope Urban VIII, convinced by the ar-
guments of various Church officials, stopped its distribution; the case is
referred to the Inquisition and Galileo was summoned to Rome despite his
infirmities.

if?a

Galileo Galilei (Feb. 15, 1564-1642). Born near Pisa,
Italy, died near Florence, Italy. In 1581 he matriculates
as a student of the Arts at the University of Pisa (his fa-
ther's wish is that he study medicine) and he is first in-
troduced to Euclid's Elements while studying in Florence
under 1 he court mathematician Ost.ilio likci. In 1585 he re-
turns to Florence without a degree. He gives private lessons
in mathematics until 1589; he begins his studies in physics.
In 1588 he obtained a lectureship of mathematics at the
Univ. of Pisa where he taught until 1592; he publishes On
motion. In 1592 Galileo obtains the chair of mathematics
at the University of Padua in the Venetian Republic where
he remains until 1610.

In 1599 he enters a relationship with Marina Gamba with
whom he had three children, two daughters and one son.
The daughters were placed in a convent as Galileo could
not provide adequate dowries; he eventually managed to
have his son legitimated. In 1613 Marina Gamba married
Giovanni Bartoluzzi, it appears that Galileo kept cordial
relations with Gamba and Bartoluzzi.

In 1609, he observes (using telescopes of his construction)
the Moon, and discovers 4 satellites around Jupiter. In this
year he was also appointed (for life) "Chief Mathematician
of the University of I'isa and Philosopher and Mathemati-
cian to the Grand Duke of Tuscany". In 1611 he is admitted
to the Lycean Academy and came to the attention of the
Inquisition for the first time . In 1615 he is denounced to
the the Inquisition, he defends himself in the Letter to the
Grand Duchess Christina. In 1616 the Copernican doctrine
is declared heretical, Galileo is warned against supporting
this theory either orally, but lie is allowed to write about
it as a mathematical hypothesis. In 1621 Galileo is elected
Consul of the Accademia Florentine In 1625 a complaint
to the Inquisition in Galilc puMii 1 1 ion The Assayer

is lodged by a person unknown; the complaint charges that
the atomistic theory embraced in this book is heretical;
Galileo is cleared.

In 1630 completes his book Dialogue Concerning the Two
Chief World H\ ^t i i out isi i i to 1 1 maic and Coper-
nican models. The book was printed in 1632 but the Pope
Urban VIII stopped its distribution; the case is referred to
the Inquisition and Galileo was summoned to Rome despite
his physical infirmities. A year later Galileo is formally in-
terrogated by the Inquisition. He recants of his support
of the Copernican model and is ordered held under house
arrest where he would remain until his death; also in 1633
he begins writing his Discourse on Two New Sciences. His
health deteriorates steadily, in 1631 lie suffers a painful her-
nia, by 1638 he is totally blind. Galileo dies in Arcetri on
8 January 1642.

Galileo also invented several objects of great practical in-
terest such as an hydrostatic balance (1608), a horse-
driven water pump (1593), a geometric and military com-
pass (1597), various telescopes (1609) and a thermometer
(1606). In 1641 he conceives of the application of the pen-
dulum to clocks.

In 1633 Galileo was formally interrogated for 18 days and on April 30
Galileo confesses that he may have made the Copernican case in the Di-
alogue too strong and offers to refute it in his next book. Unmoved, the
Pope decides that Galileo should be imprisoned indefinitely. Soon after,
with a formal threat of torture, Galileo is examined by the Inquisition and
sentenced to prison and religious penances, the sentence is signed by 6 of
the 10 inquisitors. In a formal ceremony at a the church of Santa Maria
Sofia Minerva, Galileo abjures his errors. He is then put in house arrest
in Sienna. After these tribulations he begins writing his Discourse on Two
New Sciences.

Galileo remained under house arrest, despite many medical problems
and a deteriorating state of health, until his death in 1642. The Church
finally accepted that Galileo might be right in 1983.

4.3 Isaac Newton

On Christmas day 1642, in the manor house of Woolsthorpe, a weak child
was born and christened Isaac. He was to become the most influential scien-
tist of the next 250 years. Isaac Newton discovered the laws that explained
all phenomena known at the time, form the motion of the stars to the behav-
ior of dust particles. It was his extremely successful model that lead people
to believe that humanity was on the verge of understanding the whole of
Nature.

Newton's life can be divided into three quite distinct periods. The first
is his boyhood days from 1642 up 1665 when the Plague forced him to leave
Cambridge. The second period from 1665 to 1687 was the highly productive
period in which he became Lucasian professor at Cambridge. The third
period (nearly as long as the other two combined) saw Newton as a highly
paid government official in London with little further interest in science and
mathematics.

I will talk about Newton quite a bit because his view of the world together
with the mathematical formalism he developed lasted for 200 years: the first
experimental results incompatible with it were obtained at the end of the
XlX-th century and the whole structure was shown not to be fundamentally
correct by 1925. One nonetheless should be aware of the fact that, while not
perfectly correct, the results using the Newtonian are exceedingly accurate
in all every-day applications. Newton's theory is not "wrong" it's just that
it has a limited range of validity.

Isaac Newton (1643-1727). Born in the manor house of
Woolsthorpe, near Grantham in Lincolnshire on Christmas
Day 1642. Newton came from a family of farmers; his father
died before he was born. His mother remarried, moved to a
nearby village, and left him in the care of his grandmother.
Upon the death of his stepfather in 1656, Newton's mother
removed him from grammar school in Grantham where he
had shown little promise in academic work. His school re-
ports described him as 'idle' and 'inattentive'. Legend has
it that one day the si ndcul jus] ahead of him in class kicked
him in the stomach, Newton won the fight and he also de-
cided to get ahead of this student in class ranking. He
succeeded admirably. An uncle decided that he should be
prepared for the university, and he entered his uncle's old
College, Trinity College, Cambridge, in June 1661.
Instruction at Cambridge was dominated by the philosophy
of Aristotle but some freedom of study was allowed in the
third year of study. Newton's aim at Cambridge was a law
degree, yet he also studied the philosophy and analytical
geometry of Descartes, Boyle's works, and the mechanics
of the heliocentric astronomy of Galileo.
His scientific genius flourished suddenly when the "Black
Death" plague closed the University in the summer of 1665
and he had to return to Lincolnshire. There, in a period of
less than two years, while Newton was still under 25 years
old, he began revolutionary advances in optics, physics,
and astronomy. In mathematics he laid the foundation for
differential and integral calculus several years before its in-
dependent discovery by Leibniz, (this work, De Methodis
Serierum et Fluxionum, was written in 1671 but appeared
only 60 years later).

Impressed with Newton's abilities, Barrow resigned the Lu-
casian chair in 1669 recommending that Newton (still only
27 years old) be appointed in Ids place Newton's first work
as Lucasian Professor was on optics. Newton was elected a
fellow of the Royal Society in 1672 after donating a reflect-
ing telescope. In that year he published his first scientific
paper on light and color in the Philosophical Transactions
of the Royal Society.

Newton's relations with the influential scientist Robert
Hooke deteriorated and Newton turned away from the
Royal Society and mainstream science; he delayed the pub-
lication of a full account of his optical researches until after
Hooke's death in 1703: Newton's Opticks appeared in 1704.
Newton's greatest achievement was his work in physics and
celestial mechanics, which culminated in t lie theory of uni-
versal gravitation. His results are summarized in his trea-
tise of physics Piiiiosopliiae ;Yaluni (is I'viucipin Miithcmnt-
ica which appeared in 1687.

After suffering a nervous breakdown in 1693, Newton re-
tired from research to take up a government position in
London becoming Warden of the Royal Mint (1696) and
Master (1699). In 1703 he was elected president of the
Royal Society and was re-elected each year until his death.
He was knighted in 1708 by Queen Anne, the first scientist
to be so honored for his work. Newton died in 1727; his
tomb in Westminster Abbey is inscribed with these words:
" Mortals! Rejoice at so great an ornament to the human

4.3.1 Mechanics.

During the years of the Plague Newton constructed what was to become
an remarkably successful model of Nature. In it he proposed three laws
that describe the motion of all material bodies (at least for all phenomena
within reach at the time). These were not mere descriptions but actual
calculational tools, and the enormous accuracy in the predictions achieved
by this theory resulted in its universal acceptance that lasted more than two
centuries. ..until Einstein came along.

After returning to Cambridge, Newton lost interest in mechanics until
1684. In this year Halley, tired of Hooke's boasting, asked Newton whether
he could prove Hooke's conjecture that planets moved in ellipses because the
sun attracted them with a force decreasing as the square of the distance.
Newton told him that he had indeed solved this problem five years earlier,
but had now mislaid the proof. At Halley's urging Newton reproduced the
proofs and expanded them into a paper on the laws of motion and problems
of orbital mechanics. Halley then persuaded Newton to write a full treatment
of his new physics and its application to astronomy. Over a year later (in
1687) Newton published the Philosophiae Naturalis Principia Mathematica
or the Principia as it is commonly known. It is one of the greatest scientific
books ever written.

Newton laid in his Principia three laws which describe the motion of
bodies. These laws have an immense range of applicability, failing only at
very small distances (of 10 _8 cm or less), for very strong gravitational fields
(about 10 8 stronger than the Sun's), or for very large speeds (near 10 8 m/s).

The first of Newton's laws addresses the motion of free bodies. The
second law states quantitatively how a motion differs form free motion. The
third law states the effect experienced by a body when exerting a force on
another object.

• 1st law. Every body continues its state of rest or uniform motion in a
straight line unless it is compelled to change this state by forces acting

on it. Free bodies move in straight

lines or remain at rest

• 2nd law. The effect of a force F on the motion of a body of mass m is
given by the relation

F = ma

where a is the acceleration: a body in the presence of a force F attains

an acceleration equal to F/m. Force=massx acceleration

• 3rd law. Every body exerting a force on another, experiences a force
exerted by the second body equal in magnitude and in opposite direc-
i mii.

These three laws constitute Newton's basic hypothesis He asserted that
they are valid in all circumstances and to all bodies, in particular for heav-
enly bodies as well as for earth objects; this marks the final passing of Aris-
totelian physics. All experimental evidence of the time (and for the next
two centuries) was to support these hypothesis, Newton's theory became the
theory of Nature.

I will now discuss some of the features of these laws.

1st Law and Newtonian space and time.

One of the most important consequences of the First Law is that it defines
what we mean by an inertial frame of reference.

An inertial reference frame is a reference frame where isolated
l reference frame bodies are seen to move in straight lines at constant velocity.

is a reference frame where
isolated bodies are seen to
i straight

An observer at rest with respect to an inertial frame of reference is
nt velocity called an inertial observer. The laws of physics devised by Newton take a

particularly simple form when expressed in terms of quantities measured by
an inertial observer (such as positions, velocities, etc.). For example, an
inertial observer will find that a body on which no forces act moves in a
straight line at constant speed or is at rest.

All motion occurs in space and is measured by time. In Newton's model
both space and time are unaffected by the presence or absence of objects,
imed that space That is space and time are absolute, an arena where the play of Nature
• absolute unfolds. In Newton's words,

Absolute space in its own nature, without relation to anything
external, remains always similar and immovable.

...absolute and mathematical time, of itself, and from its own
nature, flows equally without relation to anything external,
and by another name is called duration.

Space and time were taken to be featureless objects which served as a uni-
versal and preferred reference frame (see Fig. 4.9 for an illustration). A
consequence of this is that a given distance will be agreed upon by any two

observers at rest with respect to each other or in uniform relative motion,
for, after all, they are just measuring the separation between two immovable
points in eternal space. In the same way a time interval will be agreed upon
by any two observers for they are just marking two notches on eternal time.

Figure 4.9: Illustration of Newton's concept of space. The grids represent
space which are unaffected by the presence and properties of the objects in
it.

Newton's assumptions about space and time are the foundation of his
theory of Nature and were accepted due to the enormous successes of the
predictions. Eventually, however, experimental results appeared which dis-
agreed with the predictions derived from Newton's theory. These problems
were traced to the fact that these basic assumptions are not accurate de-
scriptions of space and time (though they do represent a very good approxi-
mation): space and time are not absolute (Chaps. ??, ??) 4 . The realization
that Newton's theory required revisions came to a head at the beginning of
the XXth century. In the two decades from 1905 to 1925 a completely new
framework was constructed and has now replaced Newton's ideas. These
theories comprise the special and general theories of relativity and quantum
mechanics.

4 F — ma is also not u
at very small distances an

ersally valid but deviations from this expression occur only
an be understood in the framework of Quantum Mechanics.

20

Do we know that the current theories of space and time are the truth?
The answer is no: we do know that the current theories explain all the data
(including the one explained by Newton and more) , but we cannot determine
whether they represent the ultimate theories of Nature. In fact, we expect
them not to be the last word as there are many unexplained questions; for
example, why should the proton be precisely 1836.153 times heavier than
the electron? Why should space have 3 and not 25 dimensions? etc.

But in the 17th century there was no inkling of these problems and
very few scientist questioned Newton's hypothesis. In particular Newton
constructed his mechanics to comply with Galilean relativity: an observer
in uniform motion with respect to another cannot, without looking outside
his laboratory, determine whether he is at rest or not. And even if he looks
outside, he cannot decide whether he is in motion or the other observer is.
In fact for two inertial observers moving relative to each other the question,
"which of us is moving?" is un-answerable and meaningless. The only thing
to be said is that they have a certain relative velocity.

2nd Law

The second law is of great practical use. One can use experiments to deter-
mine the manner in which the force depends on the position and velocity of
the bodies and then use calculus (which was also invented by Newton) to
determine the motion of the bodies by obtaining the position as a function
of time using the known form of F and the equation F = ma. Note that
in this equation m measures how strongly a body responds to a given force
(the larger m is the less it will be accelerated); m measures the inertia of
the body.

Suppose we choose a test body of mass, say, lgm. By measuring its
motion one can obtain its acceleration and, using F = ma, determine the
force. Once F is known the motion of any body is predicted: by measuring
the falling an apple you can predict the motion of the Moon.

3rd Law

The third law is, at first sight, almost unbelievable: if I kick a ball, the ball
kicks me back? But in fact it is so: suppose I push a friend while we are
both standing on ice (to minimize friction), then he/she will move in the
direction of the push, but I will move backward! What happens when I kick
a ball is that the push backward is countered by the friction between my
other leg and the ground, and because of this no motion backward ensues.

It is interesting to do the kick-the-ball experiment on ice, you should try it.

4.3.2 Optics

Newton's first work as Lucasian Professor was on optics. Every scientist
since Aristotle had believed light to be a simple entity, but Newton, through
his experience when building telescopes, believed otherwise: it is often found
that the observed images have colored rings around them (in fact, he devised
the reflecting telescope, Fig. 4.10, to minimize this effect). His crucial
experiment showing that white light is composite consisted in taking beam
of white light and passing it through a prism; the result is a wide beam
displaying a spectrum of colors. If this wide beam is made to pass through a
second prism, the output is again a narrow beam of white light. If, however,
only one color is allowed to pass (using a screen) , the beam after the second
prism has this one color again. Newton concluded that white light is really
a mixture of many different types of colored rays, and that these colored
rays are not composed of more basic entities (see Fig. 4.11).

Figure 4.10: Newton's first reflective telescope.

Concerning the nature of light. Newton believed that
it consists of a stream of small particles (or corpuscules)
rather than waves. Perhaps because of Newton's already
high reputation this "corpuscular" theory was accepted un-
til the wave theory of light was revived in the 19th C.

Figure 4.11: Diagram of Newton's experiments on the composition of white
light.

4.3.3 Gravitation.

One of Newton's greatest achievements was on the field of celestial mechanics
where he produced the first synthesis in the theories describing Nature: he
realized that the same force that makes things fall, gravity, is responsible
for the motion of the Moon around the Earth and the planets around the
Sun.

He reasoned (more or less) as follows. Suppose I let an apple fall form
a very high tower, it will take, say, t seconds to reach the ground. Now
suppose I throw it very hard, then again it will take t seconds to reach the
ground provided I assume the Earth is flat. But the Earth isn't flat and has
curved from beneath the apple! Hence the apple will take longer to hit the
ground. By throwing the apple with increasing force one reaches a point
where the apple never hits the ground as the distance it falls equals the
distance the earth has curved under it: the apple is in orbit! (see Fig. 4.12)

With this thought experiment Newton convincingly argued that an apple
can behave in the same way as the Moon, and, because of this it is the very
same force, gravity, which makes the apple fall and the Moon orbit the
Earth. This is consistent with the hypothesis that gravitation is universal.
In a way it represents the unification a several physical effects which appear

Figure 4.12: Newton's explanation of the equivalence between the force
making apples fall and the one responsible for the Moon orbiting the Earth.

unrelated at first sight: the falling of apples and the orbiting of planets.

Having realized this he then used the results of Kepler and showed that
if the planets and the sun are assumed to be point-like, the gravitational
force drops as the inverse distance squared: the gravitational force between
two bodies of masses m and M separated by a distance r is attractive and
directed along the line joining the bodies, its value is

where G is a universal constant, in words,

all matter attracts all other matter with a force proportional to
the product of their masses and inversely proportional to the

square of the distance between them. All matter attracts al

matter with a force
Having discovered this Newton was able to explain a wide range of previously proportional to the produc
unrelated phenomena: the eccentric orbits of comets, the tides and their

of their

..lversely proportional to the
variations, the precession of the Earth's axis, and motion of the Moon as square of the distance
perturbed by the gravity of the Sun. It also predicts the position of the between them
planets for thousands of years so that the occurrence of eclipses can be
foretold with exquisite accuracy, Moon landings can be planned without
uncertainties, etc.

Consider now the application of the second law to the case of the gravi-
tational force.

mMG

^— = -Fgrav = ma

so that the factors of m cancel (!) This implies that the motion of a body
generated by the gravitational force is independent of the mass of the body
(!!), (just as Galileo had observed). This unique feature results from -Fg rav
being precisely proportional to m. So m is seen made to play two roles:

• On the one hand m in F = ma is a measure of how strongly is a body
accelerated by a given force: it is a measure of the body's inertia. In
this role m is called the inertial mass.

• On the other hand m in F gTlLV is a measure of how strongly is a body
affected by the force of gravity and also how strong a gravitational
force is generated by m; in this role it is called the gravitational mass.

These two quantities refer to different properties of a body and need not
be equal. Extremely precise measurements, however, indicate that they are
equal (at least to one part in ten parts per trillion). Newton just stated that
this was the way of the world and kept going. Einstein, in contrast, noted
this as a very important fact of nature, which he used to give birth to his
General Theory of Relativity (Chap. ??).

Concerning the nature of gravitation, there is another in-
teresting feature of Fg rav : it is time independent, this im-
plies that if a body moves, this change is perceived instanta-
neously by all tin- bodies t hroujihoul the universe. Leibnitz
(among others) criticized \"< whin's hypothesis along these
lines, and was disregarded. But this only due to the enor-
mous success of Newtoninan gravity in making predictions
of the motions of the bodies in the solar system. In fact we
will see that this is not correct, and that the effect spreads
out from the body at a finite speed (Chap. ??).

To give an idea of the trust and excellent successes of Newtonian gravity
consider the story of the discovery of Neptune. In 1843 a young astronomer
at Cambridge, J.C. Adams discovered an anomaly in the orbit of Uranus
and by the end of 1845 had concluded that this was due, not to a failure
of Newton's law of gravity, but to the presence of a new planet. Adams
submitted his results to G. Airy, his boss, who was unconvinced and dropped
the matter. Meanwhile U. Leverrier in France had done a similar set of
calculations independently, he published in 1846. This spurred Airy into
action, but the Cambridge Observatory lacked an up to date chart of the

region of the sky were the new planet was supposed to have resided at
the time. During that time Leverrier wrote to J.G. Galle at the Berlin
Observatory who promptly located the new planet. After much discussion
this planet was called Neptune.

Chapter 5

The Clouds Gather

For more than two centuries after its inception the Newtonian view of
the world ruled supreme, to the point that scientists developed an almost
blind faith in this theory. And for good reason: there were very few prob-
lems which could not be accounted for using this approach. Nonetheless, by
the end of the 19th century new experimental evidence difficult to explain
using the Newtonian theory began to accumulate, and the novel theories
required to explain this data would soon replace Newtonian physics. In
1884 Lord Kelvin in his Baltimore lectures already mentions the presence of
"Nineteenth Century Clouds" over the physics of the time, referring to cer-
tain problems that had resisted explanation using the Newtonian approach.
Among the problems of the time (not all were mentioned by Kelvin) were

• Light had been recognized as a wave, but the properties (and the very
existence!) of the medium that conveys light appeared inconsistent.

• The equations describing electricity and magnetism were inconsistent
with Newton's description of space and time (Sect ??).

• The orbit of Mercury, which could be predicted very accurately us-
ing Newton's equations, presented a small but disturbing unexplained
discrepancy between the observations and the calculations.

• Materials at very low temperatures do not behave according to the
predictions of Newtonian physics.

• Newtonian physics predicts that an oven at a stable constant temper-
ature has infinite energy.

The first quarter of the 20-th century witnessed the creation of the revo-
lutionary theories which explained these phenomena. They also completely
changed the way we understand Nature. The first two problems require the
introduction of the Special Theory of Relativity. The third item requires the
introduction of the General Theory of Relativity. The last two items can be
understood only through the introduction of a completely new mechanics:
quantum mechanics.

As a result of these developments the formalism developed by Newton
lost its fundamental character. It is of course still a perfectly good theory
but with a very well defined range of applicability. As mentioned previously,
this does not imply that Newton was "wrong", it merely implies that his
theories, although accurately describing Nature in an impressive range of
phenomena, do not describe all of it. The new theories that superseded
Newton's have the virtue of explaining everything Newtonian mechanics did
(with even greater accuracy) while extending our understanding to an even
wider range of phenomena. In this chapter I will describe the growth of
the theory of electricity and magnetism which was to be fundamental to the
development of Special Relativity.

The replacement of Newtonian mechanics was driven by the data that
required the replacement of Newtonian physics by these more fundamen-
tal ones; the theories of relativity and quantum mechanics together explain
all the phenomena probed to date, but they might be replaced in the fu-
ture by others providing a yet deeper understanding of nature. These new
theories will have to explain everything relativity and quantum mechanics
do and provide experimentally verifiable predictions which are subsequently
confirmed.

5.1 Electricity and magnetism
5.1.1 Electricity

It was known to the ancient Greeks as long ago as 600 B.C. that amber,
rubbed with wool, acquired the property of attracting light objects. In
describing this property today, we say that the amber is electrified, (from
the Greek, elektron: amber), possesses an electric charge, or is electrically
charged. It is possible to put an electric charge on any solid material by
rubbing it with any other material (rubbing brings many points of the sur-
faces into good contact, so that, at the atomic level, electrons are ripped
from one material and transferred to the other). Thus, an automobile be-
comes charged when it moves through the air, a comb is electrified in passing

negative

through dry hair, etc.

By the end of the 18th century it was known that electricity comes in two
flavors: positive and negative; and that equal charges repel while unequal electricity c
charges attract. The manner in which this attraction and repulsion occurs
was discovered by Coulomb in 1785. He found that the force between them
is very similar in form to the gravitational force: it is proportional to the
charges of each body, directed along the line joining them, and decreases
like the distance squared. There is, however, an important difference: this The electric force is

electric force can be attractive or repulsive; the gravitational force is always P r °P° rtlonal to tne charges

of each body, directed along

attractive. the | in e joining them, and

decreases like the distance
squared

Charles Augustin de Coulomb (June 14, 1736- Aug 23
1806)). Born in Angouleme. Franco: died in Paris, France.
Coulomb spent 9 years as a military engineer in the West
Indies but his health suffered so, when the French Rev-
olution began, he retired to the country to do scientific
research. He worked on applied mechanics, but he is best
known for his work on electricity and magnetism. He estab-
lished experimentally the inverse square law for the force
between two charges which became the basis of Poisson's
mathematical theory of magnetism. Coulomb also wrote
on structural analysis, the fracture of beams, the fracture
of columns, the thrust of arches and the thrust of the soil.

5.1.2 Magnetism

The earliest observations on magnets can also be traced back to the early
Greeks (eg. Thales of Miletus; see Sect. ??). The Chinese literature also has
extensive references to naturally occurring magnets (then called loadstones).
The fact that magnets align in a unique way, together with the fact that
the Earth itself is a magnet, lead to the discovery of the compass. This was
of paramount importance to the development of civilization. The earliest
known compass appeared in China by the first century A.D.; it arrived in
Europe by the twelfth century A.D.

4

11

William Gilbert (1544-1603). Born in Colchester, Eng-
land, into a middle class family of some wealth. Entered
St. John's College, Cambridge in 1558, and obtained his
B.A. (1561), M.A. (1564) and M.D. (1569). Became a se-
nior fellow of the college, lioldiug several offices and set up
a medical practice in London becoming a member of the
Royal College of Physicians. He never married.
He published De Magnete (On the Magnet) in 1600 which
became the standard work throughout Europe on electri-
cal and magnetic phenomena. It is a comprehensive review
of what was known about the nature of magnetism, and
Gilbert added much knowledge through his own experi-
ments. He built a philosophy where magnetism was the
soul of the Earth; he believed that a perfectly spherical
lodestone, when aligned with the Earth's poles, would spin
on its axis, just as the Earth spins on its axis in 24 hours.

According to thirteenth-century philosophy, the compass needle points
towards the North star which, unlike all other stars, in the night sky, ap-
pears to be fixed. Thus, philosophers reasoned that the lodestone obtained
its "virtue" from this star. Better observations, however, showed that the
needle does not point exactly to the North Star and eventually it was shown
that it is the Earth that affects the compass. Apart from the roundness of
the Earth, magnetism was the first property to be attributed to the body
of the Earth as a whole:

Magnus magnes ipse est globus terrestris [the whole Earth is a
magnet]. William Gilbert

By the early 17th century the properties of magnets were well known
and many folk tales (such as the anti-magnetic properties of garlic) had been
debunked. Magnetism was believed to be an effect different from electricity,
their intimate relationship had not been discovered.

Careful experimentation with magnets came to a head in the late 19th
century. By then reliable batteries had been developed and the electric
current was recognized as a stream of charged particles. In 1870 0rsted
noted that a compass needle placed near a wire was deflected when a current
was turned on, that such a deflection also occurs when the wire is moved,
and he concluded that moving charges generate magnetic effects. These
results were furthered by Ampere and who rendered them into a precise
mathematical formulation.

Hans Christian 0rsted (Aug. 14, 1777 - March 9, 1851).
In 1806 0rsted became a professor at the University of
Copenhagen, where his first physical researches dealt with
electric currents and acoustics. During an evening lecture
in April 1820, 0rsted discovered that a magnetic needle
aligns itself perpendicularly to a current-carrying wire, defi-
nite experimental evidence of t lie lelat ion ship between elec-
tricity and magm phenoin had been first dis-
covered by the Italian jurist Giaii Douienico Romagnosi in
1802, but his announcement was ignored).
0rsted's discovery, in 1820, of piperine, one of the pungent
components of pepper, was an important contribution to
chemistry, as was his preparation ol metallic aluminum in
1825. In 1824 he founded a society devoted to the spread of
scientific knowledge among the general public. Since 1908
this society has awarded an 0rsted Medal for outstanding
contributions by Danish physical scientists. In 1932 the
name oersted was adopted for the physical unit of mag-
netic field strength.

Andre Marie Ampere (Jan. 20 1775-.lu.iie 10 1836). Born
in Lyon. France, died in Marseilles. France. Andre Ampere
was a Professor at the Ecole Folyteclmique trout 18.1 !. to
1828 and then at Universite de France from 1826 until his
death. He worked on electromagnetism and analysis. He
also made contributions to line geometry extending ideas
of Binet. Ampere attempted to give a combined theory
of electricity and magnetism in the early 1820's. He for-
mulated a circuit force law and treated magnetism by pos-
tulating small closed circuits inside the magnetized sub-
stance. This approach became fundamental for the 19th
Century. Ampere's most important publication is Memoir
on the Mathematical Theory of Electrodynamic Phenom-
ena, Uniquely Deduced from Experience (1827).

During the same period Faraday made various experiments with moving
magnets (as opposed to moving wires). He found that a magnet moving in
a coil of wire generates a current: moving magnets generate currents. This Moving magnets generate

result provides the principle behind electric generators, be it small house-
hold ones, or the giant ones found in Hoover Dam. The fact that charges in
motion create magnets and that moving magnets generate currents demon-
trates the intimate connection between electric and magnetic phenomena.

Michael Faraday (Sept. 22, 1791 - August 25, 1867).
Michael Faraday became one of the greatest scientists of
the 19th century. He began his career as a chemist; wrote
an important manual of practical chemistry, and discovered
a number of new organic compounds, among them benzene.
He was the first to liquefy a "permanent" gas {i.e., one that
was believed to be incapable of liquefaction).
His major contributions were in the field of electricity and
magnetism. He was the first to produce an electric current
from a magnetic field, invented the first electric motor and
dynamo. He provided the experimental, and a good deal
of the theoretical, foundation upon which Maxwell erected
classical el< el i nmagnel ic Held ! henry.

Faraday created the concept of a field. He imagined that
any magnet or charged object generates an influence that
permeates space, such emanation is called a field. If an-
other magnet or charged object draws near, it is the in-
teraction between this field and the new charged object or
magnet which the latter feels as a force. He also showed
that charge is never destroyed not created.

James Clerk Maxwell (June 13 1831-Nov 5 1819). Born in
Edinburgh, Scotland, died in ( 'ainbridgc. Cambridgeshire,
England. Maxwell attended Edinburgh Academy where he
had the nickname 'Dafty'. While still at school he had
two papers published by the Royal Society of Edinburgh.
Maxwell then went to Peterhouse ( Iambi idge but moved to
Trinity where it was easier to obtain a fellowship. Maxwell
graduated with a degree in mathematics from Trinity Col-
lege in 1854. He held chairs at Marischal College in Ab-
erdeen (1856) and married the daughter of the Principal.
However in 1860 Marischal ( 'ollcgo and King's College com-
bined and Maxwell, as the junior of the department had
to seek another post. After failing to gain an appoint-
ment to a vacant chair at Edinburgh he was appointed to
King's College in London (1860). He made periodic trips to
Cambridge and, rather reluctantly, accepted an offer from
Cambridge to be the first Cavendish Professor of Physics
in 1871. He designed the Cavendish laboratory and helped

Maxwell's first major contribution to science was to show
that Saturn's rings must consist of many solid particles
(confirmed by the Voyager spacecraft), this result won him
the Adams Prize at Cambridge. Maxwell next considered
the theory of gases and showed that temperature and heat
are related to the motion of gas molecules.
Maxwell's most important achievement was his extension
and mathematical formulal ion of Faraday's theories of elec-
tricity and magncl ism. Llis papei on 1/araday's theory was
read to the Cambridge Philosophii al Society in two parts,
1855 and 1856. Maxwell showed that a few relatively sim-
ple mathematical equations could express the behavior of
electric and magnetic fields and their interrelation. The
four equations (now known as Maxwell's equations), first
appeared in fully developed form in his book Electricity
and Magnetism (1873). They are one of the great achieve-
ments of 19th-century mathematics. Maxwell showed that
an electromagnetic disturbance travels at a speed of light
(1862) and concluded thai light is an electromagnetic phe-

Faraday also showed that charge is conserved. That is, the amount of Charge is conserved
positive charge minus that of negative charge is always the same.

The results of all these investigations can be summarized in a series
of four equations. These were studied extensively by Maxwell who noted
that they are inconsistent with charge conservation, but Maxwell himself
realized that a slight modification in one equation would get rid of this
problem. The modification proposed by Maxwell is simple, but the results
are so momentous that the modified set of four equations are known as

Maxwell's equations. Why are Maxwell's equations so important? There
are four reasons:

> They describe all electromagnetic phenomena with perfect accuracy
for distances larger than about 10 _8 cm.

> They are inconsistent with Newtonian mechanics, and so present the
first solid evidence for the modification of Newton's theory

> There are solutions of the equations which describe waves traveling at
speed c = 299, 792km/s (which is also the speed of light).

Light, electricity and
magnetism are different
aspects of the same set of
phenomena and are
described by a single theory

The last point leads to the inescapable conclusion is that light is pre-
cisely the object that was described by the wave-like solution of Maxwell's
equations (without his modification there are no wave- like solutions); in
Maxwell's own words

We can scarcely avoid the conclusion that light consists in the
transverse undulations of the same medium which is the cause
of electric and magnetic phenomena.

It is in this way that the next unification in physics occurred: light,
electricity and magnetism are different aspects of the same set of phenomena
and are described by a single theory. Because of this we now speak of
electromagnetism and not of electric and magnetic phenomena separately.

5.2 Waves vs. particles

I mentioned above the word "wave" in several occasions. Since waves will
appear repeatedly in the following I will take a short detour to explain what
waves are and what are their properties. The American Heritage Dictionary
defines wave as

A disturbance or oscillation propagated form point to point in a
medium or in space

Thus when a stone is dropped on a calm pond we see a series of circular
waves emanating form the spot where the stone hit the water, spreading out
at a certain speed. If a bigger stone is used the water the waves become more
pronounced, the distance form crest to trough becomes larger. If instead of
dropping a stone we attach it to a rod and move it up and down we find
that the faster we move it the closer together the crests and troughs of the

waves; so that if we look at one point on the pond's surface we will see the
water swelling and ebbing faster.

These characteristics of the waves have definite names; see Fig. 5.1,

• The frequency is the number of wave-crests that go through a point on

the pond every second. The frequency is the

> The wavelength is the distance between two crests.

> The amplitude is the distance between crest and trough.

mber of wave-crests that
go through a point on the
pond every second

The wavelength is the
distance between two ere:

These properties, together with the speed at which the wave spreads char- Tne amplitude is the
, . ,, distance between crest

actenze the waves. trough

Figure 5.1: Definition of the wavelength and amplitude of a wave.

Imagine a cork floating on the pond. As the wave goes by the place
where the cork is floating it will boob up and down. Suppose that you
measure the time it takes for it to go down from its highest point, down to
its lowest and then back to its highest point again, then the frequency is the
inverse of this time. So if the cork takes 0.5 seconds to go up and down and
back up, the frequency would be Q 5 1 see or 2 inverse-seconds. This is just
a way of counting the number of oscillations per second: if each oscillation
takes half a second, there will be two oscillations per second, and so the
frequency is two inverse-seconds; a frequency of 7 inverse seconds indicates
that there are seven oscillations each second, etc. There are many kinds of
waves: water waves on a pond, sound waves in air (or water or any other
medium), electromagnetic waves, etc.

Imagine now a calm pond with a few leaves floating on the surface. At
one time a child drops a stone which makes a series of expanding circular
waves. As they spread the waves eventually come to the floating leaves
which bob up and down. The notable thing about this detail is that the
leaves do not change position, even though the wave spreads, it does not
carry the leaves with it. The same thing can be said of the water itself, the
waves spread though it but do not carry the water along with them. In fact,

if you look closely at the particles suspended in water (ponds usually have
many of those) as the waves pass, they make circular motions about their
initial positions but are not carried along. These waves use water as their
propagation medium, in the same way as sound waves use air (or water or
other materials) to propagate in. Without a medium these waves simply do
not propagate: there is no sound in the vacuum. A reasonable question in
connection with these observations is whether all waves need a medium to
propagate in, the answer is (perhaps surprisingly) no!, and the way this was
discovered is the subject of many of the following sections

A particle is characterized by its mass and other measurable properties
(for example, its charge). I will assume that this is intuitively clear. Or-
dinary everyday experience shows that waves behave very differently from
particles 1 . For example, if you are taking cover behind a wall form a person
shooting peas at you, you will not be hit; yet when she screams that you
are a chicken, you hear her perfectly well. Sound waves (and all waves in
general) have the ability to go around obstacles (up to a certain extent: if
the wall is very tall and wide the insults will not reach you); particles have
no such ability.

The above properties of sound waves are well known. But, if light is a
wave, should it not behave in the same way? And if it does, how come we
do not see a person standing behind a wall (whom we can clearly hear)? I
will now consider this (apparent) paradox.

5.3 Light

It is now known that under all common circumstances light behaves as a
wave propagating at a speed close to 300,000km/s. This, however, is a
recent realization; in fact, whether light traveled at finite or infinite speed
was the subject of much debate was left unanswered for a long time. Galileo
tried to measure the speed of light by experiment: he put two persons on
hills (separated by a bit less than a mile), and then told one open a lantern,
the other was to raise his/her hand when he/she sees the light and the first
notes any lapse between his/her opening the lantern and seeing the raised
hand. No time delay was observed (which is not unnatural, the lapse is
about 10 _5 s!). So the question remained unanswered 2 .

This is not true when phenomena at very short distances are examined, at distances

below 10~ 8 cm (atomic size) the difference between waves and particles becomes blurred.

2 One can, however, use this result to get a limit on the speed of light. If the human

response time is, say, half a second, then this experiment shows that light travels faster

In 1670 the Danish mathematician Olaus R0mer observed that the eclipses
of Jupiter's moons were 11 minutes ahead of schedule when the Earth was
closer to Jupiter, and they lagged behind (also by 11 minutes) when the
Earth was farthest from Jupiter. Assuming that there are no problems with
the predictions of Newtonian physics concerning the motion of Jupiter's
moons, he concluded that the discrepancy was due to the different times
light takes to get to Earth at the two extremes of its orbit (Jupiter moves
very little during one year, it takes 12 years for it to circle the sun), see Fig.
5.2. R,0mer then calculated that the speed of light would be 210,000km/s.
The modern value of the speed of light is 299, 792km/s.

This is, of course, not the only possible explanation, Romer could have
argued, for example, that Newton's equations could not account for Jupiter's
motion. Still the hypothesis that light travels at a finite speed furnished the
simplest explanation and, following Ockham's razor (Sect. ??) it is the one
which ought to be examined first. Soon after R,0mer's argument was made
public the fact that light travels at finite speed was demonstrated in various
experiments and was universally accepted.

Figure 5.2: Diagram of the reasoning used by R,0mer to determine the speed
of light.

So light propagates at a finite speed. What is it made of? Newton
than 2miles per second.

12

believed that light was made of corpuscles, but even the weight of Newton's
opinions could not withstand the experimental evidence showing that light
behaves as a wave. This sounds preposterous: a wave, such as sound, will
"go around corners" but light does nothing of the kind... or does it? In fact,
it does! If you look very closely at a very sharp edged screen you will see
that some light actually goes behind the screen: light does behave as a wave
(see Fig. 5.3). This is not common knowledge because it is a small effect,
light dies out almost as it turns the corner, if the corner is not very sharp,
light is scattered in many ways and the effects disappears; in other words,
for light, almost any obstruction is a very tall wall.

Figure 5.3: Picture of the shadow cast by the corner of a screen. Noote that
the shadow region is not completely dark.

The wave theory of light leads to some surprising consequences. For
example, it predicts that the shadow cast by a dark circular screen should
have a bright spot in its center, and this would be absurd were it not for
the fact that the bright spot is indeed there! (see Fig: 5.4)

Figure 5.4: Shadow cast by a small opaque disk. Note the bright spot in
the center of the shadow.

By the beginning of the 19th century the hypothesis that light is a wave
traveling at large (by our standards) but finite speed 3 was proven and was
universally accepted. Being a wave we can ask what is its wavelength, ampli-
tude, frequency, etc; it turns out that visible light has very small wavelength,

3 The speed depends on the medium in which light travels; the value given abcn
responds to the speed in space.

about 10 _5 cm. Another natural question is then, do electromagnetic waves
with larger and smaller wavelengths exist?

The answer is yes. Visible light is but a member of a large family of
waves; they are all electromagnetic waves, and they are all described by the
Maxwell's equations. For historical reasons waves of different wavelengths
have different names (see Fig. 5.5). Thus we have (the symbol ~ means
"about" )

Wavelengths c

f electromagnetic waves

Name

Wavelength

Radio

~ 10cm or larger

Microwave

<~ 1cm

Infrared

~ 10" 3 cm

Visible

~ 10" 5 cm

Ultraviolet

~ 10 _6 cm

X-rays

~ lCT 8 cm

Gamma-rays

~ 10" 9 cm or smaller

1

■ Ml-

- 3. Z. 3 t

Figure 5.5: The electromagnetic spectrum.

All of these are common names. Every one of these waves travels at the
same speed in vacuum 4 equal to the speed of light (called "visible" above)
in vacuum; the only difference between them is the wavelength, the distance
between two consecutive crests in the corresponding wave trains.

So light is a wave, similar then to sound waves, or water waves. But all
these waves are produced by the undulations of some medium: water for

4 In a medium there is
i be different.

v interaction between the atoms and the

is and the speed

water-waves, air (for example) for sound, etc. Thus it was postulated that
the medium in which light undulates is called ether.

5.4 Problems

The end of the 19th century witnessed the growth of evidence against the
classical physics based on Newton's theory. I will discuss two such problems,
the first concerns the ether, which appeared to have inconsistent properties;
the second refers to an apparent contradiction between Galilean relativity
and the theory of electromagnetism. The resolution of these conflicts cannot
be achieved within Newtonian physics: it requires the theory of relativity.

5.4.1 Ether

Having postulated the existence of the ether as the medium in which light
travels it becomes interesting to determine the properties of this material.
First and foremost, since the light from distant stars does reach us, we
must assume that the ether permeates the whole universe up to its farthest
reaches. We must then imagine that the Earth plunges through this ether
as it circles the Sun. The ether must then be very tenuous, for otherwise
the friction would have stopped the Earth long ago. Let us now derive some
other predictions derived from the ether hypothesis.

As the Earth moves through this ether a kind of "ether wind" must be
present on Earth's surface. To see why this should happen consider the
following analogy. Imagine a windless day in which you take a ride in your
red convertible which, unfortunately, has no windshield. As you speed up
you will feel the air blowing, the faster you go, the stronger this wind is. In
the same way, replacing air — > ether and red convertible — > earth, a very
sensitive apparatus on the surface of the earth should detect and ether wind.

So, can the ether wind be detected? Apparently yes! The idea for the
first experiments is based on the following argument. Imagine yourself back
in your convertible (with no windshield) taking your nagging grandmother
to the store; she sits in the back seat. ..it's safer. She talks all the time, but,
fortunately, her words get blown back by the wind. In contrast she hears
everything you say, for your words get blown back by the wind, right into her
ears (good grief!). In the same way, as we stand on Earth, the ether wind
should blow back the light coming from the stars. At different times of the
year, the ether wind blows in different directions since the earth is moving in
different directions, hence the observed positions of the stars should change
(see Fig. 5.6)... and they do!

ft— .^

Figure 5.6: The shift in the observed position of the stars caused by the
ether wind.

But, wouldn't the earth drag with it some of the ether in its vicinity?
Well, since this peculiar behavior of the images of the stars were observed,
the earth must not drag the ether with it: ether goes through the earth
"much as the wind goes through a grove of trees" (as described by T. Young.)

This consequence of the ether wind is not the only prediction of the ether
hypothesis; in order to derive other consequences we need to go back briefly
to Newtonian mechanics. Suppose you are in a train moving at a speed of
lm/s with respect to a train station. Suppose now you kick a ball in the
direction of the train's velocity and which, as a result of your action moves
at 2m/s as measured in the train. Then an observer in the station will see
the ball move at 1 + 2 = 3m/s (see Fig. 5.7).

Thus the two parallel velocities (the train's and the ball's with respect
to the train) add up. In contrast if the ball were thrown up both observers
would measure the same (vertical) velocity. Consider now the same situation
but with light replacing the ball. If the train moves at speed v then light
traveling forward will move at speed v + c. If the light-beam is directed
upward both observers would measure the same vertical speed c. These
conclusions are inescapable from the Newtonian standpoint and, because
they are wrong, constitute some of the most important nails in the coffin

Figure 5.7: Addition of velocities according to Newtonian mechanics

of Newtonian mechanics. Let me examine first the following consequence
derived from it.

Suppose you consider light going in air and that the same beam is made
to enter a piece of glass. In air light will have a speed c a i r , while in the glass
it will have speed c g i ass ; these two quantities being measured at rest with
respect to the ether. The experiment I want to discuss measures the ratio
of speeds in glass and air. Now, if there is an ether, and the earth is moving
at a speed v with respect to it, then one can select the orientation for the
apparatus such that the beam happens to lie along the velocity v 5 . In
this case the speed of light in air and in glass will be altered, they become
c a ; r + v and c g i ass + v respectively; the experiment should give the result
(c a ir + v) J (cgi ass + v). If the beam is rotated 180° then the direction of
the ether wind is reversed and the experiment ought to produce the value
( c air ~ v ) / ( c glass — v )- The amazing thing is that, as first shown by Arago,
that this experiment gives the same value no matter how it is oriented with
respect to the motion of the Earth through the ether. In order to explain
this Fresnel suggested that transparent substances trapped some of the ether
and dragged it along, and the amount and manner of trapping was "just-so"
that the above experiment does not exhibit any effect. Of course the shift in
the position of the stars would then imply that the air does not trap ether
at all.

Curiouser and curiouser: the speed of light in glass depends on the color
of light, nonetheless the above experiment gives no effect for any color.
Therefore the ether trapped in glass should undulate with light precisely so

5 In practice the experiment is set on a rotating table and is repeated for a variety of
orientations.

as to compensate for this difference in speeds (note that the ether trapped
with the glass travels with it).

So the ether is a medium which goes through all objects, but some of it
is trapped by transparent substances and whose elasticity depends on the
color of light going through it. In order to test this Fizeau performed a very
important experiment. He sent light through tubes with water flowing in
different directions. The water was supposed to drag at least some ether,
which would then alter the speed of light. The results were positive and in
accordance with Fresnel's hypothesis. So we have a big contradiction: the
observation of starlight requires the Earth and the Earth's air not to drag
any ether. But the Fizeau experiment requires transparent media to drag a
significant (and measurable) amount of ether.

The most famous of the experiments made to detect the motion through
the ether was the Michelson-Morley (or M£M) experiment. This is a very
clear experiment. The idea is to send to take a light beam, to split it in
two and send the daughter beams in perpendicular directions, these are
then reflected back and recombined. The distances traveled by the daughter
beams will be different and so there will be a mismatch between the two
light wave trains resulting in a pattern of light and dark fringes after they
are recombined (see Fig. 5.8)

Figure 5.8: A diagram of the Michelson-Morely interferometer
Now suppose we rotate the table where the experiment is placed. The

speeds of the two beams with respect to the ether will change, and so will
the times taken for the beams to recombine. Because of this the mismatch
between troughs and crests in the two wave trains also changes and a shift
in the pattern of dark and bright lines should be seen. ..except that it wasn't!
No detection of the motion through the ether could be measured.

It was then claimed that the only thing proved was that the ether in the
basement where the experiment was done was dragged along with the air.
But the experiment was repeated a large number of times, in particular it
was done on a hilltop: no effects were ever obtained.

This last result was the death blow to the ether theory: MfiM's experi-
ment showed that the ether must be dragged along by the air, while stellar
observations denied precisely that!

Maxwell's equations do not
specify in which frame the
speed of light equals c

5.4.2 Galilean Relativity

Galileo formulated his principle of relativity by stating that one cannot use
any mechanical experiment to determine absolute constant uniform velocity.
Now Maxwell's equations contain a velocity c but they do not specify with
respect to what is this velocity to be measured!. We must conclude that either
absolute velocities can be determined using experiments involving light, or
else light must move at speed c in all reference frames.

But this is impossible to accept within Newtonian mechanics, for within
this theory velocities simply add. If we then have a source of light moving at
speed v, the light form it ought to travel at speed c + v in direct contradition
to Maxwell's equations which predice that light travels with speed c, no
matter how fast the speed of the source.

The properties of the ethei
were inconsistent
Newtonian physics w;
inconsistent with the

of e

-nagnf

5.5 Prelude to relativity

So this was the situation before 1905: the ether was postulated, but its
properties were inconsistent. Newton was believed to be right, but the cor-
responding mechanics was inconsistent with the results of electromagnetism.
l ts Was Newton's theory correct and all the experiments in electricity and mag-
netism wrong? If Newton was wrong, how can all the successes of his theory
be understood? How can one understand light as a wave if the thing in
which it travels cannot be described consistently?

All these problems were solved with the advent of the Special Theory of
Relativity to which I now turn.

Chapter 6

The Special Theory of
Relativity

6.1 Introduction

The puzzling properties of light and the ether remained through the turn of
the century and up to 1904: the speed of light (as described by the equations
of electromagnetism) did not depend on the motion of the observer and,
stranger still, the medium in which light propagates could not be described
consistently.

A final effort was made in order to understand in a "fundamental" way
the negative result of the Michelson-Morley experiment. It was postulated
(independently) by Fitz-Gerald and by Lorentz that matter moving through
the ether is compressed, the degree of compression being just so that there
is a negative result in the MfifM experiment. The claim was that the ether
wind does slow down and speed up light, but it also contracts all objects
and these two effects conspire to give no effect in all experiments.

A calculation shows that an object of length £ moving with velocity v
with respect to the ether should be contracted to length £' given by

(where c is the speed of light) in order to get the null result required.
1

So in order to understand the gamut of experimental results the ether
had to be a very tenuous medium that could not be felt or tasted, nonethe-
less the strongest materials would be squashed by it by an amount which
makes it impossible to see the ether's effects. The amount a material would
be squashed, though admittedly very small, would always be there and is
independent of the composition of the object going through the ether (see
Fig. 6.1). This is a situation like the one I used in the " little green men on
the moon" example (see Sect. ??): the ether has was awarded the property
that no experiment could determine its presence; the ether hypothesis is not
falsifiable.

-

Figure 6.1: The idea behind the Lorentz-Fitz-Gerald contraction.

6.2 Enter Einstein

In 1905 Einstein published three papers. The first (dealing with the so-called
"photoelectric effect") gave a very strong impulse to quantum theory, and
got him the Nobel prize in 1921. The second dealt with the movement of
small particles in a fluid (Brownian motion).

The third paper (Fig. 6.3) of 1905 was called On the electrodynamics of
moving bodies, it changed the face of physics and the way we understand
nature.

This paper starts with a very simple (and well known) example: if a
magnet is moved inside a coil a current is generated, if the magnet is kept
fixed and the coil is moved again the same current is produced (Fig. 6.4).
This, together with the difficulties in detecting the motion with respect to
the ether, led Einstein to postulate that

Figure 6.2: Albert Einstein (in his later years)

the same laws of electrodynamics and optics will be valid for all
frames of reference for which the laws of mechanics hold good

which is known as the Principle of Relativity.

In order to understand the implications of the Principle of Relativity we
need (again) the concept of an inertial observer (see Sec. ??). This is a
person which, when observing an object on which no forces act, finds that
it moves with constant speed in a straight line, or else is at rest. In terms
of inertial observers we can restate the Principle of Relativity:

all the laws of physics are the same for all inertial observers.

Galileo made a very similar statement but he referred only to the laws of
mechanics, Einstein's achievement was not only to provide a generalization,
but to derive a host of strange, surprising, unexpected and wonderful con-
sequences from it.

I the laws of physics ar
e same for all inertial
observers

Figure 6.3: The 1905 paper on Special Relativity

Figure 6.4: Illustration of one of the experimental facts that lead Einstein
to the Principle of Relativity.

The speed of light c \i
same for all inertial
observers

6.2.1 The first prediction: the speed of light and the demise
of Newton's mechanics

Now that we have stated the Principle of Relativity we can examine its
implications, and almost immediately we find reason to worry.

Maxwell's equations (the equations of electromagnetism, see page ??)
contain a quantity we called c, the speed of light, which is given without
reference to any inertial observer. So, if we accept the Principle of Relativity
and trust Maxwell's equations, we must conclude that c is the same for all
inertial observers. So if Jack measures the speed of a beam of light while
sitting at the top of the hill, and Jill also measures the speed of the same
beam of light while running up the hill, they should get exactly the same

answer, no matter how fast Jill runs. It is often said that Einstein "proved
that everything is relative" but, in fact, his first conclusion was that the
speed of light is absolute.

This property of light is very different from, say, the properties of peas
as described by the mechanics of Newton: if a person rides on a scooter
and shoots peas, these move faster than the peas shot by a person standing
by (see Sect. ??). In contrast if the person on the scooter turns on a laser
and the person standing by does the same when they coincide on the street,
these two laser beams will reach Pluto at the same time (Fig. 6.5); this
happens even if the scooter moves at 99% of the speed of light.

Figure 6.5: The pea shot from the scooter moves faster, yet both laser beams
get to Pluto (it is really a photograph of Pluto) at the same time.

Newton would be horrified by this behavior of light beams: according to
his mechanics velocities add, so that the laser beam from the scooter should
reach Pluto sooner.

Thus, once Einstein adopted his Principle of Relativity, he was faced
with a choice: either dismiss Newtonian mechanics or dismiss Maxwell's

equations. It was impossible for them both to be right. Newton's mechanics
had survived for about 250 years, it was universally accepted in the physics
community, and its predictions agreed with all experiments (done up to
1905). Maxwell's equations, in contrast, were rather new, were not tested
as thoroughly as Newton's, and were not universally accepted. Nonetheless
Einstein took the daring path of siding with Maxwell and so challenged the
whole edifice of the Newtonian theory. He was right.

Having chosen sides, Einstein assumed that Newton's mechanics were
not a good description of Nature under all circumstances: it must then be
only a good approximation. Einstein's work was then cut out for him: he
needed to find a generalization of Newton's mechanics which is consistent
with the Principle of Relativity, and which agrees with experiment as well
as (or better than) Newton's theory. He was successful.

Significant discrepancies between Newton's and Einstein's mechanics be-
come noticeable only at speeds close to c which explains why no problems
were detected with Newton's theory before 1905: all experiments were done
at speed very small compared to c. In this century a wealth of experimental
evidence has been gathered which supports Einstein's mechanics in favor of
Newton's. The best examples appear in experiments done since the 1950's
using subatomic particles which are relatively easily accelerated to speeds
approaching c. The behavior of such experiments completely vindicates
Einstein's approach while being inexplicable from the Newtonian viewpoint.

;ergy accelerators. Most of the studies in subniomic
ysics are done in enormous machines commonly called
jlliders" where electrically charged particles such as elec-
ns and protons are accelerated to speeds very close to
I of li^lii and then forced to crash into each other. The
ullinjj debris provides important clues as to the funda-
structure of matter. A popular desiju) for a collide)
of one or more concentric rings in which the collid-
ng particles are piped and accelerated using electric and
magnetic fields. Given the enormous speeds of the par-
ticles the design must be extremely accurate, even a very
small error can send all the particles crashing into the walls
of the ring. All calculations are done using Einstein's me-
chanics, and the behavior of the particles perfectly matches
the predictions of the theory; a design of a collider using
Newtonian mechanics would lead to a useless machine.

Concerning the addition of velocities, Newton's formula is, strictly speaking,
not correct even for slow moving obejcts. The corrections are, however, very samll
when the speeds are small compared to that of light. For example for the case of
the passenger in a train in Fig. ?? if the speed of the ball is u and that of the train
is v the speed measured from the platform is not u + v as Newton would claim, but

( u + v ) xc 2 /(c 2 + uv)

that is, there is a small correction factor c 2 /(c 2 + uv) which, for ordinary velocities
is very small indeed, for example for the example u = lm/s, v = 2m/s, this factor
is 0.9999999998 (Newton would have predicted 1 instead). On the other hand, if
both u and v are half the speed of light, the speed seen from the platform would
be 80% of the speed of light (and not c as Newton would have expected). For the
extreme case where either u or v (or both) are equal to c, the speed seen from the
platform would again be c.

In conclusion: the Principle of Relativity together with Maxwell's equa-
tions imply that there is a universal speed whose value is the same to all
inertial observers. This fact required several fundamental changes in the
manner we understand the world.

6.2.2 The second prediction: Simultaneity is relative

One concept which is radically modified by the Principle of Relativity is that
of simultaneity. Every-day experience indicates that the statement "two
events happened at the same time" (i.e. they are simultaneous) is universal,
and would be verified by any one looking into the matter. Thus I can say, "I
got home at the same time you got to work" and nobody (usually) wonders
about the consistency of such statement.

The surprising result is that two FBI agents looking into the matter but
moving with respect to each other (and having very accurate clocks) would
get conflicting answers. In order to illustrate this result we will consider
two murder mysteries, one set in Victorian England which is analyzed using
Newton's ideas, the other is set in outer space and is studied following
Einstein's guidance.

The first murder mystery (ca. 1890)

Sherlock Holmes is called to investigate a murder: a man was found shot in
a train car, with two bullets in his head. After much investigation Sherlock
finds a hobo who was at a station as the train wheezed by. This man saw
two men come in from opposite sides of a wagon and simultaneously fire
their revolvers at a chap sitting right in the middle of the train car. Being

a Newton acolyte, Holmes is a firm believer that simultaneity is a universal
concept, and concludes that both men fired at the same time as an absolute
fact. Inspector Lestrade (from Scotland Yard) manages to find both men,
who are found guilty of the crime and die in the gallows.

The second murder mystery (ca. 2330)

A murdered man is found in the cargo bay of the starship Enterprise with
two head wounds caused by laser beams. The tragedy was observed from
three places: a space station, the cargo bay itself, and a Klingon ship (a
"bird of prey" ) . At the time of the crime the Enterprise was moving at a
speed c/2 with respect to the space station; the bird of prey was moving
in the same direction as the Enterprise at a speed 3c/4 with respect to the
space station (and was ahead of the Enterprise). To simplify the language
we will say that both ships as seen from the space station were moving to
the right (see Fig. 6.6).

Everyone agrees that the dead man was hit on the head by two laser
beams simultaneously. These beams were fired by a klingon at the back of
the cargo bay, and by a human at the front. They shot while they stood at
the same distance from the victim. Both life-forms are arrested and put to
trial.

Captain Kirk, then at the space station, acts as the human's lawyer.
Kirk points out that the klingon must have fired first. Indeed, at the time
of the murder the klingon was placed in such a way that the Enterprise
carried the victim away from his laser bolt; in contrast, the ship carried the
victim towards the human's laser bolt (Fig. 6.7). Since both bolts hit at the
same time, and they travel at the same speed c for all observers, the klingon
must have fired first. "The klingon's guilt is the greater one!" Kirk shouted
dramatically, and sat down.

The captain of the bird-of-prey, who is (of course) acting as the klingon's
lawyer, disagrees. His ship was moving to the right of the space station,
but much faster than the Enterprise, hence, with respect to this ship, the
Enterprise was moving to the left. "I can then use my esteemed colleague's
arguments and categorically state that it was the human that fired first (see
Fig. 6.8), it is her guilt that is the greatest."

Dr. McCoy happened to be in the cargo bay at the time of the shooting
and testifies that he saw both the human and the klingon fire at the same
time: since the beams hit the victim at the same time, and they were at the
same distance, they must have fired at the same time (Fig. 6.9).

at rest

f^~

Figure 6.6: The setup for the second murder mystery. The velocities are
measured with respect to the space station (labeled "at rest").

Now, the law (in this story) states that the guilty party is the one who
fired first, but deciding who did fire first is impossible! This is so because
events occurring at different places will not be simultaneous to all observers. Events occurring at
The fact that c is the same for all observers implies that if two events sep-
arated by some distance (such as the firing of the lasers) are simultaneous
to one observer (such as McCoy) they will not be simultaneous to observers
moving relative to the first (such as Kirk and the Klingon captain). Even
the ordering in time of these events is relative

Simultaneity is relative for events separated by a non-zero dis-
tance. 1

Let me use a short-hand and let K be the statement "the klingon shoots" ,
while H denotes "the human shoots" . Then

d if Fere

multaneoi
observers

,s explained by Spock to Kirk. ..at great length.

Figure 6.7: Illustration of Kirk's argument (the murder as seen from the
space station)

Summary of the arguments

K

happens before

H

as seen from the space station

(Kirk'

argument)

H

happens before

K

as seen from space station

(Klingon capt.'

argument)

K

simultaneous w i.l 1)

H

us seen (nun Enterprise

(McCoy'

argument)

So the Principle of Relativity forces us to conclude that in this situation
the ordering of events in time is relative. But, this better not be true for all
events: if the Principle of Relativity would predict that all time orderings
are relative we could then imagine an observer who sees you, the reader,
being born before your parents!

So there are events such as birth and death of a person which should
occur in succession with the same ordering for any observer. And there are
other events, such those in the shooting mystery, whose ordering in time is
observer dependent. What is their difference?

The one clue is the following: in the story the assassins came in from

Figure 6.8: Illustration of Klingon captain's argument (the murder as seen
from the bird of prey)

opposite sides of a cargo bay and shot the victim. Since lasers travel at the
speed of light, the human will receive the image of the klingon shooting only
after she herself has fired (in order to see anything we must receive light
from some source); the same is true for the klingon. So, when they fired they
could not have been aware of each other's action.

This is not the same as for birth and death: a cat is born and then the
dog eats the cat. It is then possible for you to tell your dog, that is, to send
him a signal, that the cat was born. This signal reaches the dog before he
performs his grim action (Fig. 6.10)

So two events A (cat is born) and B (dog eats cat) are ordered in the
same way in time for all observers if we can send a signal at the time one
event occurs (A) which will reach an observer who will witness the second
event (B). In this case everyone will agree that A occurs before B, no matter
what the relative speed of the observer. An extreme case consists of those

View from the enterprise

Figure 6.9: Illustration of McCoy's argument (the murder as seen from the
Enterprise) .

events occurring at the same time at the same place will be seen to occur
at the same time by all observers (everyone agrees that the laser beams hit
the victim at the same time).

In contrast if no signals sent at the time A occurs can reach an observer
before B happens, then the ordering in time of A and B depends on the
relative velocity of the observer.

So there is no hope of going back in time with the winning Loto number
and becoming a millionaire. If you think about it, the number of paradoxes
which would arise if all time orderings were relative would be enormous: if
you could go back in time, there would be two of you: one a pauper and the
other a millionaire... but which one is you? Fortunately the Special Theory
of Relativity simply disallows such situations.

Why did all this happen? Because the speed of light is always c. Both
laser bolts will be seen to travel at the same speed by all observers, and

Figure 6.10: Illustration of events whose time ordering is the same for all
observers.

because c is not infinite, the time it takes to reach a target depends on how
the target is moving.

I will emphasize again the conclusions. Since the speed of light is the
same for every observer in an inertial frame of reference, two things that are
simultaneous to one observer will not be so according to other observers.
The inescapable conclusion is that simultaneity is not an absolute concept:
the statement "two events at different places occurred at the same time"
is true only in a certain inertial reference frame and will be found to be
incorrect in other frames.

Despite this there are events that everyone will agree are simultaneous:
any two events happening at the same time and at the same spot will be
seen to coincide by any observer. It is when the events are separated by a
distance that simultaneity is relative. If events occurring at the same time
and place for one observer were seen to occur at different times by another
observer one can imagine going to a reference frame where the bullet that
killed Lincoln went by his seat one hour before the president sat down. In
this frame he was never assassinated!

One thing that Principle of Relativity does not permit is for some events
which occur sequentially and such that the first affects the second to be
inverted in order. For example it is impossible to go to a frame of reference
in which the end of an exam occurs before it begins. It is only events that are
mutually independent whose ordering in time can be inverted: two babies
could be seen to be born one before the other or vice- versa, but only if they
are not born at the same time at the same spot, so Jacob could not be the
first born to Isaac (as opposed to Essau) in some frame of reference... the

1 J

Bible's story is, in this sense, frame independent.

6.2.3 The third prediction: The demise of Universal Time

Another peculiar and surprising consequence of the Principle of Relativity
is that time intervals are no longer universal but depend on the frame of
reference. Consider, for example, a clock consisting of a light source and
detector. The source emits a light pulse, the pulse goes up and is reflected
at a height hby a mirror. It is then detected and this determines one unit
of time. See Fig 6.11.

and detector

Figure 6.11: A clock at rest with respect to the observer

The time it takes the light pulse to come and go is to = 2/i/c. This
is precisely the time it would be measured by any observer carrying any
other clock as long as this observer is not moving with respect to the above
timepiece.

Now let's consider what an observer moving with respect to this simple
clock sees. This is shown in Fig. 6.12

It is clear that the distance traveled by the beam is larger than the up-
down trip observed by the first person. But since the speed of the light beam
is the same for both observers, the time measured by the second observer
will be larger. If we have two such clocks one is at rest with respect to us and
the other is moving, we find that the moving clock slows down, moreover,
the faster it moves the slower it ticks. This is called time dilation: a moving
clock ticks slower.

This argument was based on the simple clock of Fig. 6.11, will it be
true for all clocks? To examine this question let's assume we have another
clock (a Rollex, for example) which gives ticks same way no matter how

Figure 6.12: A clock moving with speed v to the right with respect to the

i»l >S(TY(T

observer

it moves. You go on a long trip to a near-by star taking the Rollex with
you and also a clock like the one in Fig. 6.11. Your spaceship, you will
notice, has no windows (they had to cut the budget somewhere. 1 ), but you
go anyway. You experience the effects of lift-off but after a while you appear
to be at a standstill: you are then moving at a constant speed with respect
to Earth. But remember we assumed that the Rollex still ticks the same
way as the clocks on Earth, and we have proved that your light-clock does
not. So you will see a mismatch between the Rollex and the light-clock:
this is an experiment which is done completely inside the spaceship and
which determines whether you are moving. If there were such a Rollex the
Principle of Relativity would be violated.

If we accept the Principle of Relativity we must conclude that time
dilation will occur for any clocks, be it a Rollex, a biological clock or a
Cartier. Note that this follows from the Principle of Relativity and the
validity of Maxwells' equations, no additional assumptions are required.

If an observer at rest with respect to a clock, finds that she is pregnant
and eventually delivers, the whole process taking precisely nine months,
another observer moving with respect to her (and the simple clock) will find
this claim to be wrong, he will state that she had a longer pregnancy (or a
very long delivery) but that in any case the whole thing took longer than
nine months.

Time dilation and Pythagoras ' theorem. The distance the
light has to travel in Fig. 6.12 can be determined by using
Pythagoras' theorem.

In this reference frame light travels along the long sides of
the triangles, each has a length which I call £/2; let's call
X the time it takes to complete the trip, by Pythagoras'
theorem £/2 = ^h 2 + (vT/2) 2 . On the other hand £ = cT
since light moves at speed c for any observer and it takes
a time T (according to the moving observer!) for it to get
back to the detector. Solving for T we get

(2fe/c)

To

\/l " WC) 2 y/l-(v/c)*'

Thus the observer in motion with respect to the clock will
measure a time T greater than To, the precise expression
being given by the above formula.

So how come we do not see this in ordinary life? The reason is that the
effect is very small in everyday occurrences. To be precise it an observer at
rest with respect to the clock in Fig. 6.11 measures a time To then the ob-
server which sees the clock move at speed v (and sees the situation depicted
in Fig. 6.12) will measure a time T, where T = Tq/^/1 — v 2 /c 2 (see the box
above). So the effect reduces to the appearance of the factor l/\/l — v 2 /c' 2
which in usual circumstances is very close to one (so that T is almost equal
to To). For example an ordinary man moving at, say 90miles/hr (trying to
get his wife to the hospital before she delivers), v/c = 0.0000001 = 10~ 7
(approximately) so that the above factor is essentially one (up to a few hun-
dredths of a trillionth) . This is typical of the magnitude of the new effects
predicted by Einstein's theory for everyday situations: they are in general
very small since the velocities of things are usually very small compared to

There are some instances, however, in which the effects are observable.
There are subatomic particles which are unstable and decay (the process
by which they decay is irrelevant) in a very small time interval when mea-
sured in the laboratory. It has also been found that high intensity radiation
coming from space and hitting the upper atmosphere generates these same
particles (again the process is immaterial). To the initial surprise of the
experimenters, these particles survive the trip down to surface of the earth,
which takes longer, as measured on the Earth, than the particle's lifetime!

The surprise evaporated when it was noted that the particles are moving
very fast with respect to the Earth, almost at the speed of light, so that a
time interval which is very short when measured at rest with respect to the
particle will be much longer when measured in the laboratory.

So the rate of all clocks depends on their state of motion. In this sense The rate of all clocks

depends on their state of
Time is relative. motion.

And while the effect is small in many cases, it is spectacular in others. This
is a surprising consequence of the Principle of Relativity and requires a
complete divorce from Newton's concept of time (which he assumed to flow
evenly under all circumstances, see Sect. ??): time intervals depend on the
motion of the observer, there is no "universal" time.

Time dilation is a prediction of the theory which must not be accepted
as dogma but should be verified experimentally. All experiments do agree
with this prediction. The fact that the theory of relativity makes predictions
which can be tested experimentally, is what makes this an honest theory:
it is falsifiable. It has been accpeted not because of its beauty, but because
these predictions have been verified.

6.2.4 Length contraction

So time is relative, what about distance? In order to think about this note
that when we say that the distance between two objects is £ we imagine
measuring the position of these objects simultaneously... but simultaneity is
relative, so we can expect distance to be a relative concept also.

To see this consider the above subatomic particles. As mentioned they
are moving very fast but we can still imagine Superman (an unbiased ob-
server if there is one) riding along with them. So we have two pictures:
from the observer on earth Superman's clocks (accompanying the particle)
are very slow, and so he/she can understand why it takes so long for the
particle to decay. But for Superman the particle is at rest and so it must
decay in its usual short time. ..the fact remains, however, that the particle
does reach the earth. How can this be? Only if the distance which the
particle traveled as measured in the frame of reference in which it is at rest
is very short. This is the only way the observation that the particle reaches
the earth's surface can be explained: for the observer on the earth this is
because of time dilation, for the observer riding along with the particle, this
is because of length contraction, see Fig. 6.13.

But we do not require peculiar subatomic particles in order to demon-
strate length contraction (though the Principle of Relativity requires that if

Figure 6.13: An observer measures a long life-time for the particles due to
time dilation. The particles measures a short distance between itself and
the observer due to length contraction.

it occurs for the example above it should occur in all systems, otherwise we
could determine by comparison which system has an absolute motion). So
consider the previous experiment with the moving clock (Fig 6.12).

• The observer watching the clock move with velocity v notes that in a
time T the clock moves a distance I = vT.

i The observer riding with the clock notes that the same distance is
covered in a time To; therefore the length measured by him/her is
£o = vTq (He also sees the other observer receding with speed v.)

> Therefore we have £ = vT = vT$/ \J\ — (v/c) 2 = £§f \J\ — (v/c) 2 .
Thus, the observer moving with the clock will measure a shorter length
compared to the one measured by the other observer.

19

It is important to note that these expressions are not to be interpreted
as "illusions", the an observer in motion with respect to a ruler will, when
measuring its length, find a result smaller than the result of an observer at
rest with respect to the ruler. An observer in motion with respect to a clock
will measure a time larger than the ones measured by an observer at rest
with the clock.

The question, "what is 'really' the length of a ruler?" has no answer for
this length depends on the relative velocity of the ruler to the measuring de-
vice 2 . The same as with velocity, specifying lengths requires the framework
provided by a frame of reference.

Length is relative. Length is relative

Note that this peculiar effect occurs only for lengths measured along the
direction of motion and will not occur for lengths perpendicular to it. To
see this imagine two identical trees, we sit at base of one and we observe the
other move at constant speed with respect to us, its direction of motion is
perpendicular to the trunk. In this setup as the roots of both trees coincide
also will their tops, and so in both frames of reference we can simultaneously
determine whether they have the same height; and they do.

This implies that a moving object will be seen thinner (due to length
contraction) but not shorter. Thin fellows will look positively gaunt at
speeds close to that of light.

These conclusions require we also abandon Newton's description of space:
distances are observer-dependent, no longer notches in absolute space.

6.2.5 Paradoxes.

The above conclusions can be very confusing so it might be worthwhile to
discuss the a bit.

Take for example length contraction: the Principle of Relativity implies
that if we measure the length some rod while at rest with respect to it,
and then we measure it when it is moving along its length, the second
measurement yields a smaller value. The crucial point to keep in mind is
the condition that the first measurement is made at rest with respect to the
rod.

2 One can, of course, say that the length of a ruler is the one measured while at rest with
respect to it... but this is only a convention. Once the result of any length measurement
is known (for any relative speed between ruler and measuring device), special relativity
determines unambiguously what any other observer wo

20

Similarly suppose we have two clocks labeled 1 and 2. which are in
perfect agreement when they are at rest with respect to each other. Suppose
now these clocks are endowed with a relative velocity. Then when we look
at clock 2 in the frame of reference in which clock 1 is at rest, clock 2 will
be measured to tick slower compared to clock 1. Similarly, in the frame of
reference in which clock 2 is stationary, clock 1 will run slower compared to
clock 2.

These results can be traced back to the fact that simultaneous events
are not preserved when we go from one reference frame to another.

There are many "paradoxes" which appear to imply that the Principle
of Relativity is wrong. The do not, of course, but it is interesting to see how
the Principle of Relativity defends itself.

1. Consider a man running with a ladder of length £ (measured at rest)
and a barn also of length £ (again, when measured at rest). The barn
has two doors and there are two persons standing at each of them; the
door nearer to the ladder is open the farthest is closed. Now the man
with the ladder runs fast towards the barn while the door persons have
agreed to close the first door and open the second door as soon as the
rear of the ladder goes through the first door.

This is a paradox for the following reason. The ladder guy is in a frame
of reference in which the ladder is at rest but the barn is moving toward
him, hence he will find the length of the barn shortened (shorter than
his ladder), and will conclude that the front of the ladder will hit the
second door before the first door is closed.

The barn people in contrast find the ladder shortened and will conclude
that it will fit comfortably. There will even be a short lapse between
the closing of the first door and the opening of the second, there will
be no crash and the ladder guy will sail through.
So who is right?

The answer can be found by remembering that an even simultaneous
for the barn people (the closing and opening of the doors) will not
be simultaneous for the ladder guy. So, while for the door person the
opening of the rear door and closing of the front occur at the same time,
the ladder guy will see the person at the second door open it before
the person at his rear closes that door and so he will sail through but
only because, he would argue, the door guards were not synchronized.

2. There is an astronaut whose length is 6 ft and he sees a big slab of
metal with which he/she is going to crash. This piece of metal has

a square hole of length 6 ft. (measured at rest with respect to the
slab) . From the point of view of the astronaut the hole is shrunk and
so he will be hit... and die! Prom the point of view of an observer on
the shuttle the plate is falling toward earth and the astronaut moving
at right angles toward it, hence this observer would measure a short
astronaut (5 ft) 3 and conclude that he/she will not be harmed (see
Fig. 6.14). What does really happen?

Figure 6.14: An astronaut's close encounter with a metal plate

The problem is solved in the same way as above. For the astronaut
to be hit a simultaneous coincidence of his head and legs with the two
extremes of the slab's hole should occur. In fact he is not hit. What
is more peculiar is what he sees: he will see the slab tilt in such a way
that he goes through the hole with no problem!

This story illustrates the peculiar look which big objects acquire at
very large speeds. For example, a kettle moving close to the speed of
light with respect to, say, the Mad Hatter will be observed to twist in

rresponds to an astronaut moving at about half the speed of light toward the

i very curious way indeed, see Fig. 6.15.

c%r cfc/ cfcf

Figure 6.15: A relativistic kettle. The top view shows how the three di-
mensional view is distorted due to relativistic effects. The bottom view
shows the corresponding behavior of a flat kettle which exhibits only length
contraction.

Just as for the case of length contraction and time dilation, the effect
on the kettle is not an optical illusion, but any unbiased observer (such
as a photographic camera) would detect the above images precisely as
shown. If the relative velocity between the observer and the kettle
is known, one can use the formulas of special relativity to determine
the shape of the object when at rest with respect to it. ..and we would
obtain the first of the images: a nice kettle

3. Consider two identical twins. One goes to space on a round trip to
Alpha-Centauri (the star nearest to the Sun) traveling at speeds very

Xi

close to c. The round trip takes 10 years as clocked on Earth 4 . As
seen by the twin remaining on Earth all clocks on the ship slow down,
including the biological clocks. Therefore he expects his traveling twin
to age less than 10 years (about 4.5 years for these speeds; the differ-
ence is large since the speed is close to c).

On the other hand the twin in the spacecraft sees his brother (a to-
gether with the rest of the solar system) traveling backwards also at
speeds close to c and he argues that Einstein requires the twin on
Earth to age less than 10 years. Thus each one states that the other
will be younger when they meet again!

The solution lies in the fact that the traveling twin is not always in
an inertial frame of reference: he must decelerate as he reaches Alpha-
Centauri and then accelerate back. Because of this the expressions for
time dilation as measured by the traveling twin will not coincide with
the ones given above (which are true only for observers in different
inertial frames). It is the traveling twin that will be younger.

6.2.6 Space and Time

All events we witness are labeled by a series of numbers, three to tell us
where it happened, and one to determine when it happened. All in all four
numbers are needed. These numbers are determined by some measuring
devices such as measuring rods and clocks.

According to Newton (see Sect. ??) the properties of measuring rods
and clocks can be made completely independent of the system which they
measure (if it does not look like that, you can buy a higher quality device
which will satisfy this criterion). But Einstein showed this is not the case:
even Cartier watches slow down when compared to Seiko watches when
they move with respect to each other. Even high density steel beams will
be measured to be shorter than wimpy papers when their relative velocity
is non-zero.

The measurements obtained by two observers in motion relative to each
other are not identical, but they are related. For example, the times mea-
sured by two clocks are related by the time-dilation formula given earlier.
Suppose observer A measures the location and time at which an event oc-
curs: spider-man ran the 100 yard dash in 3 seconds flat. Now observer B,
moving with respect to A, wants a description of this feat in his own coor-
dinates. In order to find how many yards spider man ran as measured by B

rresponds to a speed of 90% that of light

21

this observer needs to know his velocity with respect to A, the distance spidy
ran as measured by A (100 yds) and how long did he take as measured by
A's clock (3 sec); it is not enough to know the distance and relative velocity,
the time it took is also needed.

The fact that in order to compare results from different observers both
position and time are required is completely foreign to Newtonian mechanics.
Yet this is the way the universe is organized. Far from being independent,
space (that is, position) and time are interlinked. In fact, the mathematical
description of the Special Theory of Relativity is most naturally expressed
by combining space and time into one object: space-time. A point in space-
time determines the position and time of occurrence of an event.

Within Special Relativity space-time is unaltered by whatever is in it.
There are rules that state how the measurements of two observers are related,
but these rules are unaltered by the objects (and beings) that populate
space-time, they are the same whether we look at a pea, an elephant or a
star millions of times more massive than the Sun. Space and time are still
the arena where Nature unfolds.

We will see when we describe the General Theory of Relativity (Chap. ??)
that space-time is far from being this imperturbable object where things
just happen, it is in fact a dynamical system which affects and is affected
by the matter in it. The development of our ideas of space and time from
being independent of each other and imperturbable, to being meshed into
space-time system, to being a dynamical object is one of the most profound
developments derived from the general and special theories of relativity.

6.2.7 The top speed.

In all the above discussion all the effects would go away is the speed of light
were infinite. If there is a top speed, which by definition has and absolute
value (the same for all observers), then all the above effects return. It is
because the equations found by Maxwell involve an absolute speed, and
because they agree with experiment, and because nothing has been found
to travel faster than light in vacuum, and because all the consequences of
the Principle of Relativity are verified again and again with the top speed
equal to c, that we believe this top speed to be precisely c.

Imagine, as Einstein did when a teenager, what would happen is you
could move at the speed of light. As you go by a village (for example) you'd
move at the same speed as all the light coming form that village. So, if
you look around, you would see the same things all the time, nothing would
ever change since you are riding along with a single image: the one carried

by the light from the village at the time you passed it. In your frame of
reference time would stand still! (we will see, however, that it is impossible
for something having mass -such as you- to move at the speed of light. You
can reach speeds very close to c but never reach the speed of light itself).

Imagine now what would happen if, for example, a rat manages to travel
at a speed greater than c. Let's imagine that as the rat travels by you, you
send a short laser light pulse after it. According to you the rat will gain
on the light pulse steadily. Since the distance between the rat's tail and
the front edge of the pulse increases the rat would think that the pulse is
moving in the opposite direction. So you and the rat would disagree even on
the direction along which the light is traveling. This, of course, contradicts
the Principle of Relativity and/or Maxwell's equations and it shows that
the Principle of Relativity together with Maxwells' equations imply that
nothing can move faster than light.

This is a good feature: if a faster-than-light-rat could be found, the
vermin farme of reference would have time flowing backwards. To see this
imagine the rat going by the same village mentioned above. Since the rat
moves faster than light it will steadily gain on the light beams than come
from the village. As it looks around the rat will see the church clock strike
12, and, as it gains on some earlier images, the rat would wee the clock
strike 11, etc. So events whose time orderings wer aboslute would no longer
occur in the correct order in this frame.

6.2.8 Mass and energy.

How could it be that we cannot accelerate something to go faster than light?
Surely we could kick a ball again and again and again until it travels faster
right? No! and the reason is quite interesting.

As something is moving with respect to another object we say that the
moving thing has a certain amount of energy by virtue of its motion. Energy
is the ability to do some work, and, indeed, a moving thing can be lassoed
and made to do some work, like pulling a car (of course in so doing it looses
energy and slows down).

Now, when we have the above object moving, it will have a certain
amount of energy. Einstein argued, the only way we can insure that it cannot
be accelerated indefinitely, is if there is a universal equivalence between mass
and energy. The more energy an object has, the heavier it will be. When
we speed it up a little bit it becomes a bit heavier, and so it also becomes
a bit harder to speed it up further. In fact, the closer we are to the speed
of light, the larger the force is needed to accelerate the object; an infinite

force is needed to speed up a material object to the speed of light: it never
happens!

But there is more to the equivalence of mass and energy, for it also
implies that an object of mass m has energy, just by virtue of its existence;
the specific relationship is

E = mc 2 .
This formula plays a basic role in nuclear reactions (and in atom bombs,
for that matter): in these processes an atomic nucleus of initial mass M
is transformed (either because the environment is tailored to insure this or
because is is unstable and disintegrates spontaneously) into another object
of smaller mass m. The difference in mass is released as energy in the amount
(M -m)c 2 .

To give an idea of how powerful this is, suppose we initially have a sheet
of paper weighing 6gr, and that at the end we have something weighing half
this amount. The energy released is then so big as to turn on a light bulb
of 100W for about 86,000 years, or run a hair-drier for about 4000 years.

The energy released through the transformation of mass is also capable
of destroying a whole planet (or at 'least' all life on it). Einstein was not
aware of this application until much later in his life.

Shin's tricycle. Shin-ichi was a three year old boy who

loved his tricycle. When the bomb was dropped, he was

playing with his best friend, Kimiko. They died. They

were buried in the garden of Shin-ichi's house together. In

July 1985, 40 years later, their parents decided to move

them to a proper grave.

From the story of "Shin's Tricycle" (Translation by Kazuko

Hokumen- Jones and Jacky Copson):

Early in the morning, I began to dig open the grave with

Kimi's mother, who had come to help. After digging for a

while a rusty pipe began to show. "Oh! It's the tricycle!"

Before I realized it I had started to sob. To tell you the

truth, I'd forgotten all about the tricycle.

"Look! There's something white," someone cried. I felt like

ice. Carefully we uncovered the bones using chop-sticks and

brushes. There were a number of tiny bones.

"Shin-ichi, Shin-ichi." "Kimiko." Everyone's eyes were

glued to the little white hands of the two children. They

were still holding hands....

The principle E = mc was used during the Second World War to develop
what is now known as atomic weapons (Fig. 6.16). Shortly thereafter it

was used to develop the hydrogen bomb. Atomic bombs were used during
the Second World War in two Japanese cities, Hiroshima and Nagasaki.
Hundreds of thousands of people died. The creation of nuclear weapons was
one of the watersheds of the 20-th century, and it marks one of the most
dramatic instances in which physics has affected the social structure of the
planet. Yet the very same formulas also suggest the possibility of obtaining
vast amounts of energy which can be used for constructive purposes. It is a
burden of post-second World War physicists to deal with this issue, and to
strive for decent and environmentally safe applications of nuclear power.

Figure 6.16: An atomic explosion.

Chapter 7

The General Theory of
Relativity

The General Theory of Relativity is, as the name indicates, a generalization
of the Special Theory of Relativity. It is certainly one of the most remarkable
achievements of science to date, it was developed by Einstein with little or
no experimental motivation but driven instead by philosophical questions:
Why are inertial frames of reference so special? Why is it we do not feel
gravity's pull when we are freely falling? Why should absolute velocities be
forbidden but absolute accelerations by accepted?

Figure 7.1: Einstein

7.1 The happiest thought of my life.

In 1907, only two years after the publication of his Special Theory of Rel-
ativity, Einstein wrote a paper attempting to modify Newton's theory of
gravitation to fit special relativity. Was this modification necessary? Most
emphatically yes! The reason lies at the heart of the Special Theory of Rel-
ativity: Newton's expression for the gravitational force between two objects
depends on the masses and on the distance separating the bodies, but makes
no mention of time at all. In this view of the world if one mass is moved,
the other perceives the change (as a decrease or increase of the gravitational
force) instantaneously. If exactly true this would be a physical effect which
travels faster than light (in fact, at infinite speed), and would be inconsis-
tent with the Special Theory of Relativity (see Sect. ??). The only way
out of this problem is by concluding that Newton's gravitational equations
are not strictly correct. As in previous occasions this does not imply that
they are "wrong", it only means that they are not accurate under certain
circumstances: situations where large velocities (and, as we will see, large
masses) are involved cannot be described accurately by these equations.

In 1920 Einstein commented that a thought came into his mind when
writing the above-mentioned paper he called it "the happiest thought of my
life":

The gravitational field has only a relative existence... Because for
an observer freely falling from the roof of a house - at least
in his immediate surroundings - there exists no gravitational
field.

Let's imagine the unfortunate Wile E. Coyote falling from an immense
height 1 . As he starts falling he lets go of the bomb he was about to drop
on the Road Runner way below. The bomb does not gain on Wile nor does

* constajrt /v sp^edJn^^^ This realization is important because

this is exactly what an astronaut would experience in outer space, far away
from all bodies (we have good evidence for this: the Apollo 10-13 spacecrafts
did travel far from Earth into regions where the gravitational forces are quite
weak) .

Mr. Coyote is fated to repeat the experience with many other things:
rocks, magnets, harpoons, anvils, etc. In all cases the same results are
obtained: with respect to him all objects, irrespective of composition, mass,

1 I ignore air rcsisfaino

etc. behave as if in free space. So^jfjiejphould fall inside^cbsedi^cjie

This is reminiscent of Galileo's argument: the observer lets go of some
objects which remain in a state of uniform motion (with respect to him!).
This behavior is independent of their chemical or physical nature (as above,
air resistance is ignored). The observer (Wile), as long as he confines his/her
observations to his/her immediate vicinity (that is, as long as he/she does
not look down) has the right to interpret his state as 'at rest'. Just as
Galileo argued that experiments in a closed box cannot detcrminejbjejstate
of uniform motion of the box. Einstein argued that experiments, injijrjeely

tfee^grip^a^rayitatip^^

Why would this be true? The answeTcan be traced back to the way in
which gravity affects bodies. Remember (see Sect. ??) that the quantity we
called m (the mass) played two different roles in Newton's equations. One
is to determine, given a force, what the acceleration of the body would be:
F = ma (the inertial mass). The other is to determine the intensity with
which the said body experiences a gravitational force: F = mMG/r 2 (the
gravitational mass). As mentioned before these two quantities need not be
equal: the first "job" of m is to tell a body how much to accelerate given
any force, a kick, an electric force (should the body be charged), etc. The
second "job" tells the body how much of the gravitational force should it
experience and also determines how strong a gravitational force it generates.
But, in fact, both numbers are equal (to a precision of ten parts per billion).

What does this imply? Well, from Newton's equations we get

mMG MG

— 2 — = ma so tri at — ^ = G ;

this equation determines how a body moves, which trajectory it follows,
how long does it take to move from one position to another, etc. and is
independent of ml Two bodies of different masses, composition, origin and
guise will follow the same trajectory: beans, bats and boulders will move in
the same way.

So the equality of the two m's was upgraded by Einstein to a postulate:
the Principle of Equivalence; this one statement (that the m in ma and
the m in mMG/r 2 are identical) implies an incredible amount of new and

2 The reasons behind the requirement that the box be small will become clear soon.

The m \n F = ma is
called the inertial mass
and the m in mMG/r 2
the gravitational mass
The inertial and

surprising effects. The m in F = ma isjgalfe djhe inertiajma^ andjhg m in
mMG/llthg graxj^ionai m^gg. Then the principle of Equivalence] states
that the inertial and gravitational masses are identical.

and will fail if one can find a material for which the inertial and gravita-
tional masses have different values. One might think that this represents a
defect of the theory, its Achilles heel. In one sense this is true since a single
experiment has the potential of demolishing the whole of the theory (people
have tried... hard, but all experiments have validated the principle of equiv-
alence) . On the other hand one can argue that a theory which is based on a
minimum of postulates is a better theory (since there are less assumptions
involved in its construction) ; from this point of view the General Theory of
Relativity is a gem 3 .

The completed formulation of the General Theory of Relativity was pub-
lished in 1916 (Fig. 7.2).

Figure 7.2: Einstein's General Theory of Relativity paper.

3 The Special Theory of Relativity is equally n
inertial frames of reference are equivalent.

e, it is based on the one statement that

7.1.1 Newton vs. Einstein

I have stated that Newton's mechanics and his theory of gravitation are but
approximations to reality and whose limitations are now known 4 . So it
might be questionable to use F = ma and -F gra v = mMG/r 2 as basis to any
argument as was done above. Einstein was careful to use these expressions
only in situations where they are extremely accurate (small speeds com-
pared to c and small gravitational forces). In these cases the inertial and
gravitational masses are identical, as shown by experiment. Then he postu-
lated that the same would be true under all circumstances. This statement,
while consistent with Newton's equations, cannot, in a strict logical sense,
be derived from them.

7.2 Gravitation vs acceleration

Consider the following experiment: a person is put in a room-size box high
above the moon (chosen because there is no air and hence no air friction)
with a bunch of measuring devices. This box is then taken high above the
lunar surface and then let go: the box is then freely falling. The question
is now, can the observer determine whether he/she is falling or whether
he/she is in empty space unaffected by external forces (of course the answer
is supposed to come before the box hits the surface). The answer to that is
a definite NO! The observer can do experiments by looking at how objects
move when initially at rest and when given a kick, he/she will find that they
appear to move as is there were no gravitational forces at all! Similarly any
experiment in physics, biology, etc. done solely inside the box will be unable
to determine whether the box is freely falling or in empty space.

Why is that? Because of the equality of the gravitational and inertial
masses. All objects are falling together and are assumed to be rather close
to each other (the box is not immense) hence the paths they will follow will
be essentially the same for each of them. So if the observer lets go of an
apple, the apple and the observer follow essentially the same trajectory, and
this implies that the observer will not see the apple move with respect to
him. In fact, if we accept the priniciple of equivalence, nothing can be done
to determine the fact that the observer is falling towards the Moon, for this
can be done only if we could find some object which behaved differently from
all the rest, and this can happen only if its gravitational and inertial masses

4 For all we know our present theories of mechanics and gravitation may also be invalid
under certain conditions.

are different. The principle of equivalence then implies that the observer
will believe that he/she is an inertial frame of reference... until disabused of
the notion by the crash with the surface.

The principle of equivalence is of interest neither because its simplicity,
nor because it leads to philosophically satisfying concisions. It's importance
is based on the enormous experimental evidence which confirms it; as with
the Special Theory, the General Theory of Relativity is falsifiable.

Figure 7.3: An observer cannot distinguish between acceleration produced
by a rocket and the acceleration produced by gravity.

For any gravitational force

frame of reference in which
an observer will not
experience any gravitational
effects in his/her immediate
vicinity

The lesson is that for any gravitational force we can always choose a
frame of reference in which an observer will not experience any gravitational
effects in his/her immediate vicinity (the reason for this last qualification
will become clear below). Such a frame of reference is, as stated above,

freely falling.

Conversely one can take the box an attach it to a machine that accel-
erates it (Fig. 7.3). If an observer drops an apple in such an accelerated
box he/she will see the apple drop to the floor, the observer will also feel
hi/her-self pressed against the bottom of the box, etc. The observer can-
not distinguish between this situation and the one he/she would experience
in the presence of gravitational forces! As long as we do experiments in a
=wahjxgion., the effects produced by a gravitational force are indistinguish-
~ able from those present in an accelerated reference frame. In a small region the effects

Does this mean that the gravitational forces are a chimera, an illusion? P roduced b v a gravitational
° ' force are indistinguishable

Of course no. Consider for example Fig. 7.4, two apples fall to the Moon f rom those present in an
inside a box which is also falling. If they are separated by a sufficiently accelerated reference frame
large distance an observer falling with the apples and box will find that the
distance between the apples shortens as time goes on: this cannot be an
inertial frame he argues (or else it is, but there is some force acting on the
apples).

This same set-up can be used to distinguish between a box under the
influence of a gravitational force and one being pulled by a machine; again
we need a very big box (planet-sized). An observer places an two apples at
the top of the box and releases them, he/she carefully measures its initial
separation. The apples fall to the bottom of the box and the observer
measures their separation there. If it is the same as above, and is the same
irrespective of their initial separation, the observer is being pulled by a
machine (box and all). If the separation is different, he/she can conclude
that he/she is experiencing the effects of a gravitational force.

7.3 Light

A^vgry^smjirising^^^ is that light paths are bent by

gravitational forces! I will argue this is true in a slightly round-about way.

Consider an elevator being pulled by a crane so that it moves with con-
stant acceleration (that is its velocity increases uniformly with time). Sup-
pose that a laser beam propagating perpendicular to the elevator's direction
of motion enters the elevator through a hole on the left wall and strikes the
right wall. The idea is to compare what the crane operator and the elevator
passenger see.

The crane operator, who is in an inertial frame of reference, will see the
sequence of events given in Fig. 7.5. Note, that according to him/her, light
travels in a straight line (as it must be since he/she is in an inertial frame!).

Figure 7.4: Experiment that differentiates between a gravitational effect and
the effects of uniform acceleration: for an observer in the box the apples will
draw closer.

The elevator passenger will see something very different as shown in Fig.
7.5: the light-path is curved! Thus for this simple thought experiment light
paths will be curved for observers inside the elevator.

Now we apply the equivalence principle which implies that we cannot
distinguish between an elevator accelerated by a machine and an elevator
experiencing a constant gravitational force. It follows that the same effect
should be observed if we place the elevator in the presence of a gravitational
force: light paths are curved by gravity

That gravity affects the paths of planets, satellites, etc. is not something
strange. But we tend to think of light as being different somehow. The above
argument shows that light is not so different from other things and is indeed
affected by gravity in a very mundane manner (the same elevator experiment
could be done by looking at a ball instead of a beam of light and the same

i
*

00 eoo

View by an inertial observer

View from inside the elevator

Figure 7.5: Left: sequence of events seen by an crane operator lifting an
elevator at constant acceleration (the speed increases uniformly with time).
The short horizontal line indicates a laser pulse which, at the initial time,
enters through an opening on the left-hand side of the elevator. At the
final time the light beam hits the back wall of the elevator. Right: same
sequence of events seen by a passenger in an elevator being hoisted by a
crane. The line joining the dots indicates the path of a laser which, at the
initial time, enters through an opening on the left-hand side of the elevator.
At the final time the light beam hits the back wall of the elevator.

sort of picture would result).
gj- A^iaturaiimestkm^

dManceJghtiaTl^^

Idiejiccelexatkm^
ofjg^iyJty^on^Fjai^
Earjtli^wniwejg^

10

This does not mean, however, that this effect is completely unobservable
(it is small for the case of the elevator because elevators are designed for
very small accelerations, but one can imagine other situations). Consider
from example a beam of light coming from a distant star towards Earth
(Fig. 7.6) which along the way comes close to a very massive dark object.
The arguments above require the light beam to bend; and the same thing
will happen for any other beam originating in the distant star. Suppose that
the star and the opaque object are both prefect spheres, then an astronomer
on Earth will see, not the original star, but a ring of stars (often called an
"Einstein ring). If either the star or the massive dark object are not perfect
spheres then an astronomer would see several images instead of a ring (Fig.
7.7). This effect has been christened gravitational lensing since gravity acts
here as a lens making light beams converge.

Figure 7.6: Diagram illustrating the bending of light from a star by a massive
compact object. If both the bright objects and the massive object are prefect
sphere, there will be an apparent image for every point on the "Einstein
ring" .

How do we know that the multiple images which are sometimes seen (Fig.
7.7) are a result of the bending of light? Tlie^J^ument is by contradiction:

EBBg^aa

Figure 7.7: The Einstein Cross: four images of a quasar GR2237+0305 (a
very distant, very bright object) appear around the central glow. The split-
ting of the central image is due to the gravitational lensing effect produced
by a nearby galaxy. The central image is visible because the galaxy does
not lie on a straight line from the quasar to Earth. The Einstein Cross is
only visible from the southern hemisphere.

suppose they are not, that is suppose, that the images we see correspond to
different stars. Using standard astronomical tools one can estimate the dis-
tance between these stars; it is found that they are separated by thousands
of light years, yet it is observed that if one of the stars change, all the others
exhibit the same change instantaneously! Being so far apart precludes the
possibility of communication between them; the simplest explanation is the
one provided by the bending of light. It is, of course, possible to ascribe
these correlations as results of coincidences, but, since these correlations
are observed in many images, one would have to invoke a "coincidence" for
hundreds of observations in different parts of the universe.

E^J The bending of light was one of the most dramatic predictions of the
General Theory of Relativity, it was one of the first predictions that were
verified as we will discuss below in Sect. 7.12.

7.4 Clocks in a gravitational force.

When comparing a clock under the influence of gravitational forces with one
very far from such influences it is found that the first clock is slow compared
to the second. To see this consider the same clock we used in the Special
Theory of Relativity. For this experiment, however, imagine that the clock
is being accelerated upward, being pulled by a crane. The clock gives off a
short light pulse which moves towards the mirror at the top of the box, at
the same time the mirror recedes from the pulse with even increasing speed
(since the box accelerates). Still the pulse eventually gets to the mirror
where it is reflected, now it travels downward where the floor of the box is
moving up also with ever increasing velocity (see Fig. 7.8).

W
•^ HI.-*

n n n nflfl

t , Mr''' I I

M''" J I — I

J)®©®®®©®®

Figure 7.8: An accelerated clock. The circle denotes a pulse of light which
at the initial is sent from a source; after a time it reaches the top of the the
box and is reflected. The time it takes to do the trip is longer than for a
clock at rest.

QiUfeUijmjmJJieJ^^

olJ&eJjoxjatJ^^^

Since light always travels at the same speed, it follows that the time it
takes for the pulse to go the round trip is longer when accelerating than
when at rest: clocks slow down whenever gravitational forces are present.

This has an amazing consequence: imagine a laser on the surface of
a very massive and compact planet (so that the gravitational field is very
strong) . An experimenter on the planet times the interval between two crests
of the laser light waves and gets, say, a millionth of a second. His clock ,
however, is slow with respect to the clock of an observer far away in deep
space, this observer will find that the time between two crests is larger. This
implies that the frequency of the laser is larger on the planet than in deep
space: light leaving a region where gravity is strong reddens. This is called Light leaving a region where
the \gravitational red-shilt\ (see Fig. 7.9).

gravity is strong reddens

Figure 7.9: The gravitational redshidft. Since clocks slow down in a strong
gravitational field then light, whose oscillations can be used as clocks, will
be shifter towards the red as it leaves a region where gravity is strong.

ajSgfojUXte£ray]ty^^

bxJjhnejiilatixtfi^dj;^^
rec|uired^Joi^a^izal^^

7.5 Black holes

So gravity pulls on light just as on rocks. We also know that we can put
rocks in orbit, can we put light in orbit? Yes! but we need a very heavy
object whose radius is very small, for example, wem^djgometlih^^

object^Jj,ghtj3ioy|n^Jx3war^

SnojJgMan^ni^^j^^

a£Utorj3iJ^^^

But we.jjan v goJarjaie^^
ttMJtwgJiUrnimj^

b^jckj^w^sj^he^^ Think what this means: since no light can leave

this object it will appear perfectly black, this is a black hole. Anjjbject

(since nothing moves faster than light if an object traps light it will also
trap everything else).

The effect of a black holes, like all gravitational effects, decreases with
distance. This means that there will be a ^ujjdaocl^^
hole^NSUcliJdiatsj^

jdjejsljicklicdexj^h^ black-hole horizon see Fig. 7.10

Anything crossing the horizon is permanently trapped. Black holes are
prefect roach motels: once you check in (by crossing the horizon), you never
check out.

, , The distance from the black hole to the horizon is determined by the

^Srdss of the black hole: thej^ujjexthe^massj^^^
frornjdie^center. Forjajaladkliole^^^

is^boiiO^kmji^^ for black holes with a billion solar masses

(yes there are such things) this is increased to 3 x 10 9 km, about the distance
from the sun to Uranus. For very massive black holes the horizon is so far
away from the center that an observer crossing it might not realize what has
just happened, only later, when all efforts to leave the area prove futile, the
dreadful realization of what happened will set in.

Imagine a brave (dumb?) astronaut who decides to through the horizon

Figure 7.10: Illustration of the horizon surrounding the black hole. The
black holes is represented by the small heavy dot, the light rays or particle
trajectories which cross the dotted line cannot cross it again.

and into the nearest black hole and let us follow his observations. The first
effects that becomes noticeable as he approaches the event horizon is that
his clock ticks slower and slower with respect to the clocks on his spaceship
very far from the black hole (see Sect. 7.4) to the point that it will take
infinite spaceship time for him to cross the horizon. In contrast it will take
a finite amount of astronaut time to cross the horizon, an extreme case of
the relativity of time.

As the astronaut approaches the horizon the light he emits will be more
and more shifted towards the red (see Sect. 7.4) eventually reaching the
infrared, then microwaves, then radio, etc. In order to see him the spaceship
will eventually have to detect first infrared light, then radio waves, then
microwaves, etc.

After crossing the horizon the astronaut stays inside. Even though the
of the horizon might not be a traumatic experience the same cannot
d for his ultimate fate. Suppose he decides to fall feet first, then, when
.entry close to the black hole, the gravitational pull on his feet will be
much larger than that on his head and he will be literally ripped to pieces.

So far black holes appear an unfalsifiable conclusion of the General The-

ory of Relativity: their properties are such that no radiation comes out of
them so they cannot be detected from a distance, and if you should decide
to go, you cannot come back to tell your pals whether it really was a black
hole or whether you died in a freak accident. Doesn't this contradict the
basic requirement that a scientific theory be falsifiable (Sect. ??)?

Figure 7.11: Artist's version of a black hole accreting matter from a compan-
ion star. The Star is on the left of the picture and is significantly deformed
by the gravitational pull of the black hole; the object on the right represents
the matter which surrounds the black hole and which is being sucked into
it. The black hole is too small to be seen on the scale of this picture

Well, no, General Theory of Relativity even in this one of its most ex-
treme predictions is falsifiable. The saving circumstance is provided by
the matter surrounding the black hole. All such stuff is continuously being
dragged into the hole (see Fig. 7.11) and devoured, but in the process it gets
extremely hot and radiates light, ultraviolet radiation and X rays. More-
over, this cosmic Maelstrom is so chaotic that the radiation changes very
rapidly, sometimes very intense, sometimes much weaker, and these changes
come very rapidly (see Fig. 7.12). From this changes one can estimate the
size of the object generating the radiation.

On the other hand astronomers can see the gravitational effects on near-
by stars of whatever is making the radiation. And from these effects they

Figure 7.12: X-ray emission from a black hole candidate (Cygnus XI)

can estimate the mass of the beast. Knowing then the size, the manner in I
which matter radiates when it comes near, and the mass one can compare
this to the predictions of General Theory of Relativity and decide whether
this is a black hole or not. The best candidate for a black hole found in this
way is called Cygnus XI (the first observed X ray source in the constellation
Cygnus, the swan).

All the ways we have of detecting black holes depend on the manner in
which they affect the matter surrounding them. The most striking example
is provided by some observation of very distant X-ray sources which are
known to be relatively compact (galaxy size) and very far away. Then the
very fact that we can see them implies that they are extremely bright objects,
so bright that we know of only one source that can fuel them: the radiation
given off by matter while being swallowed by a black hole 5 . So the picture
we have of these objects, generically called active galactic nuclei, is that of
a supermassive (a billion solar masses or so) black hole assimilating many
stars per second, and in disappearing these stars give off the energy that
announces their demise.

All this from the (apparently) innocent principle of equivalence.

7.6 Gravitation and energy

Consider a beam of sunlight falling on your skin; after a while your skin
warms and, eventually, will burn: light carries energy (which is absorbed
by your skin thus increasing its temperature). Recall also that a body with

5 This is much more efficient than nuclear power which would be incapable of driving

18

mass m, by its very existence, carries and energy mc 2 (Sec. ??). There is
no way, however, in which we can associate a mass with light; for example,
we can always change the speed of a mass (even if only a little bit) , but this
cannot be done with light.

The force of gravity affects both light and all material bodies; since
both carry energy, but only the bodies carry mass, it follows that gravity
will affect anything carrying energy. This conclusion lies at the root of the
construction of Einstein's equations which describe gravity.

Note that this conclusion has some rather strange consequences. Con-
sider for example a satellite in orbit around the Earth, when the Sun shines
on it it will increase its energy (it warms up), and gravity's pull with it.
When the satellite is in darkness it will radiate heat, lose energy and the
force of gravity on it will decrease 6 .

Again let me emphasize that this argument is not intended to imply that
light carries mass, but that gravity will affect anything that carries energy.

7.7 Space and time.

When considering the Special Theory of Relativity we concluded that the
state of motion of an observer with respect to, say, a laboratory, determines
the rate at which his/her clocks tick with respect to the lab's clocks (see
Sect.??). Thus, in this sense, time and space mingle: the position of the
observer (with respect to the lab's measuring devices) determines, as time
evolves, his/her state of motion, and this in turn determines the rate at
which his/her clocks tick with respect to the lab's.

Now consider what happens to objects moving under the influence of a
gravitational force: if initially the objects set out at the same spot with the
same speed they will follow the same path (as required by the principle of
equivalence). So what!? To see what conclusions can be obtain let me draw
a parallel, using another murder mystery.

Suppose there is a closed room and a line of people waiting to go in. The
first person goes in and precisely two minutes afterward, is expelled through
a back door, dead; it is determined that he died of a blow to the head. The
police concedes that the room is worth investigating, but procrastinates,
alleging that the person was probably careless and his death was accidental.
Soon after, however, a second victim enters the room with precisely the
same results, she also dies of an identical blow to the head; the police claims
an astounding coincidence: two accidental deaths. This goes on for many

6 Needless to say this is a very small effect, of the order of one part in a trillio

i<)

hours, each time the victim dies of the same thing irrespective of his/her
age, occupation, habits, color, political persuasion or taste in Pepsi vs. Coke;
animals suffer the same fate, being insects of whales. If a rock is sent flying
in, it comes out with a dent of the same characteristics as the ones suffered
by the people and animals.

The police finally shrewdly concludes that there is something in the room
that is killing people, they go in and... But the result is not important, what
is important for this course is the following. We have a room containing
something which inflicts a certain kind of blow to everything going through
the room, I can then say that this inflicting of blows is a property of the
room.

Consider now a region of empty space relatively near some stars. Assume
that the only force felt in this region is the gravitational pull of these stars,
hence all objects, people, animals, etc. going into this region will accelerate
in precisely the same way. Then I can state that the region in space has a
property which generates this acceleration'-^

Remember however that the region considered was in empty space (it
only contains the objects we send into it), yet some property of this re-
gion determines the motion of anything that goes through it; moreover this
property is a result of the gravitational pull of nearby heavy objects. The
conclusion is then that gravity alters the properties of space, we also saw
that the rates of clocks are altered under the influence of a gravitational
force, it follows that gravity alters the properties of space and time. Space
and time is in fact very far from the unchanging arena envisaged by New- '
ton, they are dynamical objects whose properties are affected by matter and
energy. These changes or deformations of space and time in turn determine
the subsequent motion of the bodies in space time: matter tells space-time
how to curve and space-time tells matter how to move (Fig. 7.13).

Figure 7.13: An illustration of the bending of space produced by a massive
object

I assume that the objects coming into this region are not too heavy, so that their
gravitational forces can be ignored and that the start from the same spot with identical
velocities.

7.8 Properties of space and time.

Up to here I've talked little of the implications of the Special Theory of
Relativity on the General Theory of Relativity, I have only argued that in
special relativity time and space are interconnected. In a separate discussion
I argued that gravity alters space. In this section I will use what we know
about k^ngtlijcontractig^^
mineJjsw^passUg^a^
ofjipaceJMisj^s^o^^
04. Consider two identical disks one of which is made to rotate uniformly
as in Fig. 7.14. In the non- rotating disk we select a small segment of its
circumference of length £ . For the rotating disk this same segment will
be measured to have length £ which is smaller, due to length contraction
(Sec. ??), than £ . Since the little segment we focused on is no different
from any other (small) segment of the circumference, we can conclude that
the circumference of the rotating disk is smaller than the circumference of
the non-rotating disk.

Rotating disk

Non-rotating disk

Figure 7.14: A rotating vs a non-rotating disk. The bit labeled £ in the
rotating disk is shorter, due to length contraction to the corresponding bit
£ in the non-rotating disk.

Consider now a radius of the disks. This is a length that is always
perpendicular to the velocity of the disk and it is unaffected by the rotation,
thus both disks will continue to have the same radius (see Sect. ??).

So now we have one non-rotating disk whose circumference is related to
the radius by the usual formula, circumference = 2irx radius, and a rotating
disk whose circumference is smaller than this number!

How can this be?

21

jdiejspliere^JTlien^j^^
§I™aWeUtoJlIx
see Fig. 7.15). ~

radius as
measured on
the sphere

ci rcumference
as measured on
the sphere

Figure 7.15: The distance from the equator to the pole on a sphere is larger
than the radius. For being constrained to move on the surface of the sphere
this distance is what they would call the radius of their universe, thus for
them the circumference is smaller than 2nx radius and they can conclude
that they live in curved space.

We conclude that the uniformly rotating disk behaves as a (piece of a)
sphere due to length contraction. So much for the effecteols^eckil^^

Now let us go back to the pnncjpie^f^ojyiy^^ One of its conse-
quences is that, by doing experiments in a small region one cannot distin-
C J guish between a gravitational force and an accelerated system. So if we
attach a small laboratory of length £q (at rest) to the small section of the
perimeter, experiments done there will not be able to tell whether the lab.
is in a rotating disk or experiences a gravitational force (remember that a
rotating object is changing its velocity - in direction - and it is therefore
accelerating!).

Putting together the above two arguments we get

Gravitation curves space and time.

c

Conversely curved space and time generate effects which are equivalent
to gravitational effects. In order to visualize this imagine a world where
all things can only move on the surface of a sphere. Consider two beings
labeled A and B as in Fig. 7.16, which are fated live on the surface of this
sphere. On a bright morning they both start from the equator moving in a
direction perpendicular to it (that is, they don't meander about but follow
a line perpendicular to the equator).

Figure 7.16: Two beings moving on a sphere are bound to come closer just
as they would under the effects of gravity

As time goes on the two beings will come closer and closer. This effect
is similar to the experiment done with two apples falling towards the moon
(Fig. 7.4): an observer falling with them will find their distance decreases
as time progresses; sentient apples would find that they come closer as time
goes on.
f^="|So we have two descriptions of the same effect: on the one hand grav-
itational forces make the apples approach each other; on the other hand
the fact that a sphere is curved makes the two beings approach each other;
mathematically both effects are, in fact, identical. In view of this the conclu-
sion that gravity curves space might not be so peculiar after all; moreover,
in this picture the equivalence principle is very natural: bodies move the
way they do due to the way in which space is curved and so the motion is

independent of their characteristics , in particular the mass of the body Bodies move the way they
does not affect its motion. do due to the wa * in which

space is curved and so the
motion is independent of
their characteristics

Figure 7.17: Just as bugs fated to live on the surface of a sphere might find
it peculiar to learn their world is curved, so we might find it hard to realize
that our space is also curved.

Now the big step is to accept that the same thing that happened to the
above beings is happening to us all the time. So how come we don't see that
the space around us is really curved? The answer is gotten by going back
to the beings A and B: they cannot "look out" away from the sphere where
they live, they have no perception of the perpendicular dimension to this
sphere, and so they cannot "see it from outside" and realize it is curved.
The same thing happens to us, we are inside space, in order to see it curved
we would have to imagine our space in a larger space of more dimensions

8 I am assuming here that the moving things are not massive enough to noticeably curve
space on their own.

and then we could see that space is curved; Fig. 7.17 gives a cartoon version
of this.

7.9 Curvature

When considering the beings living on a sphere it is easy for us to differen-
tiate between the sphere and some plane surface: we actually see the sphere
being curved. But when it comes to us, and our curved space, we cannot see
it since this would entail our standing outside space and looking down on
it. Can we then determine whether space is curved by doing measurements
inside it?

To see that this can be done let's go back to the beings on the sphere.
Suppose they make a triangle by the following procedure: they go form the
equator to the north pole along a great circle (or meridian) of the sphere, at
the north pole they turn 90° to the right and go down another great circle
until they get to the equator, then they make another 90° turn to the right
until they get to the starting point (see Fig. 7.18). They find that all three
lines make 90° angles with each other, so that the sum of the angles of this
triangle is 270°, knowing that angles in all flat triangles always add up to
180° they conclude that the world they live on is not a flat one. Pythagoras'
theorem only holds on flat surfaces

We can do the same thing: by measuring very carefully angles and dis-
tances we can determine whether a certain region of space is curved or not.
In general the curvature is very slight and so the distances we need to cover
to observe it are quite impractical (several light years), still there are some
special cases where the curvature of space is observed: if space were flat
light would travel in straight lines, but we observe that light does no such
thing in regions where the gravitational forces are large; I will discuss this
further when we get to the tests of the General Theory of Relativity in the
following sections.

The curvature of space is real and is generated by the mass of the bodies

aJIJsodjgsjmoyin^

feq^imer&sfjdi^ Their solutions predict, given the initial positions and
velocities of all bodies, their future relative positions and velocities. In the
limit where the energies are not too large and when the velocities are signif-
icantly below c the predictions of Einstein's equations are indistinguishable
from those obtained using Newton's theory. At large speeds and/or energies
significant deviations occur, and Einstein's theory, not Newton's, describes

Figure 7.18: A path followed by a determined being living on the surface of
a sphere; each turn is at right angles to the previous direction, the sum of
the angles in this triangle is then 270° indicating that the surface in which
the bug lives is not flat.

the observations.

7.10 Waves

A classical way of picturing the manner in which heavy bodies curve space
is to imagine a rubber sheet. When a small metal ball is made to roll on
it it will go in a straight line at constant speed (neglecting friction). Now
imagine that a heavy metal ball is placed in the middle of the sheet; because
of its weight the sheet will be depressed in the middle (Fig 7.13). When a
small ball is set rolling it will no longer follow a straight line, its path will
be curved and, in fact, it will tend to circle the depression made by the
heavy ball. The small ball can even be made to orbit the heavy one (it will
eventually spiral in and hit the heavy ball, but that is due to friction, if
the sheet is well oiled it takes a long time for it to happen). This toy then
realizes what was said above: a heavy mass distorts space (just as the heavy
ball distorts the rubber sheet). Any body moving through space experiences
this distortion and reacts accordingly.

Now imagine what happens if we drop a ball in the middle of the sheet.
It will send out ripples which spread out and gradually decrease in strength.
Could something similar happen in real life? The answer is yes! When there
is a rapid change in a system of heavy bodies a large amount of gravitational
waves are produced. Thesemiy<£j^^^
thehj^un^att^^

A computer simulation of a gravitational wave is given in Fig. 7.19. The
big troughs denote regions where the wave is very intense, the black dot
at the center denotes a black hole, the ring around the hole represents the
black hole's horizon.

Figure 7.19: A computer simulation of a gravitational wave generated by a
collision of two black holes, which have now merged and are represented by
the heavy black dot in the middle.

Can we see gravitational waves? Not yet directly, but we have very strong
indirect evidence of their effects. Several systems which according to the
General Theory of Relativity ought to lose energy by giving off gravitational
waves have been observed. The observations show that these systems lose
energy, and the rate at which this happens coincides precisely with the
predictions from the theory. ( = 1

Observing gravitational waves directly requires very precise experiments.
The reason is that, as one gets farther and farther away from the source these
waves decrease in strength very rapidly. Still, if a relatively strong gravita-
tional wave were to go by, say, a metal rod, its shape would be deformed
by being stretched and lengthened periodically for a certain time. By ac-
curately measuring the length of rods we can hope to detect these changes.

The technical problems, however, are enormous: the expected variation is
of a fraction of the size of an atom! Nonetheless experiments are under way.
Gravitational waves are generated appreciably only in the most violent
of cosmic events. During the last stages in the life of a star heavier than 3
solar masses, most of the stellar material collapses violently and inexorably
to form a black hole (n the rubber sheet picture this corresponds to drop-
ping a very small and very heavy object on the sheet). The corresponding
deformation of space travels forth from this site site as a gravitational wave.
High intensity gravitational waves are also produced during the collision of
two black holes or any sufficiently massive compact objects.

7.11 Summary.

The conclusions to be drawn from all these arguments are,

• All frames of reference are equivalent, pnjvidjgd^e^are^^^
chMej^ossiblegra^^

• Space-time is a dynamic object: matter curves it, and the way in which
it is curved determines the motion of matter in it. Since all bodies are
affected in the same way by the curvature of space and time the effects
of gravity are independent of the nature of the body. Changes in the
distribution of matter change space- time deforming it, and, in some
instances, making it oscillate.

7.12 Tests of general relativity.

After Einstein first published the General Theory of Relativity there was a
very strong drive to test its consequences; Einstein himself used his equa-
tions to explained a tiny discrepancy in the motion of Mercury. Yet he most
dramatic effect was the shifting of the positions of the stars (see below).
Since 1916 there have been many measurements which agree with the Gen-
eral Theory of Relativity to the available accuracy. Here I will concentrate
on the "classical" tests of the thoery.

7.12.1 Precession of the perihelion of Mercury

A long-standing problem in the study of the Solar System was that the orbit
of Mercury did not behave as required by Newton's equations.

To understand what the problem is let me describe the way Mercury's
orbit looks. As it orbits the Sun, this planet follows an ellipse. ..but only ap-
proximately: it is found that the point of closest approach of Mercury to the
sun does not always occur at the same place but that it slowly moves around
the sun (see Fig. 7.20). This rotation of the orbit is called a precession.

The precession of the orbit is not peculiar to Mercury, all the plane-
tary orbits precess. In fact, Newton's theory predicts these effects, as be-
ing produced by the pull of the planets on one another. The question is
whether Newton's predictions agree with the amount an orbit precesses; it
is not enough to understand qualitatively what is the origin of an effect,
such arguments must be backed by hard numbers to give them credence.
Tlie^recjssJOT^

b^j^n^g^p^dj^ But Mercury seemed to be an

exception.

Figure 7.20: Artist's version of the precession of Mercury's orbit. Most of
the effect is due to the pull from the other planets but there is a measurable
effect due to the corrections to Newton's theory predicted by the General
Theory of Relativity.

As seen from Earth the precession of Mercury's orbit is measured to
be 5600 seconds of arc per century (one second of arc= 1/3600 degrees).
Newton's equations, taking into account all the effects from the other planets
(as well as a very slight deformation of the sun due to its rotation) and the
fact that the Earth is not an inertial frame of reference, predicts a precession
of 5557 seconds of arc per century. There is a discrepancy of 43 seconds of

arc per century.

This discrepancy cannot be accounted for using Newton's formalism.
Many ad-hoc fixes were devised (such as assuming there was a certain
amount of dust between the Sun and Mercury) but none were consistent
with other observations (for example, no evidence of dust was found when
the region between Mercury and the Sun was carefully scrutinized). In con-
trast, Einstein was able to predict, without any adjustments whatsoever,
that the orbit of Mercury should precess by an extra 43 seconds of arc per
century should the General Theory of Relativity be correct.

7.12.2 Gravitational red-shift.

We saw in Sec. 7.4 that light leaving a region where the gravitational force
is large will be shifted towards the red (its wavelength increases; see Figs.
7.21,7.9); similarly, light falling into a region where the gravitational pull
is larger will be shifted towards the blue. This predjctioji^was^

enormous precision since the changes in the gravitational force from the
top to the bottom of a tower are minute) and the results agree with the
predictions from the General Theory of Relativity.

The gravitational red-shift was also tested by lookmgji^jhejk^h^^
a^yp>ej3fjjtare^yh^^ The observations showed

that the light received on Earth was slightly redder than expected and that
the reddening is also in agreement with the predictions from the General
Theory of Relativity.

Figure 7.21: Illustration of the gravitational red-shift predicted by the Gen-
eral Theory of Relativity. A heavy object is denoted by a deformation of
space represented by the funnel. As light leaves the vicinity of this object
it is shifted towards the red: for a sufficiently compact and massive object
a blue laser on the surface will be seen as red in outer space.

.-ill

7.12.3 Light bending

If we imagine observing a beam of light in an accelerated elevator we will see
that the light path is curved. By the equivalence principle the same must
be true for light whenever gravitational forces are present. This was fcg§Jgs|
kX^rebdlyj£5Mdlng^

whenjdierej.s^jic^chps^^

During the eclipse the observed starlight reaches us only after passing
through a region where gravitational effects from the sun are very strong
(that is why only stars near the rim are used), but the observation half a
year later are done at a time where the gravitational effects of the sun on
starlight is negligible.

It is found that the position of the stars are displaced when photographs
of both situations are compared (see Fig. 7.23). The deviations are the same
as the ones predicted by General Relativity. Eddington first observed this
effect in 1919 during a solar eclipse. The early 20th century telegram (see
Fig. ??) announcing this observation for the frist time marks the change in
our views about the structure of space time.

Observed position
during the eclipse

Real position

as the observed position
r hen there is no eclipse]

The Sun during
an eclipse

Figure 7.22: Illustration of the effects of the gravitational bending of light:
during an eclipse the observed positions of the stars will be shifted away
from the Sun.

B'T

— r\J f ,'.

es«™tj cut U9/12 5» ii 11/ii-- KSLtrni^M^I

1 „™HMM

i=si

* -j"

Figure 7.23: Eddington's telegram to Einstein announcing the observation
of the bengin of light by a gravitational force as predicted by the General
Theory of Relativity.

7.12.4 The double pulsar

There are certain kind of gtarg, which are W$®LmhMS (see Sect. ??).
These are^ery^conrrjac^

rje^ubjjntensds.

In the early 80's, Taylor and Hules (recent Nobel prize winners for this
work) discovered a system where one pulsar circles another compact object.

a^a^clock. Moreover there are several physical effects which can be used to
determine the shape of the orbits of the pulsar and the compact object. It
was found that these objects are slowly spiraling into each other, indicating
that the system is losing energy in some way.

This system can also be studied using the General Theory of Relativity
which predicts that the system should radiate gravitational waves carrying
energy with them and producing the observed changes. These predictions
are in perfect agreement with the observations. This is the first test of
General Theory of Relativity using objects outside our solar system.

Chapter 8

The universe: size, origins,
contents

8.1 Introduction

The general and special theories of relativity discussed in the previous chap-
ters are the tools currently used in the investigation and description of the
universe. Most of the objects in the universe are somewhat mundane: stars,
planets, rocks and gas clouds. Yet in many respects the universe is far from
being a placid and peaceful place. There are stars which explode with the
energy of a billion suns, black holes with millions of times the mass of our
sun which devour whole planetary systems, generating in one day as much
energy as our galaxy puts out in two years. There are enormous dust cluds
where shock waves trigger the birth of new stars. There are intense bursts
of gamma rays whose origin is still uncertain.

These phenomena are not infrequent, but appear to be so due to the
immense distances which separate stars and galaxies; for one of the most
impressive properties of the universe is its size. The universe is so large that
just measuring it is very difficult, and finding out the distance to various
objects we observe can be a very complicated proposition.

In order to extract information about the universe a toolbox of methods
has been devised through the years. I will first discuss the most important of
these methods, and with these I will describe how measure the universe and
discuss its evolution. We need to determine sizes and distances because, as
we will see, they provide basic information about the history of the universe.

Most of the data we get from the universe comes in the form of light
(by which I mean all sorts of electromagnetic radiation: from radio waves

1

to gamma rays). It is quite remarkable that using only the light we can
determine many properties of the objects we observe, such as, for example,
their chemical composition and their velocity (with respect to us). In the
first two sections below we consider the manner in which we can extract
information from the light we receive.

But detecting light is not the only way to obtain information from the
universe, we also detect high-energy protons and neutrons (forming the ma-
jority of cosmic rays). The information carried by these particles concerns
either our local neighborhood, or else is less directly connected with the
sources: isolated neutrons are not stable (they live about 10 minutes), so
those arriving on Earth come from a relatively close neighborhood (this de-
spite time dilation - Sect. ??). Protons, on the other hand are very stable
(the limit on their lifetime is more than 10 32 years!), but they are charged;
this means that they are affected by the magnetic fields of the planets and
the galaxy, and so we cannot tell where they came from. Nonetheless the
more energetic of these particles provide some information about the most
violent processes in the universe.

In the future we will use yet other sources of information. Both gravi-
tational wave detectors and neutrino telescopes will be operational within
the next few years. Neutrinos are subatomic particles which are copiously
produced in many nuclear reactions, hence most stars (including our Sun)
are sources of neutrinos. These particles interact very very weakly, and be-
cause of this they are very hard to detect. On the other hand, the very
fact that they interact so weakly means that they can travel through very
hostile regions undisturbed. Neutrinos generated in the vicinity of a black
hole horizon can leave their native land unaffected and carry back to Earth
information about the environment in which they were born.

8.2 Light revisited

In this section I will describe two properties of the light we receive and the
manner in which it can be used to extract information about its sources.

8.2.1 The inverse-square law

A source of light will look dimmer the farther it is. Similarly the farther
away a star is the fainter it will look; using geometry we can determine just
how a star dims with distance

Imagine constructing two spheres around a given star, one ten times
farther from the star than the other (if the radius of the inner sphere is R,

the radius of the outer sphere is 10R). Now let us subdivide each sphere into
little squares, 1 square foot in area, and assume than on the inner sphere
I could fit one million such squares. Since the area of a sphere increases
as the square of the radius, the second sphere will accommodate 100 times
the number of squares on the first sphere, that is, 100 million squares (all
1 square foot in area). Now, since all the light from the star goes through
both spheres, the amount of light going through one little square in the inner
sphere must be spread out among 100 similar squares on the outer sphere.
This implies that the brightness of the star drops by a factor of 100, when
we go from the distance R to the distance 10R (see Fig. 8.1).

Figure 8.1: Illustration of the inverse-square law: all the light trough the 1
square-foot first area goes through the second one, which is 100 times larger,
hence the light intensity per square foot is 100 times smaller in the second
area. The intensity drops as 1/R 2 .

If we go to a distance of 20R the brightness would drop by a factor of
400, which is the square of 20, for 30R there would be a decrease by a factor
of 900 = (30) 2 , etc. Thus we conclude that

The brightness drops as 1/ (distance) . Light intensity drops a

1/ (distance) 2 .
This fact will be used repeatedly below.

8.2.2 The Doppler effect

We have seen that light always travels at the same speed of about 300, OOOkm/s;
in particular light emitted by a sources in relative motion to an observer
travels at this speed. Yet there is one effect on light which shows that its
source is moving with respect to the observer: its color changes.

Imagine standing by the train tracks and listening to the train's horn.
As the train approaches the pitch of the blast is higher and it becomes lower
as the train recedes from you. This implies that the frequency of the sound
waves changes depending on the velocity of the source with respect to you,
as the train approaches the pitch is higher indicating a higher frequency and
smaller wavelength, as the train recedes from you the pitch is lower corre-
sponding to a smaller frequency and a correspondingly larger wavelength.

This fact, called the Doppler effect, is common to all waves, including
light waves. Imagine a light bulb giving off pure yellow light; when it moves
towards you the light that reaches you eye will be bluer, when the bulb
moves away form you the light reaching your eye will be redder. If you have
a source of light of a known (and pure) color, you can determine its velocity
with respect to you by measuring the color you observe. Qualitatively, if
one observes a redder color (longer wavelength than the one you know is
being emitted) then the source is moving away from you, if bluer (shorter
wavelength that the one you know is being emitted) the source is moving
If one observes a redder toward you (see Fig. 8.2).

color (longer wavelength -j^ important point here is that knowing the frequency at the source

than the one you know is - r

being emitted) then the an d measuring the observed frequency one can deduce the velocity of the

source is moving away from source 1 If the source is moving sufficiently fast towards you the yellow light
you, i uer (s or er w -^ ^ e rece j vec [ as f or example, X-rays; in this case, however, the source

wavelength that the one you L J

know is being emitted) the must move at 99.99995% of the speed of light. For most sources the shift in
source is moving toward you frequency is small.

8.2.3 Emission and absorption lines

When heated every element gives off light. When this light is decomposed
using a prism it is found to be made up of a series of "lines", that is, the
output from the prism is not a smooth spectrum of colors, but only a few of
them show up. This set of colors is unique to each element and provides a
unique fingerprint: if you know the color lines which make up a beam of light
(and you find this out using a prism), you can determine which elements
were heated up in order to produce this light.

1 More precisely this is the velocity along the line of sight,

BLUESHlfT

Figure 8.2: Diagram illustrating the Doppler effect. The source is moving
to the left hence a receiver on the right will see a red-shifted light while a
receiver on the left will see a blue-shifted one. .

Similarly, when you shine white light through a cold gas of a given el-
ement, the gas blocks some colors; when the "filtered" light is decomposed
using a prism the spectrum is not full but shows a series of black lines (corre-
sponding to the colors blocked by the gas); see Fig. 8.3. For a given element
the colors blocked when cold are exactly the same as the ones emitted when
hot.

The picture in Fig. 8.3 corresponds to a single element. For a realis-
tic situation the decomposed light can be very complex indeed, containing
emission and absorption lines of very many elements. An example is given
in Fig. 8.4.

After the discovery of emission and absorption lines scientist came to rely
heavily on the fact that each element presents a unique set of lines: it is its Each element presents a
inimitable signature. In fact, when observing the lines from the solar light, umc i ue set of lmes
it was found that some, which are very noticeable, did not correspond to
any known element. Using this observation it was then predicted that a new
element existed whose absorption lines corresponded to the ones observed
in sunlight. This element was later isolated on Earth, it is called Helium
(from helios: sun).

Emission spectgrum

Absorption spectrum

[111 I H

Figure 8.3: Diagram illustrating emission and absorption lines: when light
given off by hot gas is decomposed using a prism it is shown to be made up of
colored lines (emission lines). When white light shines trough a cold gas the
resulting light , when decomposed is shown to have dark lines (absorption
lines). The emission and absorption lines for the same element match.

In following this line of argument one has to be very careful that the lines
are not produced by any other element. This is complicated by the fact that
some lines are observable only under extreme circumstances and one has to
take them into consideration as well. For example, after the success of the
discovery of Helium, another set of lines (not so prominent) was isolated and
associated with yet another element, "coronium". It was later shown that
the coronium lines were in fact iron lines, which are clearly observable only
in the extreme conditions present in the sun (one can also see them in the
laboratory, it's just hard to do so).

8.2.4 A happy marriage

When observing stellar light from various distant stars (decomposed using a
prism) it was found that, just as for the sun, they presented lines. But, cu-
riously, these lines corresponded to no known element! This may imply that
each star carries a new set of elements, but the simplest hypothesis (which
should be investigated first, see Sect. ??) is that the mismatch between the

Figure 8.4: Solar light decomposed by a prism exhibiting the emission and
absorption lines. At the top is one of the first of such measurements (1817);
the curve above the lines denotes the intensity of the various colors, as ex-
pected it is largest in the yellow. The second figure is a modern photograph
of the solar absorption lines.

laboratory and stellar lines is due to the Doppler effect which will shift the

lines towards the red or blue according to the motion of the star (which is

the source in this case) with respect to Earth. One can then use the shift

in the observed stellar lines to determine the velocity at which the star is

moving (with respect to us) and also the elements in it. In one fell swoop Using spectral lines we can

we determine the constitution and the speed of the stars using only the light determine botn the speed of

the star and the elements in
we get from it. it

8.3 Cosmic distance ladder

Another important piece of information regarding objects in the universe is
their distance to us. This is not an easy thing to measure since these objects
are usually very far apart. I will measure distances in light years: one light
year is the distance covered by light during one year, which is about 9.5
trillion kilometers, or about 6 trillion miles.

In order to understand why several steps are needed in measuring dis-
tances it is useful to consider a simple example. A student is in her room
sitting at her desk and would like to find the distance to the window; she
gets a ruler and laboriously measures this distance to be 3 feet. This I will
call "the first rung in the student's distance ladder"

Her next task is to find the distance to a building which she can see
through the window. This building is too far away for her to use her 12 inch
ruler. What she does is to use sound: she notices that when she claps her
hands outside her window there is an echo produced by the sound bouncing
off the building in front of her. She has a good watch and so she can
determine the time it takes for the sound to get from her window to the
building and back. Now, if she can determine the speed of sound, she could
use the formula distance = speed x time to get the distance. In order to
measure the speed of sound she closes her window and times the echo from
her desk to the window. Since we already knows the distance to the window
(which she measured using her ruler) and she now knows the time it takes
sound to go from her desk to the window and back she can determine the
speed of sound. So, using the first measurement she determines the speed
of sound and this allows her to measure things that are much farther away.
In this way she has "constructed" the second rung in her distance ladder.

The same idea is used when measuring far away things in space: one finds
a reliable method to determine the distances to near-by stars (the equivalent
of using the ruler). Then one devises another method which requires a sort
of calibration (the equivalent of determining the speed of sound); once this
calibration is achieved the second method can be used to find distances to
objects that are outside the range of the first method. Similarly a third,
fourth, etc. methods are constructed, each based on the previous ones.

Step 1: distances up to 100 Ly.

For near-by stars their distance is measured by parallax: the star is observed
in, say, December and then in June, and the direction of the star with respect
to the sun is measured in both cases. Knowing these angles and the diameter
of the orbit of the Earth around the sun, one can determine the distance to
the star (see Fig. 8.5).

As we look at farther and farther stars the angles measured come closer
and closer to 90°. For stars more than 100 Ly. from Earth one cannot
distinguish the angles from right angles and the method fails.

Earth in December

.»••** Distant star

Earth in June

Figure 8.5: Knowing the size of Earth's orbit and measuring the angles of
the light from the star at two points in the orbit, the distance to the star
can be derived. The farther the star is, the smaller the angles.

Step 2: distances up to 300,000 i.y.

In the decade 1905-1915 Hertzprung and Russel observed a group of near-
by stars whose distances they knew (using parallax). For each star they
recorded its color and calculated its brightness as it would be measured at
a distance of 1 i.y. (using the l/(distance) 2 law, see Sect. 8.2.1). Then they
plotted this brightness versus the color; what they found is that most stars
(90% of them) lie on a narrow band in this type of plot which they called
the main sequence (see Fig. 8.6).

Suppose we now obtain the HR plot for stars which are far away, say on
the other side of the galaxy, about 10 5 light years (10 18 km). If we choose
these stars such that they are not too far apart (there are good astronomical
indicators for this) the distance from Earth to any such star will be more
or less the same. It is found that, as for the near-by stars, 90% of these far
stars will again fall on a main-sequence strip in the color vs. brightness plot.

On the other hand all these stars are dimmer than the near-by stars
originally used by H&R; the decrease in brightness is due to the fact that
brightness drops as the square of the distance (Sect 8.2.1). Comparing the
two main sequences (for near and far stars) as in Fig. 8.7, we can extract

••

• Biii!? Giant?

fis'd- Siu>e-Mianre

• •

• •

•

Main fequgvze

• A • ■
• w fed gi<mts w

• ••

•

•

*• • *

* #

T

• •

Man 5ec\;Mnc»

Whta Dm,

arfs

i

•

Figure 8.6: The Hertzprung-Russel diagram. The horizontal axis corre-
sponds to the color of the star: blue to the left, red to the right. The
vertical axis corresponds to the star's brightness (brighter stars are plotted
higher). Though the diagram does not represent it, the groups labeled red
supergiants, red giants, blue giants and white dwarfs, are but a small frac-
tion of the whole stellar population, most stars are in the main sequence.

the distance to these far-away stars. This method can be used to determine
distances up to 300,000 £.y.; for larger distances the main sequence stars are
too dim to obtain a reliable estimate of their brightness.

Step 3: distances up to 13,000,000 l.y.

In 1912 Henrietta Swan Leavitt noted that 25 stars, called Cepheid stars 2
(their location in the HR diagram is given in Fig. 8.10), in the Magellanic
cloud 3 (see Fig, 8.8) are variable, that is, they brighten and dim periodically.
Many stars are variable, but the Cepheids are special because their period
(the time they for them to brighten, dim and brighten again, see Fig. 8.9)

2 The name derives from the constellation in which they were first observed.
3 This is a small galaxy (of only 10 8 stars) bound to the Milky Way.

Figure 8.7: The Hertzprung-Russel diagram for the main sequence of near
and far stars. The comparison is used to determine the distance to the far

stars.

i) regular (that is, does not change with time), and

ii) a uniform function of their brightness (at a 1 light-year distance). That
is, there is relation between the period and brightness such that once
the period is known, the brightness can be inferred.

Leavitt was able to measure the period by just looking at the stars and
timing the ups and downs in brightness,

But in order to obtain the brightness at the distance of one light year she
needed to fist measure the maximum brightness on Earth and then, using
the HR method, determine the distance from Earth to these stars (as it
turns out, the Magellanic cloud is about 10 5 light years away from us).

What she obtained is that the brighter the Cepheid the longer its period,
and that the relation between brightens and period was very simple: a
straight line (Fig. 8.11). This means that the period and brightness are
proportional to each other

Figure 8.8: The Magellanic Cloud

AAA^

2 4 6 8 10

time (days)

Figure 8.9: Illustration of the brightening and dimming of a variable star.

Measuring properties of the Cepheid variables. The color is
no problem, you just observe the starli«hl through dilfcrciil
color filters and observe the intensity; 'the' color of the
star corresponds to the filter which lets pass the highest
intensity light. The intensity at the distance of one light
year is obtained by measuring the intensity on Earth and
calculating the distance to the star, then one uses the fact
that the intensity drops as the square of the distance. For
example, suppose we observe a star which has intensity
of 1 (in some units), and which we know is at a 10 (..y.
from Earth, then at a distance of 1 l.y. (which is 10 times
smaller) the intensity will be 100 times larger (the square
of 10 is 100) and so the intensity at the distance of 1 £.y.
will be 100 (in the same units as before).

~ 10 L

1

~ 1

40000 20000 10000 5000 2500
T(K)

Figure 8.10: The location of the Cepheid variable stars in the Hertzprung-
Russel plot.

These stars are quite distinct, reasonably abundant and very bright. One
can identify them not only in our galaxy, but in many other galaxies as well.

If one requires the distance to a given galaxy one first locates the Cepheid
variables in this galaxy. From these observations one determines the period
of each of these stars. Leavitt's data states that a given period has a unique
brightness associated to it. So form the period and Leavitt's plot we get
the brightness at the distance of one light year. We can also measure the
brightness on Earth. The brightness at the distance of one light year will
be larger than the observed brightness due to the fact that this quantity
drops like the square of the distance (Sect. 8.2.1). From these numbers one
can extract the distance to the stars. This method works up to 13 million
l.y. when Earth-bound telescopes are used; for larger distances these stars
become too dim to be observed.

Much more recently the Hubble telescope has used this same type of in-
dicators to much farther distances (the Hubble is outside the Earth's atmo-
sphere and can detect much fainter stars) . Looking at a galaxy in the Virgo
cluster (the galaxy is "called" M100), Wendy L. Freedman found (1994-5)
that the Cepheid variables in this galaxy could be used to determine its
distance; the result is 56 million l.y..

o

c

3

10 4
10 3
10 2

1 I I

1 10

period (days)

100

Figure 8.11: Relationship between the brightness and period of the Cepheid
variable stars.

Young Cepheids. Recent observations of Cepheid variables
in lln >v<lax\ Mlllll from the Hubble telescope have gen-
erated some puzzling questions. Using these observations
(and the General Theory of Relativity) it is possible to de-
termine the age of the universe: we measure both the dis-
tance and the velocity of these objects (with respect to us)
and we can calculate the rate of expansion of the universe,
from this we get the time it took to get to its present size.
Curiously enough the age is in conflict with some other age
determinations: some stars are older than the number ob-
tained!

How can this be resolved? There are several possibilities,
one of the most likely ones is that, since the Cepheid is ob-
served by the Hubble telescope are very far away, the light
we get was sent out when the stars were quite young. But it
has not been shown that Leavitt's data is also valid for such
teenage stars. It is quite possible that these stars have a
different behavior and only settle into regular predictability
only as they become middle-aged.

Step 4: distances up to 1,000,000,000 l.y.

For larger distances run of the mill stars are of no use: they are too dim.
There are, however, some stars which at the end of their life blow themselves
apart and, in doing so, become anomalously bright (out-shining a galaxy
in many cases) for a brief period of time (less than a month); such an
object is called a supernova (for more details see Sect. ??). The unique
characteristics and enormous brightness of a certain type of supernova can
be used to determine distances beyond the reach of the previous methods.

There have been many measurements of the manner in which a super-
nova, whose distance to Earth is known (using one of the previous methods),
increases its brightness and then dims into oblivion. There is one type (called
type la) for which this brightening and dimming is very regular: when the
maximum brightness at a distance of 1 l.y. is calculated (using the known
distance and the 1/distance 2 rule), it is found to be the same for all cases 4 .

If the distance to a far away galaxy is required, one must first locate a
type la supernova in it (which do occur regularly) and then measure its ob-
served brightness. Comparing this result with the known maximum bright-
ness (at a 11. y. distance) achieved by all such supernovae one can determine
the distance to the galaxy in question (again using the 1/distance rule).
Since supernovae are extremely bright this method is useful to very large
distances, up to 10 9 l.y..

Step 5: distances beyond 1,000,000,000 l.y.

For very far objects none of the above methods work. The reason is inter-
esting: since we are looking at very distant objects their light has taken a
very long time to reach us, so the light we get must have left the object a
long time ago. Because of this the farther we look the earlier the images
we get: looking far away is equivalent to looking back in time. When we
look at the farthest obects we can see, what we get are images of their early
stages of their development.

In addition, since the brightness drops as the square of the distance,
these far objects must also be very bright. From this it follows that the
most distant objects we see are necessarily very bright and very young.

In order to determine the distances with any degree of accuracy we need
to know the brightness at a distance of 1 l.y., but here we hit a stone wall:
the only objects we see are much older than the ones we are interested in,

4 In doing so astronomers must select type la supernovae that exhibit no abnormalities.
else the measurements might be corrupted.

and we do not have a reliable theory of the way in which these things evolve,
we have no way of calibrating our observations using any near-by objects.

It is here, in the observation of the universe at large, that the General
Theory of Relativity must be used to measure distances. How this is done
is described in the next section.

8.4 The relativistic universe

In everyday life there are many forces that strongly affect the world around
us: friction, electric, magnetic, etc. But in the universe at large there is only
one predominant force: gravity. It is gravity that determines the structure
of the universe at large.

Figure 8.12: NASA Hubble Space Telescope image of the central portion of
a remote cluster of galaxies (CL 0939+4713).

The (visible) universe is filled with galaxies (see Fig.8.12) each containing
a billion suns (more or less) tightly bound by their mutual gravitational
atraction. Because of this we can think of a galaxy as a solid object of a
given mass (in the same way that when you look at the gravitational pull
of the Earth on the Moon you don't have to worry about the fact that they
are made of atoms; the stars are the "atoms" which make up galaxies).

Magnitudes. The typical galaxy like the Milky Way has
most of its stars in a central bulge of 10 4 l.y. diameter
or less, where about 10 11 sun-sized stars are concentrated.
The pull of these stars on out Sun is 10 4 times stronger than

thai of the nearesl sizable galaxy ( Viidromeda) which is at
about 2 X 10 6 l.y. away and also has about 10 11 sun-sized

In this simplified picture the visible matter in the universe (that which
shines) is concentrated in a dusting of galaxies. In addition the universe
can contain matter which does not shine, such as planet-sized objects, cold
dust and, perhaps, other more exotic objects (see Sect. 8.5.1). The universe
also contains electromagnetic radiation: for example, stars continuously give
off light and heat (infrared radiation) which then disperses throughout the
universe (this is why we can see them!). Finally the universe contains a
significant amount of microwaves (see Sect. 8.4.2) and neutrinos, (see Sect.
8.5.1), both relics from a very early time.

The first person to look at the cosmos through the eyes of the General
Theory of Relativity was Einstein himself. He took the above picture of a
universe filled with matter and radiation he added two assumptions

• Homogeneity: on average the universe looks the same from every van-
tage point.

• Isotropy: on average the universe looks the same in every direction

These assumptions, though reasonable, still require justification; I will come
back to them. With these preliminaries one can solve the equations of the
General Theory of Relativity and find a description of the universe and the
manner in which it evolves.

To Einstein's initial surprise there were no steady solutions: the universe
according to the General Theory of Relativity must expand or contract. He
compared this result with the best observational data of the time and found,
to his dismay, that the observations strongly favored a steady universe. He
then made what he called "the greatest scientific blunder of my life": he
modified the equations of the General Theory of Relativity by adding a
term that countered the expansion or contraction present in his initial so-
lutions 5 . With this ad hoc modification he did find a steady universe and
was (temporarily) satisfied.

5 The modification amounts to the inclusion of a uniform cosmic pressure which balances
the tendency to the universe to expand.

General Relativity then
predicts that the distance <
to an object is related to il
velocity v both measured
with respect to the Earth
by V = H d which is
called Hubble's law and
H is Hubble's constant

Not long afterwards Hubble published his now famous observations that
demonstrated that our universe is, in fact, expanding; and the manner in
which it expands agrees with the predictions of the solutions first obtained
by Einstein. It was then that Einstein, to his satisfaction, dropped his
modification of the equations. But this was not the end of this saga: the
added term, like the genie from the bottle, refused to disappear, showing
up in many models (recent observations suggest that it must be included
in order to account for the observations). I will come back to this in Sect.
8.5.2.

What Hubble did was to measure the red-shift of a group of galaxies
whose distances he knew (there were no blue-shifted galaxies, which means
that these galaxies were receding from the Milky Way) . Using the measured
red-shift and the formulas for the Doppler effect, he found the speed at
which they receded. Then he made a plot ( called now a "Hubble plot")
of velocity vs. distance and found that, as predicted by the General Theory
of Relativity all points fall in a straight line (see Fig. 8.13); the slope of
this line is called Hubble's constant. General Relativity then predicts that
the distance d to an object is related to its velocity v (both measured with
respect to the Earth) by

v = H d

which is called Hubble's law and H is Hubble's constant, its value is ap-
proximately

11„

1.5 x 10 10 years'

It is the above relation between distance and velocity that is used to
measure distances beyond 10 9 l.y.: the final step in the cosmic distance
ladder. Needless to say astronomers have verified Hubble's law for distances
below 10 9 l.y. using supernovae (Sect. 8.3). In order to find the distance
to the farthest objects in the universe one first obtains their redshift and,
using Doppler's formulas, derives the velocity v of the object. The distance
is then v/H .

8.4.1 The expanding universe

All of Hubble's (and subsequent) measurements indicate hat all galaxies are
receding from the Milky way and its neighbors. One might think that we
are being ostracized by the universe as a whole, that the Milky Way has
become a cosmic pariah; but a little thought shows that this is not the case.
According to General Relativity the universe is expanding, but this does

70,000

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1

SO .000

!■■

J

>

30.000

it

j^Corona Boreal is

10,000

j Virgo

1 1 »

Distance (in millions of light years)

Figure 8.13: Illustration of Hubble's law.

not mean that the galaxies and such are flying out into space, it means that
space itself is growing, and in so doing, it increases the separation between
the galaxies. The classical example is to imagine a balloon with dots drawn
on it; the balloon's latex represents space, the dots represent the galaxies.
As the balloon is inflated (space grows) the distance between the dots (the
galaxies) increases. An observer in any one dot would see the other dots
receding from him/her (just as we see distant galaxies receding from us).

This universal expansion represents only the average motion of the galax-
ies, the motion of a given galaxy can present deviations from this average.
For example, galaxies which are close together are bound by their mutual
gravitational pull and this distorts the Hubble flow.

The General Theory of Relativity predicts that the universe is not static,
and observations confirm this indicating that it is expanding. Thus the
universe must have been smaller in the past, and, following this idea to its
limit, must have been a point in its inception. Thus the universe began at
a point, in the distant past and has been expanding ever since. The event
marking this beginning is known (with a characteristic scientific flair for
words) as the Big Bang.

Just after the Big Bang the universe contained an extremely hot and
dense soup of matter and energy (which are equivalent in the sense of the

The Big Bang occurred
everywhere.

The universe will continue
its expansion forever, or it
will eventually stop and

slowing down to a stop at
infinite time.

Space in an eternally
expanding or open universe
is shaped like a
3-dimensional horse saddle

Space in a closed universe
which will eventually
re-contract is shaped like a
3-dimensional sphere

Space in a flat universe
which expands slowing
down to a stop at infinite
time is shaped like a

Special Theory of Relativity) under which conditions any kind of object
would melt almost instantaeously into its components. Yet the universe
expanded and cooled accordingly, and this cooling allowed for the formation
of more and more complicated structures, ranging from atoms (300, 000
years after the Big Bang) to Galaxies (10 9 years after the Big Bang) (see
Fig. 8.16).

It must be remembered that the Big Bang represent the creation of the
universe, including space and time. The Big Bang is not to be pictured as a
big explosion somewhere out in space with galaxies being spewed out from
the explosion region. Instead the picture provided by General Relativity is
of the whole universe, including space, appearing at the Big Bang and ex-
panding after that (like the balloon model described above). In this picture
the Big Bang occurred everywhere.

And now what?

The universe expanding, but what will become of it? There are three pos-
sible solutions to the equations of the General Theory of Relativity which
represent homogeneous and isotropic universes: either it will continue its ex-
pansion forever, or it will eventually stop and re-contract or it will expand
slowing down to a stop at infinite time. The contents of the universe (matter
and radiation) determine which of these is realized in our universe. In all
three cases the shape of space remains the same as the universe expands (or
in the second case, as it expands and contracts).

That the shape of space is determined by the amounts of matter and
energy in the universe is not surprising as it is matter and energy which
determine the curvature of space (see Sect. ??).

• Space in an eternally expanding or open universe is shaped like a 3-
dimensional horse saddle. In this case the angles in a triangle add up
to less than 180°.

• Space in a closed universe which will eventually re-contract is shaped
like a 3-dimensional sphere. In this case the angles in a triangle add
up to more than 180°.

• Space in a flat universe which expands slowing down to a stop at
infinite time is shaped like a 3-dimensional plane. In this case the
angles in a triangle add up to 180°.

These possibilities are illustrated in Fig. 8.14.

z\

^

Figure 8.14: The three possible shapes of a homogeneous and isotropic uni-
verse: a closed universe (left), a flat universe (center) and open universe
(right). See the text for an explanation.

These three possibilities give the average shape of space. Individual
masses produce local bumps and troughs. This is similar to the way we talk
about the Earth: we say it is a sphere, though we know it is full of bumps
(for example, Himalayas) and troughs (the Dead Sea, for example).

Of these possibilities the one corresponding to our universe is determined
by the amount of matter in the cosmos. If there is very little the initial
thrust from the Big Bang will never be stopped, if however there is a large
amount of matter, the mutual gravitational pull will be sufficient to break
the expansion and eventually cause a re-contraction. Hence there is a critical
amount of matter such that if our universe has more it will re-contract, if
less it will expand forever (if it has precisely the critical amount it will
expand forever slowing down to a stop at infinite time). These possibilities
are illustrated in Fig. 8.15.

The obvious question is then: how much stuff is in the universe? And to
that we can say: we don't know. If we count all the matter that shines (stars
and such) we get a number very low compared to the critical value. But, is
most of the matter shining? Could it not be that there is a lot of dust out
there? The latest results suggest that the universe will expand forever, but
at present its ultimate fate is unknown.

8.4.2 The Microwave Background Radiation

General Relativity not only provides a nice history of the universe, but it
also points out viable measurements which can support its validity. The
most important is the so-called Microwave Background Radiation.

When the universe began the density and temperature of the initial fire-
ball was so high that all matter dissociated into its primary components.
Note also that in this initial setting the force of gravity was enormous. As
the expansion progressed the universe cooled and the initial fundamental

The Big Bang The Big Crunch

Figure 8.15: The universe might expand forever or will re-contract

constituents formed increasingly more complicated objects. This is so be-
cause when the temperature is very high everything is jiggling very fast and
anything that can be dissociated will; as the temperature drops so does the
jiggling and, eventually, composite structures can form and survive. Thus, if
we had been able to film the contents of the universe as it cooled, and then
run the film backwards we would first see atoms which are then broken apart
into nuclei and electrons by the intense heat, then we would see the nuclei
themselves decomposing into protons and neutrons, then the protons and
neutrons decomposing into quarks 6 . The microwave background radiation
is a messenger from this primordial soup.

To understand how why is this microwave radiation present and how it
was generated I need to talk a bit about the way charged bodies interact
with light. Remember now that light is described by the same equations that
describe the physics of electric charges (Maxwell's equations), this suggests
(and it is true) that light will interact with charged objects. In fact this
is how your skin gets hot when exposed to the sun: your skin is composed
of molecules which are made of atoms. Atoms in their turn are composed
of a small heavy nucleus (with positive charge) surrounded by a cloud of

6 There are many hypotheses about the way the
quark formation, but none has been accepted yet this

looked at times before that of
area of active research.

Figure 8.16: Abbreviated history of the universe according to the Big Bang
model.

negatively charged light particles, the electrons. When light shines on your
skin it is absorbed by the electrons which get agitated, and it is this agitation
which you perceive as heat. This is not as efficient as it might be because the
electrons are not free, they are inside atoms, so that on average the atoms
are neutral. Much more light would be absorbed by a set of free electrons.
This also works in the reverse: if you jiggle electrons sufficiently rapidly they
will give off light, this is how a light-bulb works.

Suppose now that you have a box with perfectly reflecting walls and
which are kept very hot. Into that box we introduce a bunch of electrons
and nuclei and also light. Assume that the system is so hot that the electrons
are not bound to the nuclei: as soon as they come close they are wrenched
apart by the intense heat of the environment. So, on average, what you see
is a bunch of charged particles and light running amok. In this case light is
constantly being absorbed and emitted by the electrons and nuclei.

Now imagine that you cool the box by making it larger. Eventually
things will get cold enough for the electrons to stay attached to the nuclei,
the heat is not sufficiently high for them to be wrenched apart. At this point
the rate at which light is absorbed and emitted drops rather suddenly for
now the particles in the box are neutral (on average). From this point on
light will just stream forth unimpeded (until it is reflected by a wall) .

This is precisely what happens in the universe. After the big bang there
came a point where electrons and nuclei were formed. They were immersed
in intense electromagnetic radiation (light, X-rays, gamma rays, etc.). As
time progressed and the expansion of the universe continued, the system
became cooler (much as for the box when we increased its size) . Eventually a
point was reached where the universe was cool enough for atoms to form and
from this moment on most of the radiation just streamed forth unimpeded.

This happened when the universe was a mere 300,000 years old.

So, can we see this relic of the ancient universe? The answer is yes!
But before we look for it one thing must be kept in mind. The universe
has been getting bigger and bigger and less and less dense. This implies
that the average gravitational force is getting smaller with time. So the
radiation, from the moment it no longer interacted with the newly formed
atoms has been shifting from an environment where gravity's force is large to
that where gravity is small and, using (again!) General Relativity, it must be
red-shifted. In fact the prediction of General Relativity is that this radiati
should be seen mostly as microwaves... and it has been seen. This predicti
is not only of the existence of this relic radiation, but also how this radiati
depends frequency . These predictions have been confirmed to great accuray
(see Fig. 8.17). This ubiquitous sea of radiation that permeates the cosmos
is called the microwave background radiation.

The microwave background radiation was created in approximately the
same environment everywhere (remember that it came from an epoch in
which everything was a very homogeneous hot mixture of nuclei and elec-
trons) and because of this we expect it to look the same in every direction.
This is precisely what happens, but, as it turns out, it is too much of a
good thing: the microwave background radiation is the same everywhere
to a precision of 0.1%, and understanding this presents problems, see Sect.
8.5.3.

But one can go even farther. Even though the microwave background
radiation is very homogeneous, there are small deviations. These represent
nhomogeneities in the universe at the time radiation and atoms stopped
nteracting strongly. These inhomogeneities provide a picture of the universe
n its most tender infancy, see Fig. 8.18. As the universe expands and cools
atoms will conglomerate into stars and stars into galaxies; the initial seeds
for this process to start are these inhomogeneities. They correspond to
regions where the matter was slightly mode dense than the average, and
will, in the eons that follow, attract other matter to form the structures we
see today.

It is very hard to explain the microwave background radiation by any
theory other than the Big Bang. It represent one of its biggest successes.

8.4.3 Nucleosynthesis

The most abundant element in the universe is Hydrogen, the second most
abundant element is Helium. A great success of the Big Bang theory is to
be able to predict the relative amounts of these elements: after the universe

Cosmic Microwwe Background Spectrum from COBE

Figure 8.17: Radiation relics from the epoch shortly after the Big Bang. The
horizontal axis corresponds to the frequency of the radiation, the vertical
axis to the intensity. The measurements fall precisely on the curve.

cooled down sufficiently protons and neutrons were able, after a collision,
to remain in the form of heavier atomic nuclei, in this manner Helium and
Lithium were created, and also Deuterium (whose nucleus has one proton
and one neutron). The universe was Is old, its temperature was 10 10o K.

It was initially thought that all elements would be generated by the Big
Bang, but this is not the case: even at the extreme temperatures available
when Helium and Lithium nuclei were crated, this was not enough to smash
two Helium nuclei to create something heavier, the creation of the remaining
elements of the periodic table had to await the appearance of the first stars
(see Sect. ??). Deuterium and Lithium, while used up in stars through
the nuclear reactions that make them shine (see Sect. ??), are very rarely
created by them . Whatever Deuterium and Lithium we see in nature was
created about 15 billion years ago. Most of the Helium we observe (even
though it is manufactured in stars) also came from that epoch.

The Big Bang theory predicts is the relative amounts of Helium and
Lithium and Deuterium and Hydrogen. And the observations match the The Big Bang theory
predictions; for example there are about 4 atoms of Hydrogen for each one P redicts is tne relative
of Helium. These same calcualations predict that there are 3 light neutrinos, Lithium and Deuterium

Hydrogen

The Big Bang theory
predicts that there are
light neutrinos

Figure 8.18: Inhomogeneities in the microwave background radiation. These
give an idea of the way the universe looked shortly after the Big Bang.

hich all elements in the
Ddic table were created

again confirmed by observation.

Coupled with our understanding of stellar processes and evolution (Sects.
?? and ??) we now understand the manner in which all elements in the
periodic table were created. This is one of the most important predictions
of modern cosmology.

8.5 At the cutting edge

Up to now all the results presented are well accepted and verified. There
is little doubt that the General Theory of Relativity provides an excellent
description of the universe at large, nor that the universe is currently ex-
panding. Yet there are several puzzling results...

8.5.1 Dark matter

When considering the universe we observe only what we can see. Nonethe-
less there are strong indications that there is something more. Suppose
you look at how stars in the outskirts of a galaxy move. Since gravity de-
creases with distance one would expect that the stars would slow down as
the distance to the galaxy center increases, but this is not what is seen: the
speed of these outlying stars appears to be constant (see Fig. 8.19). This
is explained by assuming that the galaxy is in fact surrounded by a mass of

matter which emits no or very little light, the so-called dark matter. In fact,
calculations show that if this hypothesis is correct, this kind of matter is the
main ingredient of galaxies, and perhaps the whole universe; an illustration
of the "dark matter halo" surrounding a typical galaxy is given in Fig. 8.20

20 40 60 BO

Radial Distance (in thousartfls ci light years]

Figure 8.19: Rotation curve for stars in the Andromeda galaxy. The velocity
becomes constant far away from the center suggesting the presence of dark
matter.

What is this dark matter? No one knows! Is it perhaps a very large
number of rocks, or planets? Is it something else? Or, maybe, is there a
completely new effect which we interpret as dark matter while in reality there
are new forces in action? The only recent answer is that there are strong
indication that there are large numbers of planet-like objects in the vicinity
of our galaxy. But these are not nearly enough to account for the whole
effect. Many experiments are under way aiming at detecting the nature of
dark matter (and it very existence).

Neutrinos

The early universe produced electromagnetic radiation which reaches us in
the form of microwaves. This radiation was the result of the electromagnetic
interactions among charged particles. There are, however, other types of
interactions. We already met the gravitational interaction, and there are

Figure 8.20: Illustration of the dark matter halo surrounding a typical
galaxy.

two others called (again with a flair for words) the strong and the weak
interactions.

Strong interactions are the ones responsible for nuclear forces between
protons and neutrons (the constituents of atomic nuclei), and we will come
back to them when we look at the evolution of a star (Sect. ??). The
remaining type, the weak forces, are experienced by all types of matter,
but they are usually overwhelmed by the electromagnetic and strong forces
because the weak interactions are, well, weak!

One is used to hear about electrons and protons and, perhaps to a lesser
extent, neutrons. All these arc constituents of atoms and atomic nuclei. But
nature has a much richer population, and among its citizens one of the most
intriguing are the neutrinos.

Neutrinos are very light particles 7 and experience only the weak in-
teractions and it is because of this that they are rarely affected by other
types of matter. Only in the densest of environments are neutrinos strongly
disturbed. These occur in the center of neutron stars (Sec. ??) or in the
early universe. In this last case neutrinos were originally extremely energetic

7 It had been assumed for a long time that they were massless, recent results however,
indicate that neutrinos have a very small mass, of a billionth of a proton mass or less.

but, just as in the case of radiation, there came a time when the universe
expanded to the point that the environment wasn't dense enough for the
neutrinos to be affected by it. From that point on the neutrinos have been
just cruising along, interacting only very rarely.

Initially these neutrinos lived in a very hot environment, which implies
that each of them had a lot of energy and they were in a situation where
very large gravitational forces were present. Nowadays they are in an envi-
ronment where the gravitational forces are very weak. To understand what
this implies consider the following analogy.

Imagine that you throw a ball up from the earth: initially the ball has
a lot of kinetic energy, that is, energy due to its motion, but as it rises it
slows down losing kinetic energy. Of course, this energy does not disappear,
it is stored in potential energy (see Fig. 8.21). As the ball falls it will pick
up speed so that when you catch it will be moving at the initial velocity (or
close to it). In the same way the neutrinos in the present universe will have
lost most of their kinetic energy.

Slow baseball I Slow neutrino I

(small kinetic energy I (small kinetic energy *

Fastbaseball Fastneutrino

(large kinetic energy I (large kinetic energy J

Figure 8.21: Neutrinos from the early universe have smaller kinetic energy
now than in earlier epochs just as a baseball has lower kinetic energy the
farther it is from Earth.

So another prediction of the Big Bang theory is that the universe is
filled with neutrinos of very small kinetic energy. Unfortunately, out current

technology is not sufficiently sophisticated to be able to detect them directly,
but this might improve in the future.

8.5.2 The cosmological constant

When Einstein first studied the universe at large using the General Theory of
Relativity he discovered that his equations predicted a universe which was
either expanding or contracting, and this was contradicted with the best
astronomical observations at the time. He then modified his equations to
satisfy the observations. This modification corresponds to the assumption
that the whoie universe is permeated with a constant pressure (which in
his case balanced the expansion yielding a steady universe), this universal
pressure is called the cosmological constant

Though subsequently the data showed that the universe is in fact ex-
panding and Einstein rejected the modification, on a philosophical basis the
question still remains whether the measured cosmological constant is indeed
zero (remember that on philosophical grounds Aristotle rejected heliocen-
trism: one must eventually back assumptions with observations). For many
years the best value for the cosmological was assumed to be zero since no
measurement gave positive indication to the contrary. Yet even a very small
pressure can be important if it permeates the whole universe.

For many years the best value for the cosmological was assumed to be
zero since no measurement gave positive indication to the contrary. Yet even
a very small pressure can be important if it permeates the whole universe.

Recent measurements of the expansion rate of the universe (see Sect.
8.4.1) using type la supernovae (Sect. 8.4.1) favor an open universe with
a small but non-zero cosmological constant. If these results are confirmed,
Einstein's "blunder" will prove to be one more piece in the jigsaw of nature.:

8.5.3 Homogeneity and isotropy

One of the central simplifying assumptions of Einstein's cosmology is that,
on average, the universe is the same in every direction (isotropy) and in every
location (homogeneity), this does not mean, however, that the universe is
a boring tapioca-like thing. The distribution of galaxies is far from smooth
with most of them concentrated in relatively narrow sheets separated by
large voids, see Fig. 8.22. The situation is reminiscent of a series of soap
bubbles where the soapy water corresponds to the galaxies, the air inside
the bubbles to the voids.

There are a few hipotheses which explain the origin of this type of struc-
ture. These must account not only for the voids, but also for the inhomo-
geneities in the comsic background radiation; and they must also predict a
reasonable time-line for the development of galaxies. All these constraints
are difficult to satisfy, making this an area of very active current research.

Figure 8.22: Large scale bubble-like structures in the universe. The image
contains about 4000 galaxies each representing one luminous point.

Inflation

When we look at the microwave background radiation it looks the same in
every direction, even from opposite sides of the sky, to a precision of 0.1%.
Since they are so nicely correlated one would naturally assume that at some
time all points in the observable universe were in close contact with each
other, for otherwise it would be an unbelievable coincidence for all of them
to look so much the same (at least through a microwave detector).

Now, a perfectly reasonable question is whether the Big Bang model has
this property: will the Big Bang model predict not only the existence of
the microwave background radiation, but also its exquisite uniformity? The
answer is "yes" but only with additional assumptions.

This seems confusing: is the Big Bang theory to be modified and tuned
every time a new piece of data comes along which does not agree with its

:',_>

predictions? Isn't this cheating? Doesn't this sound like Ptolemy adding
epicycles every time things weren't quite accurate?

Fortunately this is not the case. The Big Bang theory determines the
evolution of the universe provided the matter and energy content is known,
and their behavior at very extreme conditions is well understood. The fact
is, however, that we are not certain of all the matter and energy in the uni-
verse, nor do we know, for example, how they behave at temperatures above
10 15 °K. Hence these "modifications" of the Big Bang theory correspond to
different hypothesis of the behavior of matter at very high temperatures and
densities, not of the general description provided by the General Theory of
Relativity.

The simplest version of the Big Bang model which predicts a very uni-
forms microwave backgound goes by the way of Inflation. The idea is the
following: the simplest way of getting uniform background radiation is if all
the observable universe was in very close contact at an early time. Granted
that, inflation provides a mechanism for increasing the size of this initially
tiny region to the very large universe we see. Though mathematically in-
volved what is assumed is that at a very time (about 10 -35 s after the Big
Bang) a new force comes into play which forces an exponential increase in
the size of the universe (hence the name 'inflation'). After a fraction of a
second this force is balanced by other interactions and the universe resumes
a more dignified, if ponderous, expansion (see Fig. 8.23).

Figure 8.23: Time evolution of the size of the inflationary universe
One tantalizing conclusion derived from the inflationary hypothesis is

3:5

that there are regions in the universe which we have not yet seen and which
might look very different. Since no light has reached us from those regions
we are currently unaware of their existence, only our inheritors will see the
light coming from these distant reaches of the universe.

It is a challenge for current researchers to produce models that generate
the intergalactic voids, yet with the same amount of dark matter required
to understand the rotation of stars (Sect. 8.5.1) and using the inflation
hypothesis such models actually exist. The corresponding computer simu-
lations produce results such as the one shown in Fig. 8.24 which should be
compared to the observations (Fig. 8.22).

Figure 8.24: Simulation of the generation of structures in the universe as-
suming the presence of dark matter and an early epoch of inflation.

8.5.4 Summary

Though the General Theory of Relativity has produced a generous amount of
verified predictions, its application to the universe at large has also generated
a set of puzzles which, coupled to recent observations, are the topics of
intense research. Whether there is a cosmological constant, whether the
universe is filled with dark matter and the nature of this stuff and whether
our current models of the universe are accurate enough to understand physics
to the very earliest of times are issues currently addressed by researchers.
The near future will provide more puzzles and some answers leading us, we
hope, to a better understanding of the universe, our home.

Chapter 9

The lifes of a star

9.1 Introduction.

When stars are plotted in the H-R diagram, the number of stars in and out
of the main sequence, together with models of stellar evolution provides a
description of the possible ways in which stars are born, evolve and, even-
tually, die. During this process the star "move about" in the HR diagram
(see Fig. 9.1). Since most stars are in the main sequence it is reasonable to
suppose that during their life most stars stay in the main sequence, evolving
into it when they are born and out of it when they are about to die. Models
of stellar evolution confirm this.

For large objects (such as stars, galaxies, etc) the one ever-present force
is gravity. This is always an attractive force which tends to condense stars For large objects the one
and such into smaller and smaller objects. There are (fortunately) other ever -P resent force is gravity
effects which, at least temporarily, can balance gravity and stop this con-
traction. These effects are generated by the material which makes up the
star and are always associated with various kinds of pressure (which tends
to enlarge objects); a familiar example is the usual gas pressure

A less known type of pressure is produced by electrons 1 when they are
brought in very close contact. Under these circumstances there is a very
strong repulsion between the electrons, not only because they have equal
charges (and hence repel each other), but because electrons, by their very
nature, detest being close to each other: they require a relatively large
breathing space. This repulsion between electrons is called degenerate elec-
tron pressure 2 . This effect has a quantum origin and has many interesting Due to their dislike of being

i close contact e

Everything is made up of atoms. Atoms consist of a very dense and small nucleus and produce a degenerate
a bunch of electrons surrounding it. electron pressure

Figure 9.1: Diagram illustrating the evolution of a sun-like star. Born from
a gas cloud it moves towards the main sequence (1) where it spends most of
its life. After all Hydrogen is consumed in its core, the star burns Helium
and becomes a red giant (2). Finally, when the Helium is consumed nuclear
reactions subside and the star becomes a white dwarf (3) where it will spend
its remaining (billions of) years.

consequences, to mention two, thanks to this strong dislike of electrons for
occupying near-by locations, the floor supports your weight, and atoms have
different chemical properties.

Electrons are not the only kids of particles that dislike being in close
contact with one another. For example, the nucleus of a Hydrogen atom,
called a proton also exhibits this property. Finally, and this is important for
stellar evolution, other particles called neutrons also dislike being close to
each others. Neutrons have no electric charge and are slightly heavier than
protons; they are also found in atomic nuclei and are, in fact, a common
sight in nature. All the atomic nuclei (except for Hydrogen) are made of
protons and neutrons, with the neutrons serving as buffers, for otherwise the

2 This is just a peculia:
character of the electron;

tnd should not be interpreted as a judgment on the moral

electric repulsion of the protons would split the nuclei instantly. When close

to each other neutrons produce a degenerate neutron pressure and protons Neutrons in close contact
a degenerate proton pressure (see Fig 9.2). also Produce a degenerate

neutron pressure
Protons in close contact
produce a degenerate
proton pressure

\\ Protons, neutrons,
S\ electrons and quarks
produce degenerate
pressure.

Figure 9.2: List of the most important particles which generate a degenerate
pressure when in close contact. Also in the picture, the places where these
particles are most commonly found.

The various stages of stellar evolution are classified according to the
origin of the pressure which counterbalances gravity's pull. For most stars The
a balance is reached in the final stages of the star's life; there are some
objects, however, for which gravity's pull overwhelms all repulsion in the
stellar material, such objects are called black holes.

The mass of the star largely determines its history, light stars (such as our
Sun) will end in a rather benign configuration called a White Dwarf] heavier
stars (with masses below 3-4 solar masses but larger than one solar mass)
end as neutron stars after some spectacular pyrotechnics. Very massive stars
end their lifes as black holes. This will detailed below, but before we need
to understand what makes stars tick.

stages of stellar
evolution are classified
according to the origin of
the pressure which
counterbalances gravity's

9.2 Stellar Power

The main power source for all stars is furnished by nuclear reactions. Pos-
sibly the most familiar of these reactions are the ones used in nuclear power
plants; these, however, are not the ones or relevance in stellar processes. The
relevant reactions present inside stars go under the name of nuclear fusion.
Recall that all atoms are made of a very dense and small nucleus which
is positively charged and a bunch of electrons, which are negatively charged,
and which surround the nucleus. At the center of the stars temperatures as
very high (at least a few million degrees Celsius); pressures are also high 3
Under these circumstances the electrons are stripped off the nuclei and float
around. In this large-temperature environment both electrons and nuclei
very high speeds, so high that when two nuclei collide they often overcome
the the repulsion produced by the fact that they both have positive charge.
But when the nuclei come in such close contact with each other they will
"stick" . The result is a new nucleus and also energy is released. For example
one can imagine slamming Hydrogen nuclei to produce the nucleus of a new
element, Helium (see Fig. 9.3).

The result of the nuclear reaction in Fig. 9.3 is the depletion of Hydrogen
in the star, the creation of Helium, and the release of energy in the form
of radiation. Some of the radiation will heat the environment encouraging
more nuclear reactions of the same type, but a small fraction of this energy
will make its way to the star's surface and escape into space. Knowing the
equivalence of mass and energy this implies that the stars become slightly
lighter through this process. For our sun the loss is of "only" 1.35 x 10 14
(135 trillion) tons per year (which is only about 7 x 10~ 12 - 7 trillionths - of
a percent of the total solar mass). The jargon is that this reaction "burns"

The main nuclear reaction Hydrogen and that resulting "ashes" are mainly Helium.

in stars burns Hydrogen rp^ g^ove is just one of a vcrv large number of fusion reactions but it

and that resulting "ashes" J J fa

are mainly Helium is the most common, and is present in all stars at souk stage; of their lifes.

Other reactions are also important, I will talk about them later.

As time goes on the amount of Hydrogen drops and, eventually, there
is not enough left to generate appreciable amounts of energy. There are
nuclear reactions involving Helium (which is now quite abundant), but they
require higher temperatures. So, when the Hydrogen is used up, the nuclear
reactions turn off and the star continues to contract due to the gravitational
pull. But, just as before, as the contraction proceeds, the temperature at the

Remember that the pressure must balance gravity's pull. A star is a very r
body, hence gravity's pull will be very large; the pressure must then be also very large to

Figure 9.3: Illustration of the nuclear reactions which create Helium from
Hydrogen. At very high temperatures Hydrogen atoms are slammed to-
gether, as a result a new element, Helium is created, the amount of Hydro-
gen is slightly depleted and energy, in the form of radiation, is released (in
the "intermediate steps" some unstable nuclei are created).

core raises, eventually reaching the threshold of nuclear reactions involving
Helium.

9.3 The lifes of a star
9.3.1 In the beginning

It all begins with a swirling cloud of dust and debris (perhaps some old-star It all begins with a swirlin
remnants). Gravitational attraction causes this cloud to slowly contract. As cloud of dust and debris
it contracts the cloud speeds up its rotation (much as an ice-skater turns
faster when he/she draws her hands towards his/her body), and it heats

up. The cloud becomes unstable and separates into blobs, some might be
ejected due to centrifugal force, others condense into planets. The center of
The center of the cloud the cloud condenses into a big blob of matter (mainly Hydrogen since this

condenses into a big blob of j s ^ e mos i abundant element in the universe). This process takes about
one billion years to complete and produces a primitive planetary system:
a protostar (which is very big but too cold to produce nuclear reactions)
circled by protoplanets. As time goes on, the protoplanets in their orbits
will "sweep-out" the remaining debris from the cloud.

matter

temperatures at the center
are so high that nuclear

The main reaction
occurring at this stage
consume Hydrogen and

9.3.2 A rising star

Through the evolution of the star the only force opposing the gravitational
collapse is the pressure of the stuff the central blob is made of; this pressure
is initially very small compared to the pull of gravity. This means that the
blob will contract until pressures and temperatures at the center are so high
that nuclear reactions turn on. At this point the energy release from the
fusion reactions heats up the stellar material, this in its turn increases the
pressure and the contraction stops. As mentioned above, the main reaction
occurring at this stage consume Hydrogen and produce Helium: the star
"burns" Hydrogen into Helium. This goes on for a long time: if the star is
light (as our sun) it proceeds for about 10 billion years, much heavier stars
use up Hydrogen much faster (for the heaviest ones it takes 'only' 1 million
years).

When burning Helium the
star becomes a red giant

9.3.3 A Giant appears

After the supply of Hydrogen in the core is depleted the corresponding nu-
clear reactions stop (there are other fusion reactions, but they can occur
only at higher temperatures than the ones present at the center of the star
at this stage). Then the pressure drops and the gravitational collapse pro-
ceeds. During this process the center of the star is compressed more and
more, increasing the central temperature until, finally, it becomes so hot that
nuclear reactions involving Helium start up: Helium atoms slam together,
and, after a complicated reaction produce Carbon. When these reactions
turn on the energy output is enormous, the core becomes extremely hot and
radiates a very large amount of energy. This radiation pushes out the outer
layers of the star, and as they are pushed out they become a bit cooler and
thus look redder. The star then becomes a red giant a bloated result of
the burning of Helium Our sun will eventually go through this process and
will grow to the point that it will engulf the orbits of Mercury, Venus and,

possibly, the Earth.

9.3.4 And so it goes

What happens when the supply of Helium is used up? The story is repeated:
gravitational contraction takes over and the star collapses further. Eventu-
ally other nuclear reactions become viable, power increases until the various
nuclei are depleted, then contraction takes over again. In this manner the
star produces, Oxygen, Silicon and, finally, Iron, this is, in fact, the way in stars create Oxygen,
which these elements are manufactured in nature. Every bit of Carbon in SlllC0n and . finally, Iror
a flower's DNA, every bit of Oxygen we take in every breath, every bit of
Silicon in a sandy beach was created in a star.

When the core of the star turns into Iron all nuclear reactions stop,
permanently. The reason is that Iron is a very stable nucleus so that if two
Iron nuclei are slammed together they will only stick if energy is supplied (in
contrast, two Hydrogen atoms stick and also release energy). When nuclear
reactions stop gravitational contraction continues again and will proceed
until the electrons in it are closely squashed together. As mentioned above
electrons dislike being in close contact with each other and when squashed
will generate a pressure which opposes gravity; whether this pressure is
sufficient to stop collapse depends on how heavy the star is.

Light stars

For stars lighter than 1.4 solar masses the electron degenerate pressure will
balance gravity. The star has by now contracted from its red-giant size to
the size of a small planet (like Earth). The material of this star is so dense
a teaspoon of it would weight 1 ton on Earth.

When this final contraction occurs there is a certain amount of overshoot
and bouncing back and forth before stability is achieved; in this process all
the outer layers of the star are ejected. The end result is a beautiful ring of
stellar material which spreads out, at the center of which a small star, called
a white dwarf, remains (see Fig. 9.4). White dwarfs are is stable and their
racy days of nuclear reactions are forever gone; they slowly radiate their
remaining heat little by little and eventually become dark cinders. This is
the end of a star whose mass is smaller than 1.4 times the mass of the Sun;
this process is summarized in Fig. 9.5

It is interesting to note that the theory predicts that these objects will
always be lighter than 1.4 solar masses. Observations have confirmed this.
This theory is a combination of quantum mechanics and gravitation and, in

Figure 9.4: Photograph of a ring nebula. The central white dwarf has, in its
last throes, expelled its outer layers appearing here as a ring surrounding
the small remnant.

fact, it provided the first application of quantum physics to stellar objects.
For heavier stars the pull of gravity overcomes the degenerate electron
pressure and collapse continues.

Electrons, protons i
tions is composed of electrons, protons and neutrons. Elec-
trons are negath < I h.u d ud i ' " 10 k.n- pi

weight 1.8 X 10 -27 kg and have positive charge, exactly op-
posite to that of the electrons. Neutrons weigh as much as
protons and have no charge. Usually protons and neutrons
are bound together in atomic nuclei and are surrounded
by a cloud of electrons so that the whole systems is neu-
tral. If, however, matter is subjected to higher and higher

l>r '-.in v.. (-will ually I lie atoms are crushed together to the

point that the electrons can jump around from the vicinity
of one nucleus to another.

If the pressure is increased still further the nuclei them-
selves are brought into close contact and lose their identi-
ties. At this point the protons undergo a reaction in which
they absorb an electron and turn into protons while emit-
ting a neutrino (yet another subatomic particle). Because
of this process most of the matter turns into neutrons and
neutrinos. The latter interact very seldom and just leave
the system; because of this what remains is essciii.iallv an
enormous number of neutrons

The life of a star lighter than 1.4 solar masses

Hydrogen burning

Initial condensation

Figure 9.5: Time and life of a star of mass below 1.4 times the solar mass
(less than about 3 x 10 tons).

Medium-size stars

For stars heavier than 1.4 solar masses but lighter than about 3-4 solar
masses (the calculations are still a bit uncertain), the electron pressure is
not strong enough to balance gravity. The contraction then goes crushing the
electrons together and braking apart the Iron nuclei into their constituents.
These constituents, neutrons and protons, also detest being close to each
other and, as mentioned above, produce a (degenerate) pressure which op-
poses gravity. For a star in the present mass range this pressure is sufficient
to stop further collapse, but is effective only when the material is extremely
dense which occurs only when the star has contracted to an object a few
kilometers in diameter.

The contraction of these stars from their initial solar size to the size of
a city is one of the most spectacular events in the heavens: a supernova.
Imagine an object weighting 5 x 10 27 tons (that is five thousand trillion-

in

trillion tons, or about 2.5 solar masses), which contracts from a size of 10 6
(one million) kilometers to about 10 kilometers, and it all happens in a
fraction of a second. During collapse the amount of energy generated is
fantastic, part of it goes into creating all elements heavier than Iron, part
into creating neutrinos and part is transformed into light.

Radioactive elements are also created during the collapse. These ele-
ments rapidly decay, and the resulting radiation is so intense it produces
a fantastic flash of light. At this point the supernova will out-shine a full
galaxy of normal stars (several billion or up to a trillion of them!).

After the collapse there is a violent overshoot before equilibrium sets in,
at this time all the outer layers of the star are ejected at speeds close to that
of light. When this material goes trough any planets around the star (if
any) it vaporizes them. In the middle of this cloud the core of the original
star remains, a rapidly rotating remnant, protected against further collapse
by it neutron degenerate pressure.

The overshoot is so violent that the elements created will be strewn all
over the region surrounding the star, part of this material will end up in
dust clouds which will become stellar systems ( the shock produced by the
supernova material colliding with a dust cloud may initiate the formation
of a stellar system); this is how the Earth acquired all elements aside from
Hydrogen and Helium. Every bit of tungsten used in our light bulbs came
from a supernova explosion, as all the uranium, gold and silver. All the iron
in your hemoglobin got there through a supernova explosion, otherwise it
would have remained locked into the deep interior of some star.

The most famous supernova was observed by Chinese astronomers more
than one thousand years ago (see Sect. ??), its remnants are what we call
the Crab nebula (Fig. ??). We also met another important supernova (see
Sect. ??) observed by Tycho Brahe in 1572 (Fig. 9.6). In 1987 a star in
our galaxy "went" supernova, since then we have observed the ejecta from
the star and the remnant of the core (Fig 9.7. There are, of course, many
known supernova remnants (see, for example, Fig. 9.9). The evolution of a
middle-size star is illustrated in Fig. 9.8.

Figure 9.6: An X-ray photograph of the remnant of Tycho's supernova.

The Crab

lebula. The Crab nebula in Fig.

7? .

the

nant of a s

ipernova explosion. The explosioi

was

obse

rved

on July 4,

1054 A.D. by Chinese astronomers.

and

was

perhaps about as bright as the Full Moon,

and

was

visi-

ble in daylight for 23 days. It was probably

a 1st.

reco

rded

by Anasaz

Indian artists (in present-day Ne

wM

'xio

and

Arizona),

s findings in the Chaco Canyon

National Park

(NM) indie

After gravity is balanced, and after the exterior shells are ejected the
star stabilizes forever. But not without some fancy footwork: the remains
of the star usually rotates very rapidly (up to 30 times per second!) and it
also possesses a very large magnetic field. These two properties cause it to
emit X-rays in a directional fashion, sort of an X-ray lighthouse. Whenever
the X-ray beam goes through Earth we detect an X-ray pulse which is very
regular since the star's rotation is regular. This is called a pulsar. As time
goes on the rotation rate decreases and the star dies a boring neutron star.
Neutron stars are very compact objects having radii of about 10 km (6 miles)
so that their density is enormous, a teaspoon of neutron-star material would
weigh about 10 12 (one trillion) tons on the Earth's surface.

Figure 9.7: Left: a picture of the supernova 1987A remnant (the most recent
supernova in our galaxy. Right: photograph of the core.).

9.3.5 The heavyweights

But what happens for stars heavier than about 3-4 solar masses? In this
case the pressure from the squashed nuclei cannot stop the gravitational
attraction and collapse continues. In fact no known effect can stop the
collapse and it will go on and on until the star collapses to a point. This
how a black hole is created (see Sect. ??).

For this object the gravitational force is so big that even light cannot
leave its vicinity: as mentioned in section ??, if a light beam comes too close
to the center of such an object, the bending effect is so severe that it spirals
inwards. Light emitted from up to a certain distance will be bent back into
the star. This distance defines a horizon: nothing inside the horizon can
ever come out, nothing that crosses the horizon ever leaves the black hole.
The more massive the black hole, the larger the horizon.

For a very massive black hole an astronaut may cross the horizon without
feeling any personal discomfort, only later he realizes that he is inside a
cosmic Venus fly-trap (or roach motel 4 ) out of which there is no escape.

General relativity together with our knowledge of subatomic physics
guarantees that a sufficiently large star will eventually collapse to the point
where a horizon appears. The manner in which such a star evolves thereafter
is impossible to know since no information from within the horizon can be

4 You check in... but you never check out

The life of a star between 1.4 and 3 solar masses.

Hydrogen burning

Initial condensation

Rapidly rotating pulsar

Figure 9.8: Time and life of a star of mass between 1.4 and 3-4 times the
solar mass (between about 3 x 10 27 and 7 x 10 27 tons).

sent to the outside universe. There might be some new kind of effects which
will stop the collapse of even the most massive stars, but even then the
horizon will remain. The point is that our present knowledge of physics pre-
dicts the existence of black holes, even if we do not know all physical effects
in Nature. The fact that we have several excellent black-hole candidates
supports (albeit indirectly) our understanding of gravitation and physics in
general.

The detection of black holes is difficult: one looks not for the object itself
but for certain characteristics of the radiation emitted by matter falling
into the black hole; see Fig. ??. Anything coming near the black hole
will be strongly attracted to it, it will swirl into the black hole, and in
the process it will heat up through friction, this very hot matter emits
electromagnetic radiation in a very characteristic way and it is this patter
what the astronomers look for (see Sect. ??).

Figure 9.9: A radio picture of the Cassiopeia A nebula, a supernova remnant.

The best candidate for a black hole was, for a long time an object in the
constellation Cygnus and is called Cygnus XI. Very recently (May 1995) an
object with the name GRO J1655-40 in the constellation of Sgittarius be-
came an excellent black-hole candidate. In this object a star is accompanied
by an object that emits no light, there is material falling into the compan-
ion and the X-rays from this material are unique to black-holes. Moreover,
the mass of the companion can be determined to be heavier than 3.35 solar
masses. The companion has then all the properties of a black hole.

Black holes are also supposed to be the engines at the center of active
galactic nuclei and quasars (see Fig. 9.11). These are very distant objects
which, by the mere fact of being detectable on Earth, must be immensely
luminous. So much so that nuclear energy cannot be the source of that
much radiation (you'd need more nuclear fuel than the amount of matter in
the system). On the other hand, a black hole of several million and up to
a billion solar masses can, by gulping down enough stellar material (a few
suns a year) generate in the process enough energy.

The life of a star heavier than 3 solar masses.

™ £ IIIB

Hydrogen burning

burning: red giant phase

w

^ii%1

Black hole with material swirling into it

iiuiiiiaiiiiiii

A***ffr\£

Catastrophic collapse

Figure 9.10: Time and life of a star of mass heavier than 4 solar masses
(8 x 10 27 tons).

Figure 9.11: A disk of accreting matter onto a very compact object believed
to be a black hole. The object at the center of M87 (located 50 million light-
years away in the constellation Virgo) weights about three billion suns, but is
concentrated into a space no larger than our solar system. The black hole is
surrounded by a disk of matter which is being sucked into the center; as the
matter falls in it radiates, and the emission from two regions are measured.
Using the Doppler effect one can calculate the velocity of the material falling
in; the region label;ed "approaching" emits blue-shifted light, while light
from the "receding" region is red-shifted. The speed of the gas is enormous:
1.2 million miles per hour (550 kilometers per second).