I . B . KJf R I P L V I C H
General
Relativrh 7
.pringer
General Relativity
LB. Khriplovich
General Relativity
4y Springer
I. B. Khriplovich
Budker Institute of Nuclear Physics
Novosibirsk, Russia 630090
LB. Khriplovich@inp.nsk.su
Library of Congress Control Number: 2005926814
elSBN: 0-387-27406-5
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Preface
The book is based on the course on general relativity given regularly at the
Physics Department of Novosibirsk University. The course, lasting for one
semester, consists of 32 hours of lectures and 32 hours of tutorials, plus home-
work of 10 - 12 problems. The exam is passed by 30 - 35 students. The results
of the homework and exam give good reasons to believe that at least 20 - 25
of these students really digest the subject.
The course requires of students the knowledge of analytical mechanics and
classical electrodynamics, including special relativity. Only chapters 7 and 10
of the book are in this respect exceptions: the acquaintance with the notion
of spin is useful for studying chapter 7, the fundamentals of thermodynamics
and quantum mechanics are necessary for the last chapter. But these parts of
the book can be skipped without any loss for understanding all other chapters.
The book (as well as the course itself) is influenced essentially by the
monograph by L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields,
(Butterworth - Heinemann, 1975). However, I strived to make the exposition
as close as possible to a common university course of physics, to make it
accessible not only for theorists.
The book is also influenced by the course of lectures by A.V. Berkov and
I.Yu. Kobzarev, The Einstein Theory of Gravity, (Moscow, MEPhI, 1989, in
Russian). In particular, I borrowed from it the derivation of the equations of
motion from the Einstein equations (going back to P.A. Dirac and L.D. Lan-
dau), the derivation of the Schwarzschild solution (belonging to H. Weyl), as
well as the discussion of cosmology.
However, the book contains a lot of material absent in the above sources.
Of course, the selection of these topics was determined to a large extent by my
own scientific interests. Among these subjects are the gravitational lensing,
the signal retardation in the gravitational field of the Sun, the Reissner -
Nordstrom solution, some spin effects, the resonance transformation of an
electromagnetic wave into a gravitational one, the gravitational radiation of
ultrarelativistic particles, the entropy and temperature of black holes.
The book contains many problems.
vi Preface
In fact, a considerable part of the content of the book was not presented
at the lectures, but was discussed at the tutorials. Moreover, in some cases
the succession of presentation is dictated by the necessity to create in good
time a necessary basis for tutorials and homework.
It is worth mentioning also that some questions considered in the book are
sufficiently difficult, though they require no extra knowledge. Usually these
questions are discussed neither at the lectures nor at the tutorials. There are
also difficult problems which are not obligatory. All this material is intended
for an independent work of those students who are most seriously interested
in the subject.
One cannot overestimate the imprint made on the book by the collabo-
ration with A.I. Chernykh and V.M. Khatsymovsky in teaching general rel-
ativity, this collaboration lasted for many years. In particular, some prob-
lems in the book belong to them. A.I. Chernykh, V.M. Khatsymovsky, and
V.V. Sokolov also made many useful comments on the manuscript.
The lively interest of numerous students was extremely important for me.
Some original results presented in the book were obtained in collabora-
tion with R.V. Korkin, A. A. Pomeransky, E.V. Shuryak, O.P. Sushkov, and
O.L. Zhizhimov.
To all of them I owe my deep and sincere gratitude.
In the fall of 2003, I lectured on general relativity at Scuola Normale
Superiore, Pisa, Italy. The major part of translating the book into English was
done during this visit. I recall with gratitude the warm hospitality extended
to me at Scuola Normale and the lively interest of its students to my lectures.
Novosibirsk, Iosif Khriplovich
October 2004
Contents
1 Introduction 1
2 Particle in Gravitational Field 5
2.1 Electrodynamics and Gravitation 5
Problem 6
2.2 Principle of Equivalence
and Geometrization of Gravity 6
2.3 Equations of Motion of Point-Like Particle 7
2.4 The Newton Approximation 9
3 Fundamentals of Riemann Geometry 11
3.1 Contravariant and Covariant Tensors. Tetrads 11
Problems 13
3.2 Covai'ianl Diiierent iat ion 13
Problems 15
3.3 Again Christoffel Symbols and Metric Tensor 15
Problems 17
3.4 Simple Illustration of Some Properties
of Riemann Space 18
3.5 Tensor of Curvature 19
Problems 21
3.6 Properties of the Riemann Tensor 21
Problems 23
3.7 Relative Acceleration of Two Particles
Moviu; 1 , Alon-; Close Geodesies 24
4 Einstein Equations 27
4.1 General Form of Equations 27
4.2 Linear Approximation 28
4.3 Again Electrodynamics and Gravity 29
4.4 Are Alternative Theories of Gravity Viable? 31
viii Contents
5 Weak Field. Observable Effects 33
5.1 Shift of Light Frequency
in Constant Gravitational Field 33
5.2 Light Deflection by the Sun 34
5.3 Gravitational Lenses 35
Problem 38
5.4 Microlenses 38
Problem 39
6 Variational Principle. Exact Solutions 41
6.1 Action for Gravitational Field.
Energy-Momentum Tensor of Matter 41
Problems 44
6.2 Gravitational Field of Point-Like Mass 44
Problems 46
6.3 Harmonic and Isotropic Coordinates.
Rclal ivistic Correction to the Newton Law 46
Problems 48
6.4 Precession of Orbits in the Schwarzschild Field 49
Problems 51
6.5 Retardation of Light in the Field of the Sun 51
Problems 53
6.6 Motion in Strong Gravitational Field 54
Problems 56
6.7 Gravitational Field of Charged Point-Like Mass 57
7 Interaction of Spin with Gravitational Field 61
7.1 Spin-Orbit Interaction 61
Problem 62
7.2 Spin-Spin Interaction 63
Problems 65
7.3 Orbit Precession Due to Rotation of Central Body 66
Problems 67
7.4 Equations of Motion of Spin in Electromagnetic Field 67
Problems 71
7.5 Equations of Motion of Spin in Gravitational Field 71
Problems 75
8 Gravitational Waves 77
8.1 Free Gravitational Wave 77
Problems 80
8.2 Radiation of Gravitational Waves 80
Problems 84
8.3 Gravitational Radiation of Binary Stars 84
Problems 85
Contents ix
8.4 Resonance Transformation of Electromagnetic Wave
to Gravitational One 86
Problem 87
8.5 Synchrotron Radiation of Ultrarelativistic Particles
Without Special Functions 87
Problem 90
8.6 Radiation of Ultrarelativistic Particles
in Gravitational Field 90
Problems 92
9 General Relativity and Cosmology 93
9.1 Geometry of Isotropic Space 93
Problems 96
9.2 Isotropic Model of the Universe 96
Problems 99
9.3 Isotropic Model and Observations 99
Problem 101
10 Are Black Holes Really Black? 103
10.1 Entropy and Temperature of Black Holes 103
Problem 109
10.2 Entropy, Horizon Area, and Irreducible Mass.
Holographic Bound. Quantization of Black Holes 109
Problems 114
Index 115
Introduction
General relativity (GR) is the modern theory of gravity relating it to the
curvature of the four-dimensional space-time.
In its so to say classical version, the theory of gravity was created by New-
ton as early as in the seventeenth century and up to now it serves mankind
iai Mi fully. It is quite sufficient for many, if not most, problems of modern
astronomy, astrophysics, and space research. Meanwhile, its inherent flaw of
principle was clear already to Newton himself. This is a theory with action at
a distance: in it the gravitational action of one body on another is transmit-
ted instantaneously, without any retardation. The Newton gravity is related
to general relativity in the same way as the Coulomb law is related to Maxwell
electrodynamics. Maxwell has expelled action at a distance from electrody-
namics. In gravity it has been done by Einstein.
One should start with the remarkable work by Einstein of 1905 where
special relativity was formulated and development of the classical electrody-
namics was completed. Certainly this work had its predecessors, one cannot
but mention among them Lorentz and Poincare. Their papers contain many
elements of special relativity. However, clear understanding and a complete
picture of the physics of high velocities appeared only in the mentioned work
by Einstein. This is no accident that up to now, in spite of the existence of
many excellent modern textbooks, this work can be recommended for the first
acquaintance with the subject even for freshmen.
As to GR, all its fundamentals were created by Einstein.
However, the anticipation that physics is related to the curvature of
space can be found in the works by remarkable scientists of nineteenth cen-
tury: Gauss, Riemann, Helmholtz, Clifford. Gauss came to the ideas of non-
Euclidean geometry somewhat earlier than Lobachevsky and Bolyai, but did
not publish his investigations in this field. Gauss not only believed that "the
geometry should be put in the same row not with arithmetics that exists
purely a priori, but rather with mechanics". He tried to check experimentally,
by means of precision (for his time) measurements, the geometry of our space.
The ideas by Gauss inspired Riemann who believed that our space is really
2 1 Introduction
curved (and even discrete at small distances). Strict bounds on the space
curvature were obtained from astronomic data by Helmholtz. And Clifford
thought of matter as a sort of ripples on a curved space.
However, all these brilliant guesses and predictions were evidently pre-
mature. Creation of the modern theory of gravity was inconceivable without
the special relativity, without deep understanding of the structure of classical
electrodynamics, without deep realization of the unity of space-time. As men-
tioned already, GR was created essentially by the efforts of a single person.
The Einstein way to the construction of this theory was long and torturous.
While his work of 1905 "On the Electrodynamics of Moving Bodies" had ap-
peared as if immediately in a complete form, leaving out of sight of readers
long reflections and heavy work of the author, with GR it was the other way
around. Einstein started working on it in 1907, and his way to GR took a
few years. It was a way of trial and error that can be traced at least partially
through his publications during those years. The problem was finally solved
by Einstein in two works presented at the meetings of the Prussian Academy
of Sciences in Berlin on 18 and 25 November 1915.
At the last stage of the creation of GR, Hilbert contributed to it by formu-
lating the variational principle for the gravitational field equations. In general,
the importance of mathematics and mathematicians for GR is truly great.
Its apparatus, the tensor analysis, or the absolute differential calculus, was
developed by Ricci and Levi-Civita. Mathematician Grossmann, a friend of
Einstein, introduced him to this technique.
However, GR is a physical theory and a completed one. It is completed in
the same sense as classical mechanics, classical electrodynamics, and quantum
mechanics. Like those theories, it gives unique answers to physically reason-
able questions and gives clear predictions for observations and experiments.
However, the applicability of GR, as well as that of any other physical theory,
is limited. So, beyond its applicability limits the superstrong gravitational
fields remain where quantum effects arc of importance. Complete quantum
theory of gravity does not exist.
GR is a remarkable physical theory because it is based essentially on a
single experimental fact, and this fact had been known for a long time, well
before the creation of GR (all bodies fall in the gravitational field with the
same acceleration). It is remarkable because it was created essentially by a
single person. But first of all, GR is remarkable because of its unusual internal
harmony and beauty. It is no accident that Landau said: one can recognize
in a person a true theoretical physicist by his admiration experienced at the
first acquaintance with GR.
Until about the middle of 1960th GR was to a considerable degree beyond
the main stream of the development of physics. And the development of GR
itself in no way was too active, being confined mainly to clarification of some
subtleties and theoretical details, as well as the solution of important, but
still sufficiently special problems. I recall a respectable physicist who did not
recommend young theorists to work in GR. He said: "This is a science for
elderly people" .
Possibly one reason is because GR arose in a sense too early, Einstein
was ahead of his time. On the other hand, already in his work of 1915 the
theory was formulated in a sufficiently complete form. No less important is
the fact that the observational base of GR for a long time remained very
narrow. Correspoiuliii; 1 , experiments are extremely difficult. It is sufficient to
recall that experimentalists succeeded in measuring the red shift only in a half
a century after it had been predicted by Einstein.
However, at present GR is devektpiuj 1 , rapidly. This is a result of tremen-
dous progress of observational astronomy and development of the experimen-
tal technique. On the other hand, researches in quantum gravity are in the
forefront of the modern theoretical physics.
I hope that the present volume will serve as a comprehensible introduction
to this exciting field of exploration of Nature.
Particle in Gravitational Field
2.1 Electrodynamics and Gravitation
We start with the comparison between the equations of motion of a point-like
particle in the electromagnetic and gravitational fields. We will compare as
well the equations for these fields.
The equation of motion for a particle of a mass m and a charge e in an
electromagnetic field F^ v is well known:
m^— = eF» v u v . (2.1)
ds
Here u^ = dx^ /ds is the four-velocity of the particle; ds 2 = rj^dx^dx" is the
four-dimensional interval in the Minkowski space; the diagonal metric tensor
in this space is chosen as -q^ = diag (1, — 1, — 1, — 1); the velocity of light is
put to unity, c = 1 .
The equations of the electromagnetic field are
Q^ F »v = AlT f, Ffiv = q^ Av _ q vA ^ _ (2.2)
Here A^ is the electromagnetic vector-potential, and the four-dimensioual
current density of point-like particles (marked by index a) is
r=X> J(r-r (t))<^. (2-3)
In the Lorentz gauge d^A^ = the Maxwell equation (2.2) reduces to
UA V = A-Kf; □ = d^df, = d\ - A . (2.4)
Equations (2.1) and (2.2), taken together with initial conditions for charges
and fields on a space-like surface, determine completely the evolution of a
system. The equations of electrodynamics are linear, for electromagnetic fields
(lie supcTposit ion principle is valid.
6 2 Particle in Gravitational Field
The equations of motion of a point-like particle in an external gravitational
field are
£-*.«'•"■ (»)
In the case of a weak gravitational field the symbol r ' n, V x is expressed as
follows through its potential, symmetric second-rank tensor h^:
r», v *= ^(duh^ + d^h^-df.h^). (2.6)
The equations for a weak gravitational field (in a gauge analogous to the
Lorentz one) are
Uh^ = -167rfc (T M „ , n liV T^) . (2.7)
k = 6.67390(9) • 10 -8 cm 3 • g" 1 • s~ 2 (2.8)
is the Newton gravitational constant, and the energy-momentum tensor of
point-like particles is
r„„ = £„,.»- MO) «^
dt ' v ' '
The similarity to electrodynamics is evident, however the distinction from the
latter is in fact very essential.
The point is that it is the charges that serve as a source of the electromag-
netic field. But the electromagnetic field by itself is neutral, it bears no charge.
As to the gravitational field, its source is energy, however, the gravitational
field possesses energy by itself. Therefore, the gravitational field equations arc
in fact nonlinear. The linear equations (2.6) and (2.7) are valid, as has been
pointed out already, for weak fields only.
2.1. What is the behavior of the current density (2.3) and the energy-
momentum tensor (2.9) under the Lorentz transformations? How does
<5(r — r a (£)) transform?
2.2 Principle of Equivalence
and Geometrization of Gravity
GR is based on a clear physical principle, on a firmly established experimental
fact known already to Galileo: all bodies move in the field of gravity (if the
2.3 Equations of Motion of Point-Like Particle 7
resistance of the medium is absent) with the same acceleration, the trajecto-
ries of all bodies with the same velocity are curved alike in the gravitational
field. Because of this, in a freely falling elevator no experiment can detect
the gravitational field. In other words, in the reference frame freely moving
in a gravitational field there is no gravity in a small space-time region. The
last statement is one of the formulations of the equivalence principle. This
property of the gravitational field is far from being trivial. It is sufficient to
recall that for the electromagnetic field the situation is completely (liitereut.
There exist for instance non charged, neutral bodies that do not feel at all
the electromagnetic field. However, there are no gravilal ionalh -neutral bod-
ies, there exist neither rulers nor clocks that would not feel the gravitational
field. There are no objects that could be identified in this field with straight
lines, as this is the case in the Euclidean geometry. Therefore, it is natural to
believe that the geometry of our space is non-Euclidean.
Still, in the local frame connected with a freely falling elevator the met-
ric remains the Minkowski one, and intervals of time and space coordinates
are measured by usual clocks and rulers. However, it cannot be done in
the whole space-time if the gravitational field is present. The coordinates
x° = t , x 1 , x 2 , x 3 are just space-time labels. They are continuous, i.e. close
values of x M correspond to two close points. The general form of the interval
ds 2 = g^dx^dx"; (2.10)
here and below the summation over repeated indices is implied. The symmetric
second-rank tensor g^ u (x) deli nes i lie Niemann space. Since in a locally inert ial
frame it reduces to 7? M „ = diag (1,-1,— 1,-1), the rank of the matrix g^, v {x)
is 4. and its signature is (—2).
A reasonable physical realization of a coordinate frame in the Riemann
space is collisionless dust. Each dust particle has a space label x m ,m = 1,2,3,
as well as an arbitrary going clock. The coordinates are continuous, and on
some space-like surface we put x° = for all clocks. In such a physically
reasonable metric g 00 > 0, the matrix g mn of the metric on the surface a; =
has the rank 3 and the signature (—3).
The metric created by a well-localized distribution of gravitating masses is
asymptotically flat. However our Universe as a whole could be non-Euclidean.
2.3 Equations of Motion of Point-Like Particle
In special relativity the trajectory of a free point-like particle, moving between
two points A and B, is determined by the variational principle
where ds is the interval in the Minkowski space. Since the action of a grav-
itational field reduces to a change of the space-time metric, in this field the
8 2 Particle in Gravitational Field
variational principle has the same form (2.11), but now ds is the interval in
the Riemann space and is defined by formula (2.10). In other words, in both
cases, in the Minkowski space and in the Riemann space, a point-like particle
moves along a geodesic.
We start with the variation of ds 2 :
Sds 2 = 5(g tlv dx tl dx v ) = d\g fl , / S-<" X dx , 'd.r' / + g fJ _ u (d6x^dx l/ + dx^d5x u ).
By shifting d from dSx^ and d5x u to other factors, i.e. in fact integrating by
parts, and by changing the summation indices, we obtain
2ds5ds = 5x x [{dxg^ — d^g\ v — d^gx^dx^dx" — 2gx v d 2 x v \ .
Then, by going over to four- velocities v/ 1 = dx^ /ds, we obtain in this way
f B 1 f B X
S ds = - / Sx ds[u> i u v {dxg^ -d^gxv- d v g XlJ ) - 2g^ x u^].
Finally, we arrive at the followin;; equation of motion for a point-like particle
in a gravitational field:
^-+r£ A «*u*=0, (2.12)
where
Kx = \ ffH^A + d x g VM - d v g„ x ) , (2.13)
and the contravariant metric tensor g^ v is related to the covariant tensor g vx
as follows: g^ ' g v>c = 5„. The quantity T^ A is called the Christoffel symbol.
One can easily check that in the case of a weak gravitational field, when
the metric deviates weakly from the flat one, g M „ = n^ v + h^ v , |/i M „| <C 1,
these equations go over into relations (2.5) and (2.6), written previously in
section 2.1.
It is useful to introduce the covariant four- velocity vector u^ = g^u" .
Using relations (2.12) and (2.13). as well as the identity
dg^ dx*
— =d„g»„—=d«g, v u ,
one can easily demons! rate that the covariant four-velocity satisfies the equa-
du R _l_dg^
ds ~ 2 dx» UU ■ (2 - 14)
From it, the quite natural assertion follows: in a gravitational field indepen-
dent of some coordinate x^, the corresponding covariant component of the
four-velocity u M is conserved, and with it the covariant component of the
four- momentum p^ = mu^. For instance, in a gravitational field indepen-
dent of time t, the energy E = po is conserved, in an axially symmetric field
independent of <fi, L z = p$ is conserved.
2.4 The Newton Approximation 9
A locally inertial frame at a given point corresponds to such a choice of
coordinates when g^ v = ?7 M „ , r^ x = . There are many such systems, they
are related to each other by Lorentz transformations.
It is sufficiently evident physically that a locally inertial frame can be cho-
sen not only at a point, but on a geodesic as well, i.e. on the whole trajectory
of a point-like particle moving in a gravitational field. Such coordinates are
called normal coordinates on geodesic.
2.4 The Newton Approximation
How is equation (2.12) related to the usual equation of motion of a nonrel-
ativistic particle in a weak gravitational field? Let the particle velocity be
small, »«1, the deviation of the metric from the flat one be small,
9fj,u = Vnv + hpv, | hp V \ -C 1,
and in addition the fields vary slowly with time, i.e.
\dh^ v /dt\<^\dh fJtv /dx m \.
In this approximation equation (2.12) reduces to
-^- - ~r 00 --- d m g o.
Now we require the validity of the Newton law:
here <j> is the gravitational field potential. The natural boundary condition for
a well-localized source of a gravitational field is:
goo — > 1, 4> — > for \x m \ — > oo.
Then
g 00 = 1 + 2(f).
In particular, at a large distance r from a source with a mass M we obtain
2kM
c 2 r
(we have recovered explicitly in this formula the velocity of light c).
The quantity
_ 2k M
r 9 - ~^~
is called the gravitational radius. For the Sun (its mass M Q = 2 • 10 33 g) the
gravitational radius is r ffQ sa 3 km. With the radius of the Sun Rq sa 7 • 10 10
cm, even on its surface the deviation of the metric from the flat one is tiny:
r g Ql 'Rq <, 10 -5 . As to the the gravitational radius of the Earth, its value is
r gm w 1 cm.
Fundamentals of Riemann Geometry
3.1 Contravariant and Covariant Tensors. Tetrads
The considerations presented in the beginning of this chapter are valid for
spaces more general than the space of GR. To emphasize this fact, we use
here for tensor indices the Latin alphabet, but not the Greek one.
Let us consider a change of variables x % = f l {x n ). Under it, the differentials
of coordinates transform as follows:
dx i=^-dx ,k . (3.1)
dx' k x '
A collection of n quantities A 1 , that transform under a change of coordinates
as the differentials of coordinates:
is called a contravariant vector.
Let be a scalar. Its partial derivatives transform otherwise:
di = dx* d^_
dx l dx l dx' k
A collection of n quantities A t transforming under a change of variables as
derivatives of a scalar,
A-g4, (3.4)
is called a covariant vector.
Tensors of higher ranks are defined in an analogous way. Thus, a con-
travariant tensor of second rank transforms as
A%3= dx^dx^ A ' H ' (3 - 5)
().!■'
dx"
' Ox-'
Ox'
Ox"
Ox' k
Ox'
12 '■> Fundamentals of Rjemaui) Geometry
a covariant tensor of second rank as
>k p>„a
(3-6)
a mixed tensor of second rank as
(3.7)
Let us go back now to the interval (2.10). Since ds 2 = g i jdx l dx : > is an
invariant, it is clear that gij is a covariant tensor. It is called the metric
tensor. The tensor g h inverse to g t j, i.e. related to it as
*"«* = *},
is called a contravariant metric tensor.
Let us find now the volume element in curvilinear coordinates. We in-
troduce vector dr, connecting two infinitely close points x % and x l + dx 1 :
dr = G.idx l . Here e^ is the vector tangential to the coordinate line i going
through the initial point x. It is clear that the infinitesimal vector dr can be
described by its components dr" in the local Lorentz (or Cartesian) frame.
The expression for the vector dr can be rewritten as dr a = e^dx 1 . The set of
four linearly independent vectors e° in a four-dimensional space, labeled by
a, is called tetrad.
Obviously, the length squared of the vector dr is dr 2 = (e i ej)dx q, dx J . On
the other hand, it is nothing but ds 2 = g^dx'dx-' . Then it is clear that
g ij = {e i e i ) = e a i e ja . (3.8)
It is well-known that the volume element dV, built on the vectors eida; 1 ,
e2dx 2 , ... , is expressed via the Gram determinant:
dV = Jdet(e i dx' t e j dxi)
(here there is no summation over i, j), or
dV = A /det(e i ej) dx x dx 2 ... dx n
= Jdet(gij) dx 1 dx 2 ...dx n = ^fgdx x dx 2
In GR, where g = det(gy) < 0, the volume element is
dV = \f^g dx 1 dx 2 dx 3 dx A .
3.2 Covariant Differentiation 13
Problems
3.1. Is the coordinate x l a vector?
3.2. Prove by direct calculation that A l Bi is an invariant. Prove the same for
A ij Bij.
3.3. In a Euclidean space covariant tensors do not differ from contravariant
ones. To what property of the rotation matrix does this coincidence corre-
spond?
3.2 Covariant Differentiation
The differential of a vector dA l (x J ) = A'(x J + dx J ) — A'(x J ) is the dillWvnct'
between two vectors taken at two different points. In curvilinear coordinates
vectors transform in different ways at different points (dx l /dx' k in (3.2) are
functions of coordinates). Therefore, here, as distinct from the Euclidean co-
ordinates. dA' is not a vector. To generalize the notion of a differential dA %
in such a way as to make it a vector, one should transport at first the vector
A'(x J ) parallel to itself to the point x J + dxK Let us denote by 5 A 1 its vari-
ation under this parallel transport. Now the difference DA 1 = dA 1 — 5 A 1 is a
vector.
The variation 5 A 1 should be linear not only in dxJ , but in A 1 as well. The
last point is clear from the fact that the sum of two vectors is also a vector.
Thus, 5 A 1 can be presented as
SA i = - rlj A k dx j , (3.11)
where the coefficients r\, are themselves functions of coordinates. In the
Cartesian coordinates all r' lk = 0.
In line with T* fe , the coefficients
r Ujk = g u r) k (3.12)
are used.
Scalar products of vectors, as well as any scalars, do not change under the
parallel transport. Then, from 6 (AiB 1 ) = it follows that
B l 5A t = -A t 5B l = A t rl B k dx j ,
or, since B l are arbitrary,
SAi = r k jA k dx 3 . (3.13)
14 3 Fundamentals ol Riemann Geometry
DA 1 = dA l + r i kj A k dx j = (d j A i + r l kj A k ) dx j ,
DA, = dA, - r k 3 A k dx ] = (djAi - r k i:j A k \ dx j .
Since DA 1 , DA 1 and dx 3 are vectors, the expressions in brackets in these
equations are tensors. These tensors,
A% = — = d,A l + r l kl A k , (3.14)
DA
Ai,j = -j^r = djAi - r%A k , (3.15)
are called covariant derivatives of the vectors A 1 and A,. Of course, in the
Cartesian coordinates, where r k j = , covariant derivatives coincide with
usual ones.
Since the transformation properties of second-rank tensors are the same as
those of a product of vectors, one can easily obtain the following expressions
for the corresponding covariant derivatives:
A il . j = djA il + r l kj A kl + r l kj A ik ; (3.16)
A\.. = djA\ + r l kj A k - rfjAi; (3.17)
Au-j = djA u - r%A kl - r%A lk . (3.18)
The generalization to tensors of arbitrary ranks is obvious. We note that for
a scalar the covariant derivative coincides with the common one.
Since the index referring to a covariant derivative is of a tensor nature,
one can raise it with the contravariant metric tensor and obtain in such a way
the so-called contravariant derivative. For instance,
A^' 1 = g lk A\ k , A: l = g lk Ai. k .
How do the coefficients T ^ transform under a transition from one coordi-
uat c system to another? Comparing the transformation laws for the right-hand
side and left-hand side of equation (3.14), we find
„ k „,] 0x k Ox"" Ox'" d 2 x' r dx k
r k = r n _i_ (3 iq^
13 mn dx a dx* dxJ dx'dxi dx' r ' y ' ;
It is clear now that the coefficients F '^ • behave as tensors only under linear
transformations of coordinates, just as in this case the inhomogeneous term
in the right-hand side vanishes.
Let us note that this inhomogeneous term in the right-hand side of (3.19)
is symmetric in i,j. Therefore, the antisymmetric in i,j combination S k j =
r • • — r ,j transforms according to
3.3 Again Christoffel Symbols and Metric Tensor 15
Q k Q/i ® xk dx' m dx' n
ij= mn d^ ~dx~^ ~dx7 '
and is thus a tensor. S^ is called the torsion tensor.
In virtue of the principle of equivalence, the geometry of our space-time
has a remarkable- property: the torsion tensor vanishes. Indeed, in the locally
inertial frame the space of GR does not differ from the flat, Minkowski one.
In other words, in this frame all the coefficients r* v , together with their
antisymmetric parts 6"f ; , vanish. And since S^ is a tensor, if it turns to zero
in some reference frame, it vanishes identically. The spaces where the torsion
tensor vanishes are called the Riemann spaces. For coordinates and tensors of a
Riemann space we use the Greek indices. In a Riemann space both r* = T*
Problems
3.4. Prove relation (3.19).
3.5. How many independent components has r* ?
3.6. Let a locally inertial frame be given at the point x l = 0. Prove that under
the transi'ormal ion
locally inertial. Calculate
dr% dr) k
dx a dx l
at the point x l = 0.
3.3 Again Christoffel Symbols and Metric Tensor
A covariant derivative of the metric tensor vanishes. Indeed, on the one hand,
DAf, = D{g llv A v ) = Dg^ A v + g^ v DA V .
But on the other hand, as well as for any vector,
DAf, = g^ DA" .
Hence, since the vector A v is arbitrary,
Dg liv = 0, or ff/11/;A = 0. (3.20)
The explicit form of the last equality, with the account for (3.18), is:
l(i :> Fundamentals of Riemam) Geometry
d x g liV -r litVX -r ViliX = o.
Interchanging indices, we obtain
d v g x „ - /\ „„ - r Mi Xv = 0, d^g vX - f\ „„ - r„, a m = 0.
Now, recalling the symmetry r^ yl/X = r^ Xv , we find easily
r x , ^= l - {d v9x ^ + d^g vX - d x g„u) (3.21)
and, correspondingly,
r%, = \ g" X {d„g X » + d»g vX - d x9llv ). (3.22)
Thus, in a Riemann space the coefficients T* coincide with the Christoffel
symbols (2.13) that arise in the equations of motion of a point-like particle
following from the variational principle (2.11). And this is quite natural since
equation (2.12), which can be written as
£)/ = cW + r^ x ii*dx x = 0,
is. in accordance with the principle of equivalence, a covariant general iza I ion
for a Riemann space of the common equations of free motion
We derive now a useful relation for T^ . From the definition of the Christof-
fel symbol, it follows that
n» = \ f X {d v gx„ + d^g Xv - d x9llv ) = \ g» x d v g x>i .
The metric tensor g Xil can be considered a matrix. Let us perform the following
transformations with an arbitrary matrix M:
<51ndetM = lndet(M + SM) - IndetM = lndet[M _1 (M + 8M)]
= In det(J + M- l 5M) = ln(l + Sp M^SM) = Sp M^SM.
Thus,
SpM~ 1 d„M = ^lndetM (3.23)
and
r;=^s^. (3.24)
M V=g
We present two other useful relations:
g^d x g^ = -g^dxg^; (3.25)
3.3 Again Christoffel Symbols and Metric Tensor 17
sTr 1 ^ = - -L= dniy/^gg"") . (3.26)
The covariant generalization of the divergence of a vector is:
(3.27)
It follows in particular from the last equation that in a Riemann space the
application of the Dalembert operator to a scalar <fi is as follows:
</> ; % = -i= d^(^9 9^d v cf>). (3.28)
The Gauss theorem is now
/ d 4 xyf^A^. ll = I dS^y^A". (3.29)
One more useful relation is:
A K v - A v . „, = d u A^ - d^A v . (3.30)
The covariant divergence of an antisymmetric tensor A^ v = — A v>i is
^.^-L^l^iH. (3.31)
V^9
Besides, we have for this tensor
-V; a + A vX , „ + A Xfl . „ = d x A^ + d^A vX + dvA^. (3.32)
3.7. Prove formulae (3.25) - (3.32).
3.8. Is A = det(A flv ) a scalar? Here A^ is a second-rank tensor.
3.9. Calculate the Christoffel symbols for cylindrical and spherical coordi-
nates.
3.10. Present the explicit form of formulae (3.27) and (3.28) in cylindrical
and spherical coordinates.
3.11. Write the Maxwell equations in a Riemann space.
18 3 Fundamentals of Riemann Geometry
3.4 Simple Illustration of Some Properties
of Riemann Space
A transparent intuitive idea of some properties of a Riemann space can be
given with the simplest example of a sphere. Let us consider on it a spherical
triangle, whose sides are arcs of great circles. An arc of a great circle connecting
two points on a sphere is known to be the shortest path between them, i.e. this
is a geodesic. Here we choose as these arcs those of the meridians, differing
by 90° of longitude, and of the equator (see Fig. 3.1). The sum of the angles
of this triangle in no way coincides with n, the sum of the angles of a triangle
Fig. 3.1. Spherical triangle
a plane, but equals to
We note that the excess of this sum of the angles over ir can be expressed via
the area S of the triangle and the radius R of the sphere:
a + /3 + 7 - 7 r=A. (3.34)
Thin relation can be demonstrated to hold for any spherical triangle. We note
as well that the common case of a triangle on a plane follows also from this
formula: a plane can be considered as a sphere with R — > oo.
Let us rewrite formula (3.34) otherwise:
v l a + 13 + J-TT
K= W = ~ s • (3.35)
It is clear from this relation that one can determine the radius of a sphere while
being confined to it, without going from the sphere to the three-dimensional
space into which the sphere is embedded. To this end, it is sufficient to n
3.5 Tensor of Curvature 1!)
the area of a spherical triangle and the sum of its angles. In other words, R
and K are in fact internal characteristics of a sphere. The quantity K, which
is called the Gauss curvature, is generalized in a natural way to an arbitrary
smooth surface:
a -L Q -L -y — 7T
^-fe s • (3 - 36)
Here the angles and area refer to a small triangle on the surface, bounded by
geodesies on it, and the curvature is a local characteristic that changes gener-
ally speaking from point to point. In a general case, as well as for a sphere, K
is an internal characteristic of a surface, independent of its embedding into the
three-dimensional space. The Gauss curvature of a surface does not change
under bending of a surface without tearing or stretching it. So, for instance,
a cylinder can be unbent into a plane, and thus for it K = 0, as well as for a
plane.
It is instructive to look at relations (3.35) and (3.36) somewhat otherwise.
Let us go back to Fig. 3.1. We take at the pole a vector directed along one
of the meridians, and transfer it along this meridian, without changing the
angle between the vector and the meridian (which is zero in the present case),
to the equator. Then we transfer the vector along the equator, again without
changing the angle between them (which is ir/2 now), to the second meridian.
And at last, wo conic back in the same way along the second meridian to the
pole. It can be easily seen that, as distinct from the same transport along a
closed path on a plane, the vector will be finally rotated with respect to its
initial direction by 7r/2, or by
a + [3 + 1 - 7T = KS. (3.37)
This result, rotation of a vector under its parallel transport along a closed
path by an angle proportional to the area inside the contour, is generalized in
a natural way not only to an arbitrary two-dimensional surface, but to mul-
tidimensional non-Euclidean spaces as well. However, in the general case of
an n-dimensional space the curvature does not reduce to a single scalar func-
tion K(x). It is now a more complicated geometrical object — the curvature
tensor, or the Riemann tensor. That is what we will now examine.
3.5 Tensor of Curvature
If x/'is) is a parametric equation of a curve (here s is the distance along it),
then v? = dx^ /ds is the unit vector tangential to the curve. If this curve is a
geodesic, then along it Dw M = 0. In other words, if m m is parallel transported
from the point x^ on the geodesic to the point x^ + dx 11 on it, then it will
coincide with the unit vector u^ + du^, tangential to the geodesic at the point
xJ l + dx 11 . Thus, under the motion along a geodesic the tangential unit vector
is transported along itself.
20 :', Fundamentals of Riemam) Geometry
By definition, under a parallel transport of two vectors the "angle" between
them remains constant. Therefore, under a parallel transport of any vector
along a geodesic, the angle between it and the vector tangential to the geodesic
does not change, i.e. the projections of the transported vector onto geodesic
lines at all points of the path remain constant.
We have seen already that a vector on the surface of a sphere does not
coincide with itself at the inil Lai poinl alter a parallel transport along a closed
contour. Now we will consider a more general problem: we will find the change
AA p of a vector A^ under a parallel transport along an infinitesimal closed
contour in a Riemann space. In the general case this change is written as the
integral § SA^ along the contour. With the account for (3.13), we obtain
AA ll = j> SA p = j> r v pX A v dx x . (3.38)
We transform this integral by means of the Stokes theorem. To this end we
need the values of the vector A p inside the infinitesimal contour of integration.
Strictly speaking, these values are not functions of a point, but depend them-
selves on the path by which this point is reached. However, for an infinitesimal
contour this ambiguity is an infinitesimal quantity of second order. Thus, one
can neglect the ambiguity and define the vector A^ inside the contour via its
values on the contour itself, by means of derivat ives:
Q a A v = r p va A p . (3.39)
Now, recalling again that the area Af pT inside the contour is infinitesimal,
we obtain with the Stokes theorem
AA U = l - [d p (r p T A p ) - drir^AJ] Ar
= - [d p r^ T Af, - d T r<i, p a p + r p T d p A„ - r^ p d T ^] Af pT .
With the account for (3.39), we obtain finally
AA V = l - R^ pT A p Af pT , (3.40)
where
R? vpT = d p r p T - d T r p p + r p p r a VT - r p r r a vp (3.41)
is the curvature tensor, or the Riemann tensor.
An analogous formula is valid also for a covariant vector A v ' . Since scalars
do not change under a covariant transport, we have
A{A V B V ) = AA V B V + A V AB V = AA V B V + A»\ R" vpT B p Af pT
3.6 Properties of the Riemann Tensor 21
= B p (AA^ + l - R p vpT A v Af pr ) = 0.
Since the vector B p is arbitrary, it means that
AA P = -- R^ vpT A v Af pT . (3.42)
The operations of the covariant differentiation do not commute. In partic-
ular,
A P - w „ - A p . „. „ = # T W „ A T , (3.43)
A". Ai;!y -.4' , . I , ;/i = -.R' 5 rAU ,A T , (3.44)
-V; w i, - A pa . v , p = R r pilv A Ta + R T a)lv A pT . (3.45)
These tensor relations can be easily proven in a locally inertial frame.
In a flat space the Riemann tensor vanishes. Indeed, in such a space one
can choose the coordinates in such a way that F^, = everywhere, and hence
i? T = 0. And a tensor vanishing in one coordinate frame, vanishes in any
The inverse statement is also true: if the Riemann tensor vanishes, the
space is flat. Indeed, locally, at a given point one can choose a Euclidean frame
in any space. And with R T pplJ = the parallel transport of the Euclidean
coordinate frame from a given point to any other one is path- independent.
Thus, the Euclidean frame can be built in a unique way in the whole space.
And this means in fact that the space is flat.
Problems
3.12. Prove formulae (3.43) - (3.45).
3.13. What is the form of the Dalembert equation (2.4) in a gravitational
field in the covariant Lorentz gauge where A p . = ?
3.14. In the flat space-time the electromagnetic field strength F pv satisfies
the equation OF pl , = 0. What is the form of the corrcspoudin;^ equation in a
gravitational field?
3.6 Properties of the Riemann Tensor
The antisymmetry of the Riemann tensor in the last two indices,
is obvious from its definition (see (3.40) and (3.41)). To investigate its other
symmetry properties, it is convenient to go over from the mixed components
to covariant ones:
22 '■> Fundamentals of Riemam) Geometry
Rt P „v = 9t„R%^ ■
Going over again into the locally inert ia I frame, one can prove the following
symmetry properties of the tensor R Tpllv :
R rpilv = R^ Tp . (3.47)
The antisymmetry in the first two indices of (3.46) is sufficiently obvious: it
guarantees the conservation of the length of a vector under its transport along
a closed contour. Less obvious is the symmetry under the permutation of the
pairs of indices in (3.47). since the meaning of these pairs is different. The
first one refers to the vector we transport, and the second refers to the site
around which this vector is transported.
Then, the cyclic sum of the Riemann tensor components over three indices,
with the fourth one fixed, vanishes:
Rrpp.1, + Rrpup + Rrvpp = . (3.48)
And at last, there is the Bianchi identity:
R°ppv; r + R a prp; u + R" pvr; „ = 0. (3.49)
By contracting the Riemann tensor in two indices one obtains a second-
rank tensor, or the Ricci tensor. We define it as follows:
r^ = rp wv = d p r p ^ - d v r p w + r p ap r^ - r p av r a w . (3.50)
Any other contraction of the curvature tensor either turns to zero or coincides
with this one up to the sign. The Ricci tensor is symmetric:
The contraction of the Ricci tensor gives an invariant — the scalar curva-
ture of the space
R = /"iJ F . (3.52)
We point out also the differential identity
R^. u = -df.R, (3.53)
that arises under contracting the Bianchi identity (3.49).
Let us find the number of independent components of the Riemann tensor
for a space of an arbitrary dimension n. The tensor R TpinJ is antisymmetric
under the permutations r < — > p, fi < — > v. Therefore, the total number of
independent combinations in an n-dimensional space for both pairs rp and
\iv is n(n- l)/2. On the other hand, the tensor R Tpflv is symmetric under the
3.6 Properties of the Riemann Tensor 23
permutation of these pairs, rp < — > \iv. Hence the total number of independent
combinations of the indices is
1 n(n - 1) [ n(n-l) 1
2 2 [2 J '
However, one should also take into account the cyclic conditions (3.48):
B TPI1V = R Tppv + R Tpup + R Tvpp = .
To find the number of them, note that the tensor B Tppu is totally antisym-
metric. For instance,
B pTpv = R pTpv + R ppvT + R pvrp = -R rppv - R Tpvp - R Tvpp = -B rppu .
It can be easily seen therefore that the total number of independent cyclic
conditions (3.48) is n(n - l)(n - 2)(ra - 3)/4! . Finally, the total number of
independent components of the Riemann tensor is
1 n(n-l) [ n(n-l) 1 _ n(n-l)(n-2)(n-3)
2 2 [ 2 J 4!
n 2 (n 2 -l)
- 12 ' ^ 54 )
In particular, the numbers of independent components of the Riemann tensor
are: 20 for n = 4, 6 for n = 3, and 1 for n = 2.
However, the number of independent components of the curvature tensor
at any given point can be made even smaller. Indeed, the locally inertial (or
locally Euclidean) system at a given point is defined up to rotations. By a
corresponding choice of rotation parameters one can turn to zero n{n — l)/2
components more of the curvature tensor. As a result, the curvature of a four-
dimensional space is characterized at any point by 14 quantities, and that of
a three-dimensional one — by 3 quantities. This consideration does not apply
to two dimensions, where one can choose as the only characteristic the scalar
curvature: a scalar cannot be turned to zero by any rotations.
In a four-dimensional space, under the condition R pv = (it will be
demonstrated in the next chapter that this is the property of the Riemann
tensor in an empty space), the curvature tensor has 10 independent compo-
nents. For any given point of this space the coordinate frame can be chosen in
such a way that all the components of R Tppv are expressed via no more than
4 independent quant it ies.
Problems
3.15. Prove formulae (3.46) - (3.4
2J :> Fundamentals of Rjemaui) Geometry
3.16. Express the Riemann tensor in a two-dimensional space via the scalar
curvature.
3.17. Express the Riemann tensor in a three-dimensional space via the scalar
curvature and the Ricci tensor.
3.18. How is the scalar curvature of a sphere related to the radius of this
sphere?
3.19. Calculate the Riemann tensor, the Ricci tensor, and the scalar cur vat urc
of the surface of a torus.
3.20. Calculate the Riemann tensor of the surface of a cone. Investigate the
integral / y/g d 2 x R near the top of the cone as follows: approximate the top
of the cone by a spherical cap and then let the radius r of the cap tend to
3.21. Choose a locally inertial frame at some point, with this point taken £
! lie origin. Prove that the metric tensor in the vicinity of this point can b
expressed through the Riemann tensor as follows:
3.7 Relative Acceleration of Two Particles
Moving Along Close Geodesies
Let a particle a move in a gravitational field. In the normal coordinates on its
geodesic, the motion of this particle is free:
d 2 x*i
The equation of motion of a particle 6 moving along a neighboring geodesic,
f< +r »(x b )^^ =o,
ds 2 pr ds ds
reduces, to first order in the difference of the coordinates 7/ M (,s) = ' '^'(s) —.v'' (.s)
(this difference is called geodesic deviation), to
ds 2
Then in the normal coordinates on the geodesic of the particle a, the equatio
for the geodesic deviation rf is
g + w^'-o.
3.7 Relative Acceleration of Two Particles Moving Along Close Geodesies 25
This equation can be rewritten in a covariant form valid in an arbitrary ref-
erence frame. We note to this end that the usual derivative of any order along
a geodesic coincides with the covariant one, so that one may write D 2 r] fi /Ds 2
instead of cPij^/ds 2 . Then, in the normal coordinates the Christoffel symbol
on a geodesic vanishes, so that
d T r^u T = —^ = .
Hence in the second term in (3.55) one may substitute
<9„r£ T u T -> (d v r^. - d T r" pu )u T .
The last expression is written in the normal coordinates, and its covariant form
is W pvT u r . In result, we arrive at the following generally covariant equation
of the geodesic deviation:
This equation describes in fact the tidal forces acting on a system of two
particles in an inhomogeneous gravitational field.
Einstein Equations
4.1 General Form of Equations
It is natural to assume that the generally covariant equations of the gravita-
tional field should be second-order differential equations, and that the energy-
momentum tensor T^ v should serve as a source in them. An additional as-
sumption is that these equations should be linear in the Riemann tensor. Then
their general structure is
aR^ + bg^R + cg^ = T fny .
The condition T» v . v = and identity (3.53) dictate that b = -a/2. In this
way we arrive at the Einstein equations
R^ - X - g^R = MkT^ + Ag"" . (4.1)
The coefficient Hnk at T M!/ (k is the Newton constant) guarantees, as will
be demonstrated below, the agreement with the common Newton law in the
correspond in;; approximation. The so-called cosmological constant A is at any
rate extremely small, according to experimental data; therefore, the last term
in the left-hand side of equation (4.1) is usually omitted.
We note that if nevertheless ^4/0, the cosmological term in (4.1) can be
presented as an effective additional contribution
y, V _ A ^
T ~ $^k 9
to the energy-momentum tensor of the matter T M!/ . This contribution is quite
peculiar. As distinct from the energy-momentum tensor of particles with a
rest mass, for t' 11 ' there is no reference frame where only the component r 00
diii'ci/s from zero. As distinct from the energy-momentum tensor of massloss
particles, the trace of t^ v does not vanish: t£ = A/2nk.
On the other hand, in the locally geodesic frame
28 4 Einstein Equations
r""= — if v = — diag(l, -1,-1,-1).
With this diagonal tensor r M1/ , the corresponding effective energy density p A
and pressure pa are as follows 1 :
r^ = dia.g(p A ,p A ,pA,p A ). (4.2)
Clearly, such a peculiar "matter" has also quite a peculiar equation of state:
PA = -p A = -r m = -^- k , (4.3)
i.e. its pressure is negative! Modern data of the observational astronomy give
serious reasons to believe that the cosmological term does not vanish. It is
quite possible that, though being tiny on the usual scale, the cosmological
term is very essential for the evolution of the Universe.
In the absence of matter T^ v = and the Einstein equations (4.1) reduce
to
£"" = 0. (4.4)
The spaces with metric satisfying condition (4.4) are called the Einstein
spaces. Equation (4.1) (in the absence of the cosmological constant) can be
rewritten as:
R^ = 8nk(r^- \gTT^\. (4.5)
The Einstein equations are in essence the content of general relativity.
Problem
4.1. Prove relation
R c0^ v = Swk \ T l«*i0 _ T M/3;«_ ^f g ^ T X;0 _ g n0 T \;a\\
(A. Lichnerowicz, 1960).
4.2 Linear Approximation
In the linear approximation, g^ u = r}p V + h^ v , \hp V \ -C 1, the Ricci tensor is
R^ = - [dpdvhpp + dpdpKp - nhp V - dpd v h pp \.
See, for instance, L.D. Landau and E.M. Lifshitz, Tht ( 'lassii al Theory of Fields,
§35, formula (35.1).
4.3 Again Electrodynamics and Gravity 29
We use for the metric the gauge
fy V - X - d v hw = , (4.6)
analogous to the Lorentz condition d^A^ = in electrodynamics. In this
gauge the Einstein equation reduces in the linear approximation to the usual
wave equation (of course, for a massless field)
-□V = 16^fc ( V " \ V^T XX J . (4.7)
As well as in electrodynamics, in the linear approximation there is no real
difference between upper and lower indices.
Let us consider the case when the source of the field is a body at rest
with density p, i.e. when the only nonvanishiii,", component of the energy-
momentum tensor is Too = P- Then
Ah 00 = 8irkp
aud
hoo (r)
Thus, at large distances from the gravitating mass M we find, as expected,
"J \T-f\-
2k f , „ , 2kM
-—J P( r ) dr = — •
In this gauge, other components of the metric far from the gravitating mass
h 0n = 0, hmn = - 2 -^5 mn . (4.9)
Of course, equation (4.7) has nontrivial wave solutions even in the absence
of sources. The existence of gravitational waves is an important prediction of
general relativity.
4.3 Again Electrodynamics and Gravity
In section 2.1 we pointed out some similarity between electrodynamics and
gravity. Now we wish to turn attention to an essential difference between them.
It is well known that the Maxwell equal ions result in only one scalar condil ion,
that of the electromagnetic current conservation. In no way does the vector
equation of motion of the charge, which has four components, follow from
them. Indeed, when applying to d^F^ = Attj u the operator d v , we obtain
duj v = 0. This single scalar continuity equation tells us not so much about
30 4 Einstein Equations
the motion of the charged particle: only that its world-line does not break
anywhere.
Now, by applying the covariant derivative D/Dx v to the Einstein equation
(4.1), we arrive at the vector equation
T> tv . v = 0. (4.10)
As distinct from the current conservation law, the four equations (4.10)
(fi = 0, 1,2,3 therein) determine completely the motion of particles. Let us
demonstrate it with the example of dust, i.e. a cloud of point-like noninter-
acting particles of small mass, moving in an external gravitational field. The
energy-momentum tensor of dust is T^ v = pu^u v , where p is the invariant
energy density initially defined in the comoving frame. Equation (4.10) can
be rewritten here as follows:
T^. v = (pu»u v ), v = (pu»), v u» + pu».„u v = ( J „ i >'). vV r + p—-=Q. (4.11)
Mull iplj ing the ob1 ained identity by u^ and taking into account that u^u^ = 1
and therefore u^Du^/Ds = 0, we obtain first of all the continuity equation
for the current density of the dust particles
and then the required equation of motion
£-•
The example of dust was chosen for simplicity sake only. For a single particle
as well one can prove that its equations of motion are contained in the Einstein
equations.
This remarkable property of the equations of gravity was formulated by
Einstein as follows: "Matter dictates to space how to bend; space dictates to
matter how to move."
As to electrodynamics, its equations are linear, the superposition principle
is valid therein, the sum of the fields of particles at rest is the solution as well
as the field of each of them. Therefore, if the equations of motion of charged
particles in the electromagnetic field were not given, the charges initially at
rest could stay at rest further. But since the equations of GR are nonlinear,
there is no superposition principle here, so that bodies initially at rest should
start moving. In fact, this argument is closely related to the above derivation
of the equations of motion, based on the existence of four conservation laws for
the tensor equations of gravitational field. The point is that the nonlinearity
of the field equations is an inevitable consequence of their tensor structure.
4.4 Are Alternative Theories of Gravity Viable? 31
4.4 Are Alternative Theories of Gravity Viable?
First of all, the long-range nature of gravity is firmly established, so that it
should be described by a massless field (or at least the rest mass of this field
should be extremely small).
The simplest alternative to the Einstein gravity, one could think about, is a
scalar theory. The relativistic invariance demands that the scalar field should
interact with a scalar characteristic of matter. Such a reasonable characteristic
is the trace Tjf of its energy-momentum tensor. However, for massless parti-
cles, light included, T£ = 0. Thus, in a scalar theory light will not interact
with a gravitational field. However, the light deflection by the gravitational
field of the Sun, the retardation of light in this field, as well as the frequency
shift by the gravitational field of the Earth are firmly established experimental
facts.
The situation with a vector theory is no better. The interactions of particles
and antiparticles with the vector field (as well as in the common electrody-
namics) have opposite signs. But certainly it is not so. Besides, here as well
the neutral photon will not interact with a gravitational field.
To summarize, general relativity, where the gravitational field is described
by a symmetric second-rank tensor, is the simplest theory of gravity consistent
with experiment.
With the best accuracy, of about 0.2% , the predictions of GR have been
checked experimentally for the retardation of light in the field of the Sun (see
section 6.5). Strictly speaking, one cannot exclude that on this level there is
an admixture of a scalar field to the tensor one.
Weak Field. Observable Effects
5.1 Shift of Light Frequency
in Constant Gravitational Field
We start with an estimate for the possible magnitude of the effect. If the
gravitational field of the Earth is meant, then it is quite natural to assume
that the frequency shift of light Au>/u>, as measured by a detector situated at
the height h above the source, should be proportional to this height as well
as to the free-fall acceleration g. Then simple dimensional arguments give
Auj gh
~uj~ ~ ~(? '
where c is the velocity of light.
And now the quantitative consideration. In a constant field (i.e. indepen-
dent of the world time t) the energy E is conserved. It is well-known to be
related to the action S as follows: E = — dS/dt. Exactly in the same way, in
a constant field the wave frequency u> is conserved, and it is related to the
eikonal <f as follows:
UJ ~~~dt'
However, both the clock that is at rest together with the source of light, and
the clock that is at rest together with the detector of light show the proper
time, each one its own. The frequency in the proper time r
m _ m dt uo
dr dt dr y/g^
in the weak »,):avitat ioual Held of the Earth reduces to
34 5 Weak Field. Observable Effects
If the detector is situated at the height h over the source, then the frequency
fixed by the detector will be red-shifted as compared to the frequency of the
source. This shift is (A. Einstein, 1907)
kMh gh
uv(r + /0-ov(r) = - W -^- = - W ^-.
In the final expression we have recovered explicitly the velocity of light c. The
agreement with the initial simple-minded estimate is obvious.
The relative magnitude of the correction is extremely tiny. Even for h ~
100 m it is
^~io--.
For the first time, the effect was measured in the Mossbauer transition in 57 Fe.
The theoretical prediction is confirmed within the experimental error that is
about 1%.
5.2 Light Deflection by the Sun
An obvious dimensional estimate for the deflection angle 6 is
where p is the impact parameter of the wave packet. The result 6 = r g /p
follows also from the naive calculation based on the picture of a fast part iclc
scattered by a small angle by the usual Newton potential.
The weak-field approximation is quite sufficient for the quantitative cal-
culation of the discussed effect. In this approximation the generally covariant
eikonal equation
g> lv d ll Vd v V=
reduces in the centrally symmetric field to
(l + ^) (d t V) 2 " (l " ^) \{dr*) 2 + ^ {d^A = . (5.1)
We use here the solution (4.8) and (4.9) for the metric far away from the
gravitating mass; in this approximation the nonvanishing contravariant com-
ponents of the metric are
.^, g ™ = -s mn (i- r -f).
(o.2)
Then we go over to the spherical coordinates and assume that the motion
takes place in the plane 6 = w/2.
For small r g /r, equation (5.1) is conveniently rewritten as follows:
5.3 Gravitational Lenses 35
(l + 2 ^) (d t Vf - \(d r <P) 2 + i (d^)A = . (5.3)
We look for the solution in the form
<F = -ivt + ujp<f> + il>{r),
where w is the frequency of light. The correspondence of the impact parameter
p to the common integral L of the orbital angular momentum is obvious:
p — > L/fko (we put here the velocity of light c = 1).
The radial part of the eikonal is
ojdrjl
- ^=i&o(r)+2ty(r).
Here i/)o(r) describes the unperturbed rectilinear motion of the packet, and
the small gravitational correction to it is
As usual, the trajectory of the packet is found by differentiating the total
eikonal over the integral of motion:
Thus obtained deviation of the beam of light from the straight line, when its
distance r to the Sun changes from —R to p, and then from p to R (R — ¥ oo),
18^ d 2R 2r g
= — = -2r g — In — = — y - . 5.4
u dp dp p p
For the minimum p close to the Sun radius, the deflection angle 9 is 1.75".
This prediction of GR {A. Einstein, 1915) is confirmed now by observations
with an accuracy of about 1%.
Let us recall that the naive calculation of the effect, based on the picture of
a fast particle deflected by a small angle in the usual Newton potential, gives
a result (see the beginning of the section) that is two times smaller than the
correct one. The discrepancy is no occasion: in the considered ultrarelativistic
problem not only the Newton potential is at work, i.e. the deviation of goo
from unit. Exactly the same contribution to the deflection is given by the
space metric g mn (see (5.1) - (5.3)).
5.3 Gravitational Lenses
Since a star deflects rays of light, it can be considered as a peculiar gravita-
tional lens. Such a lens shifts the image of a source (i.e. of a star) with respect
:',(,
5 Weak Field. Observable- Effects
to its true position. In the simplest case, when the source, lens, and observer
are on the same axis, the image of the source looks as a circle ( O.D. Chwolson,
1924; A. Einstein, 1936). It is convenient to consider at once a more general
problem when the source S is shifted by a distance ( with respect to the axis
lens - observer, L - O (see Fig. 5.1). For simplicity sake, we have approxi-
mated in this figure the real trajectory by a broken line. Since the deflection
angle 6 is small, the distance £ coincides approximately with the impact pa-
rameter p. Then, recalling again that the angles and cd are small, we find
the following relation for the true deflection:
. I
. I
C = £ r -i.e = Z r -is-
2r„
(5.5)
In the mentioned simplest case, when the source, lens, and observer are on
Fig. 5.1. Gravitational lens
the same axis, i.e. when £ = 0, we obtain from (5.5) that the fictitious radius
of the ring, that is the image in the plane of the lens, is
and its angular size equals
Cojiliarv to a possible naive dimensional estimate, this angle falls down not
as the inverse characteristic distances themselves, but only as the square root
of them. Still, the observation of the effect is practically impossible even if
stars serve as both the source and the lens. However the effect gets observable
when the source is a nebula, and the lens is a galaxy (F. Zwicky, 1937). Let
5.3 Gravitational Lent
37
us estimate the angular size of the ring for the case when this lens consists of
10 10 stars with masses on the order of the Sun mass. Let the lens be situated
at a distance on the order of 10 6 light years, or 10 19 km, from us, and the
distance to the source is much larger (i.e. I ~ l s ^> l )- Then
/6- 10 10 _ 4
q> ~ \ r^— ~ 10 rad ~ 10 angular seconds.
Such a resolution is quite accessible for astronomers.
Let us address now a more general case when the lens does not lie on
the axis source — observer. It is convenient here to go over to dimensionless
variables
Z C lo
x= F' y= Tl-
so so «
In these variables equation (5.5) redu(
wil ii i lie oi)\ itiii:-. >\)\\\A ion
(5.6)
Thus, in the general case, when the source S is shifted with respect to the
direction to the lens L, the picture is different. Two images arise (see Fig. 5.2),
one of them, 7i, is situated beyond the ring corresponding to the axisymmetric
picture, another one, I2, is inside the ring. The distance between them,
A = x+ - x- = \Jy 2 + 4 ,
Fig. 5.2. Two images
38 5 Weak Field. Observable Effects
is minimum for y = 0, i.e. for the axisymmetric position of the source, the lens
and the observer. Since for such axisymmetric position both images should
coalesce into a circle, it is clear that for y <C 1 these images appear as arcs.
For the first time a gravitational lens was discovered in 1979. This was
indeed a galaxy creating a double image of a quasar with the angular distance
between its components of about 6 angular seconds. At present, few sources
of radio waves are known w hicli look like two arcs.
5.1. Consider a common optical lens that imitates the deflection of a ray of
light by the gravitational field of a star. How does the thickness of such a lens
change with its radius?
5.4 Microlenses
If the mass of an object, which acts as a lens, is not large, say, less than
the mass of the Sun, to resolve the angle between the images is practically
impossible. Nevertheless, even in this case the effect of gravitational lensing
can be detected, due to the fact that when the images get closer, their total
brightness increases. The bright ncss amplification K results from the growth
of the total solid angle of the observed image as compared to the solid angle
of the real source.
To estimate the effect, let us note that both £ and (, as well as x and y,
are in fact two-dimensional vectors that lie in the planes of the lens and the
source, respectively. Evidently, the vector form of equation (5.6) is
y = X-J. (5.7)
Let us introduce coordinate axes in the planes of the lens and the source. We
will label by the index 1 the axes, that lie in the plane passing through the
source, lens, and observer, i.e. in the plane of Fig. 5.1; they are parallel to
one another. We ascribe the index 2 to the axes orthogonal to the axes 1. The
discussed ratio of the solid angles is obviously
Here <5£i 2(^1,2) are the sizes of the image (source) along the axes 1 and 2.
In the dimensionless variables this ratio is
: \pp\ = \dx 1 /dy 1 \\dx 2 /dy 2 \.
\oyioy2\
5.4 Microlenses 39
Both partial derivatives are taken at y 2 = 0. Therefore, in virtue of (5.7), X\
and yi are related by the same equation (5.6):
1
yi = xi ,
Xl
and the relation between x 2 and y 2 is
x 2
V2 = X 2 - -j .
Thus, for the two different images the discussed relation of the solid angles is
K ± = v r —- / . (5.8)
For both images this ratio grows for small y:
Therefore, the total brightness of the images increases as well:
K = K + + K_ ~ - .
y
What happens when a star, acting as a gravitational lens, passes close to
the line directed from the observer to the source? Even if one cannot resolve
the arising double image, the observed brightness of the source grows as the
lens approaches the line source — observer. This phenomenon, so-called mi-
crolensing, is of a rather special character: the increase of the brightness and
its subsequent decrease are symmetric in time. Then, the brightness changes
in the same way for all wave lengthes (the deflection angle (5.4) is independent
of the wave length). And at last, since the phenomenon is extremely rare, it
has one more distinctive feature: the repetition of the "flash" of a star caused
by microlensing is practically excluded.
Not only the microlensing effect was detected. In this way a new class
of celestial bodies was discovered — dwarf stars of low brightness, so-called
brown dwarfs, which acted as microlenses.
Problem
5.2. Derive relation (5.8).
Variational Principle. Exact Solutions
6.1 Action for Gravitational Field.
Energy-Momentum Tensor of Matter
The action S g for the gravitational field should be an integral over the four-
dimensional space, invariant under any coordinate transformations. It is nat-
ural to require that the field equations, resulting from variation of the action,
should contain derivatives of the metric tensor g^ v not higher than of second
order. Then the integrand of S g should contain derivatives of the metric not
higher than of first order. In other words, it may depend on g^ and _T^„ only.
However, one cannot construct a scalar from these variables. Indeed, by going
over into a locally inertial frame, one can make at any given point the met-
ric flat and the Christ oji'cl symbols equal to zero. However, in fact the scalar
cur vat lire R can serve as the integrand. Though it contains second derivatives
of the metric, it depends linearly on them, so that one can get rid of these
derivatives by means of integrating by parts.
Thus, let us demonstrate that the variation of the action K J d A x y/—gR,
with an appropriate choice of the constant K , results indeed in the Einstein
equations. The variation of the integral gives
5 J d^x^f^R = 5 d 4 x V^gg^R^
= j d^x^-g (sg^R^ + ^p R + g^SR^) . (6.1)
Then (see section 3.3),
Using the identity g^ v g^v = 4, we present the second term in (6.1) as follows:
- g^Sg^R .
12 (i Variational Principle. Exact Solutions
Let us consider the last term of (6.1) in a locally inertial frame. Interchanging
the operations of variation and differentiation, we find in this frame
V^ggTSRp,, = V=gg^S(d p r^ - d v r p w ) = V^dpWSr^ - g pp sr;„).
Of course, as follows from (3.19), the Christoffel symbol is no tensor since
it transforms inhomogeneously under coordinate transformations. However,
according to the same relation (3.19), the variation of the Christoffel symbol
transforms homogeneously,
p ox p ax^ ox"
and therefore is a tensor. Thus, the quantity
u p = g^sr^ - g pp 8r^
is a vector. Therefore, its divergence d p U p , which was written above in a
locally inertial frame, can be rewritten in a generally covariant form:
u p . p = -±=d p (^u p ).
p V=g
In result, the last term in the variation of the action (6.1) reduces to the
integral of a total divergence J d 4 x d p {yf zr gU p ) and hence can be omitted.
In this way we obtain the following variation of the gravitational action:
6S g = KS I d 4 xy/=g~R=K f cPxy/^g (r^ - ^g^Rjdg^. (6.3)
To determine the constant K in it, we need the Einstein equation with the
right-hand side, i.e. with the source. Therefore, let us find the variation of the
action of matter by example, say, of a material point:
SS m = —m 5 I ds = —m 5 I \J 'g p , y dx^dx v
1 f 5g liI/ dx> i dx v 1 f , „ „ c
2 J v /g pT dxPdx T 2 J yM
The last integral over ds transforms with the obvious identity
/"-/"
dt
- l - J d 4 x^g-puV6g^ = - \ j d^x^T^S,
6.1 Action for Gravitational Field. Energy-Momentum Tensor of Matter 43
is the generally covariant mass density, and
T pv = pu p u v (6.5)
is the energy-momentum tensor of a point-like particle. At last, with the
identity
T^ -11 5g pl , = T pT g pp g Tl '5g Pil , = —T pT g pp g pv bg TV = —T pv &g p,v ,
we find
SS m = X - J d 4 Xy/^gT^5g^. (6.6)
Now it follows from identities (6.3) and (6.6) that the variational principle
S(S g + S m ) =
leads to the Einstein equation (4.1) (with vanishing cosmological constant)
under the condition
Let us use relation (6.6) to derive the generally covariant expression for
the energy-momentum tensor of electromagnetic field. The covariant action
for this field is as follows:
S em = f d 4 xy/^L em ; L em = -— F pv F pv = - — F pp F VT g pv g pT .
J WTT WTT
ation (6.6) and
-^ (f^F^ - I g^F pT F"^j . (6.7)
/ d 4 xy/^g R ,
to exclude from it second derivatives. The terms with derivatives of Christoffel
symbols in the integrand
^g-R = ^g-g^R^ = V^gg^idpr^ - d v F p pp + r p p r° v - F p av F a w ) ,
after integrating by parts and omitting total derivatives, reduce to
-r^MV^ggn + nMy/=99? v )-
With the identities
Variation with the account for relation (6.6) and formula (6.2) gives
_ 2 dy/=gL em
Let us come back to the action
*, = "-
I J (> Variational Piiuriplo. Exact Solutions
a^x = sT tX + r \r9 TV + r Ar5 MT = o ,
the last expression reduces to
2V=g~g^(r^ T ri p - r%r pr ) .
Thus, after eliminating second derivatives, the action for the gravitational
field is as follows:
S9 = ~\bk J ^^^"(C^P - ^^pr) ■ (6-8)
Sg can be trans
Problem
6.1. Prove that the action S a can be transformed also to
6.2 Gravitational Field of Point-Like Mass
To solve the problem of the field of a point-like mass, we use the action in
the form (6.8). We express the integrand through components of the metric
possessing the spherical symmetry, and then obtain the field equations by the
direct variation of the action with respect to the functions it depends on. In
this way we do not need to calculate the Ricci tensor entering the Einstein
equation (4.1).
As usual, we take the source for the origin of the reference frame. The
spherical symmetry of our problem means that one can introduce the coor-
dinates xi, X2, X3 in such a way that ds 2 will go into itself under the trans-
formations that look as Euclidean rotations of these coordinates. In this way
we map the three-dimensional physical space onto the three-dimensional Eu-
clidean one. Now the rotations in the physical space arc mapped onto rotations
in the Euclidean space that leave the quantity r = \/x\ + x\ + x\ invariant.
In the Euclidean space there is no difference between co- and contravariant
vectors, so that the use of the coordinates Xi with lower indices does not lead
to confusion. In line with r = Vx 2 , one can construct from x and dx two more
scalars: dx 2 and xebc. Therefore, in the static spherically symmetric case the
interval can be written as
ds 2 = a 2 {r)dt 2 - b{r)dx 2 - c(r)(xdx) 2 . (6.9)
With the change of variables of the type x — > /(r)x, one can make b(r) = 1.
Then the spherically symmetric metric is expressed through two unknown
functions of r:
6.2 Gravitational Field of Point-Like Mass 45
ds 2 = a 2 {r)dt 2 - rfx 2 - c(r)(xdx) 2 ,
5oo = a 2 (r), g mn = S mn - c{r)x m x n . (6.10)
With this choice of the reference frame, the space metric
dl 2 = dx 2 + c(r)(xdx) 2 = dr 2 + r 2 {d6 2 + sin 2 6d(f> 2 ) + cr 2 dr 2
= d 2 (r)dr 2 + r 2 (d6 2 + sin 2 6 # 2 ), d 2 (r) = 1 + c(r)r 2 ,
is such that the infinitesimal arc of a circle in the plane 6 = ir/2 is dl = rd(f>,
i.e. the length of a circle, with its center in the origin, is 2?rr as usual. Sim pic
calculations demonstrate that in metric (6.10) only the following components
of the Christoffel symbol r p _ M „ do not vanish:
Amo = -A, oo = aa' ^ , r iJk = - (c Xi 6 jk +\°- x iXj x k j .
Obviously, 5 00 = 1/a 2 , so that
r o _ «^
a r '
To find _T 00 and r'- k , let us consider the expression g km g m n x n- On the one
hand, due to the identity g km g mn = S k , it equals x k . On the other hand,
a direct calculation gives g m nX n = — {5 m n + cx m x n )x n = —d 2 x m , so that
g km g m nX n = —d 2 g km x m . It is clear now that
Thus we find easily
r , aa' x t Xi ( . ^ 1 d \
At last, let us note that due to the spherical symmetry of the problem,
it is sufficient to calculate the integrand of the action at a single point,
x\ = r, X2 = X3 = 0. Only the following components of the space metric
and the Christoffel symbols are nonvanishing in this point:
511 = -d 2 , g 2 2 = 533 = -1 , 9 11 = -~h, g 22 = g 33 = -1,
Jll ~ d 2 [ C+ 2 Cr )~ 2d* ~ d-
When substituting these expressions into formula (6.8), the terms
46 6 Variational Principle. Exact Solutions
9 m in T n p -r p m r T pT ) and g^in^-r^r^)
cancel, and other terms produce
S a = -±f **(«**)' 5 = - 4 / dtdriadYr (l - l) .
It is convenient now to introduce new independent functions u = r (1 — 1/d 2 ) ,
w = ad. Then the variation of the action
kl<
is trivial and gives w = ci, u = ci- Coming back to the old functions, we find
easily
da= ( 1 -7)" 1 ' a2=c K 1 -?)-
Since a 2 enters the interval ds 2 only through a 2 dt 2 , by changing the scale of
time we can put c\ = 1. And finally, recalling that at large distances from a
gravitating mass M g o = 1 — 2kM/r, we obtain c 2 = 2fcM = r ff . In this way
we arrive at the metric for a gravitating point-like mass (K. Schwarzs child,
1916):
dg2 / _ 2^M\ rfi2 _ / _ 2^M\ X ^ 2 _ r2{d()2 + g . n2 ^ 2) (f; n)
6.2. Find the surface of rotation on which the geometry is the same as that
on the "plane" passing through the origin in the Schwarzschild solution.
6.3. Find the spherically symmetric solution of the Einstein equations with
the cosmological constant. Estimate the upper limit on the value of this con-
stant, following from the fact that for Pluto (the radius of the orbit of this
planet is ~ 10 15 cm) the Kepler laws are valid with an accuracy better than
10~ 5 . Formulate this upper limit for the corresponding effective mass density
t 00 (see section 4.1).
6.3 Harmonic and Isotropic Coordinates.
Relativistic Correction to the Newton Law
Let us note now that the space part of the metric (6.11) does not go over for
r — ¥ oo into the solution g mn = —5 mn (l + 2kM/r), obtained in section 4.2
for the case of a weak field. The reason is that metric (6.11) and the above
and Isotropic Coordinates. Relativistic Correction to the Newton Law
weak-field solution correspond to different choices of the radial coordinate.
The simplest way to reproduce that weak-field limit, starting with metric
(6.11), is to shift in (6.11) the radial coordinate as follows:
r^-r + kM. (6.12)
With this shift we arrive at the interval
ds2 = rrl^ dt2 -T^^ dr2 - (1+fcM/r)2( ^ 2+sin2 ^ 2) ' (6J3)
or
ds2 = l T^r dt2 -^ kM >^ d * 2
(kM\ 2 1 + kM/r frdrV
The metric
1 - kM/r
900= 1 + kM/r' 5 °" = °'
of interval (6.14) (or (6.13)) not only agrees with the linear harmonic gauge
dfihfn, — j 9„ft w = of section 4.2. It satisfies a more general harmonic
condition
d lt (y/=gg'"')=0, (6.16)
that is not confined to the weak- [it-Id a ppr< >xi in at ion. The coordinates satisfy-
ing condition (6.16) are called harmonic.
On the other hand, by substitution
we obtain from (6.11) such an expression for the interval, where the space
metric is isotropic. It differs from the Euclidean space metric by an overall
factor only (i.e. is conioruiall.v Euclidean):
Obviously, the asymptotics of this metric for r 3> r g also coincides with that
found in section 4.2.
Let us find now the relativistic correction to the gravitational interaction of
two bodies with comparable masses mi and m 2 . Dimensional arguments (re-
call that km/c 2 has the dimension of length) combined with the requirement
IN (i Variational Principle. Exact Solutions
of symmetry under permutation mi f-> m 2 , dictate that the corresponding
velocity-independent correction to the Newton law should have the structure
To find the dimensionless numerical constant a in this expression, we ex-
pand the Lagrangian for a light particle of mass m\ in the gravitational field
of a heavy body with mass rri2 in harmonic coordinates to first order in 1/c 2 :
, 2 miv 2 fcm 1 m 2 1 k 2 m 2 m2
L = -m 1 ^fg tiu u^u u = -m x c + — h ^2~
Thus, in the case of a heavy mass m^, the static gravitational potential
t,r(0)( r )_ fcTOlW2
acquires the relativistic correction k 2 m\m,\l'lc 1 r 1 .
For comparable masses m\ and m2, restoring the symmetry between m\
and rri2, we arrive at the relativistic correction (^4. Einstein, L. Infeld, B. Hoff-
mann, 1938; A. Eddington, G. Clark, 1938)
^ )(f)= uV^ 1 + ro2) | (62Q)
For the derivation of correction (6.20), it was rather crucial to use the
harmonic coordinates satisfying subsidiary condition (6.16) since this con-
dition does not violate the required symmetry under m\ f-> m 2 . As to the
Schwarzschild coordinates, with their origin chosen at one of the particles,
they are not appropriate for this problem.
Let us note, however, that one can arrive at correction (6.20) starting with
the isotropic coordinates.
6.4. Prove that metric (6.14) satisfies the harmonic condition (6.16).
6.5. Derive transformation (6.17) that changes the Schwarzschild coordinates
into the isotropic ones.
6.4 Precession of Orbits in the Schwarzschild Field 49
6.4 Precession of Orbits in the Schwarzschild Field
A simple-minded dimensional estimate for the relative magnitude of the pre-
cession is again ~ r g /r, where r is the characteristic radius of the orbit. In
other words, during one unperturbed turn of the radius-vector (by the angle
2-k) the semi-axis of the elliptic orbit precesses by the angle
5<f>^ -IH. (6.21)
We start the quantitative consideration of the particle motion with the
equation connecting its energy E = po with the three-dimensional momentum
g^p^-m 2 = 0. (6.22)
For the solution of the problem, it is convenient to use the isotropic coordinates
(6.18). For a diagonal metric its contravariant components g^ v are inverse to
the covariant ones, so that the explicit form of equation (6.22) is here as
follows:
The motion of a particle in a centrally symmetric gravitational field, as well
as in any other central field, takes place in a plane passing through the origin.
We choose for this plane the plane = tt/2. The energy E and the orbital
angular momentum L are integrals of motion.
Here we go beyond the linear approximation and include terms of second
order in r g /p. Multiplying equation (6.23) by (1 + r g /(4p)) 4 and cxpandiu; 1 ,
thus obtained coefficients in r g / p, we get
Now we put E = m + e, where e is the nonrelativistic integral of energy, and
keep the terms not higher than second order in 1/c (c is the velocity of light,
here we do not write it down explicitly). In the relation, arising in this way,
one can drop e 2 as compared to 2ms. and 4;//: as compared to m 2 in the factor
at r g l p. Obviously, neither of these corrections contributes to the precession
of the orbit. The resulting expression can be rewritten as
50 (> Variational Principle Exact Solutions
Thus, the problem is reduced to the motion in the Newton potential with the
perturbation
P 2
Just the same result follows from the solution of the problem in the Sclnvarz-
schild metric (6.11). A simple calculation 1 demonstrates that this perturbation
results in the rotation of the semi-major axis a by the angle
(6.25)
(1 - e 2) (l-e 2 )
during one turn. Here e is the eccentricity of the unperturbed elliptic orbit.
Our initial estimate (6.21) is confirmed (up to a factor 3/2).
As to the planets of our solar system, the maximum effect should be ex-
pected for Mercury, since the radius of its orbit is the smallest one. However,
even for it the effect is tiny: formula (6.25) gives for the shift of the Mercury
perihelion only 43.0" per century. Nevertheless such an anomaly in the Mer-
cury motion on the level of 45" ± 5" per century, incomprehensible at that
time, had been known to astronomers before Einstein. Its natural explanal ion
was the first triumph of GR. For a long time the rotation of the Mercury
perihelion was the only really observed nonlinear effect of GR. At present this
prediction of GR is confirmed by radar measurements with an accuracy of
about 1%.
Below we present the predictions of GR (first number) and the results of
measurements (second number) for Mercury and other objects. The units are
the same: angular seconds per century.
Mercury: 43.03 , 43.11 ± 0.45 .
Icarus: 10.3, 9.8 ±0.8.
Large eccentricity of the orbit of the asteroid Icarus enhances the effect (see
formula (6.25)), and at the same time allows one to measure the effect with
better accuracy.
One may expect that the effect will be much more pronounced in the mo-
tion of binary stars, since the gravitational fields in these systems are much
stronger. Indeed, careful investigations of the binary pulsar B1913+16 (B
means binary pulsar, numbers refer to the coordinates on the celestial sphere:
See, for instance, L.D. Landau and E.M. Lifshitz, Mechanics, §15, Problem 3.
6.5 Retardation of Light in the Field of the Sun 51
the direct ascension is 19'' 13'", the inclination is 16°) have shown that in this
binary the orbit periastron rotates by 4.2° per year. By the way, in such a
way the masses of the binary components were measured with high accuracy:
1.4414±0.0002 and 1.3867±0.0002 solar masses, respectively. It is no wonder
that the periastron rotation is so large here: though the masses of the com-
ponents are quite comparable to the solar mass, the distance between them,
1.8 x 10 6 km, is small as compared, say, to the radius of the Mercury orbit,
0.6 x 10 8 km.
6.6. Find the orbit precession, due to the relativistic correction, in the at-
tracting Coulomb potential.
6.7. Find the orbit precession, due to the relativistic correction, in the at-
tracting scalar potential, assuming that this potential is introduced into the
equation p^pf, = rn 2 by means of the substitution m — > m + <f>.
6.5 Retardation of Light in the Field of the Sun
The effect discussed in the present section is linear in r g , and from this point
of view should be considered in the previous chapter. However this effect is of
interest not only in relation to the experimental check of GR. Its detailed con-
sideration is quite instructive in the sense of comparison of the Schwarzschild
and harmonic coordinates. Due to it, this section is included in the present
chapter.
So, let us consider i Sir propagal ion of a signal from the point E, i*i = (xi,y),
to the point V, r2 = {x2,y), in the gravitational field created by a mass M.
situated at the point S, ro = 0, (see Fig. 6.1). We mean in fact the influence
of the gravitational field of the Sun on the propagation of a radar signal sent
from Earth to Venus. Hence the notation of the points in Fig. 6.1.
At first we solve the problem in the Schwarzschild coordinates. To this end
the interval (6.11) is rewritten as follows:
ds 2 = ( 1 - ^) dt 2 - dr 2 - — (l - — ) dr 2 = .
With the identity dr = (r • dr)/r, we obtain to first order in r g
( r a r a x 2 \
dt = dx[l+ — + -?-=- .
\ 2r 2r 3 J
The total transit time is
52 6 Variational Principle. Exact Solutio
(*i, y)
(^y)
Fig. 6.1. Radar signal from Earth to Vei
„ln
X2 + T2
xi + n '
(0.20)
(for the location of the planets as in Fig. 0.1, x\ < 0). Obviously, the retar-
dation of the signal AT is described by the terms proportional to r g in this
expression.
In the harmonic coordinates (labeled now with primes to distinguish them
from the Schwarzschild ones) we have, correspondingly,
ds 2 =
1-kM/r' 2 , f „,,.2,/2 fkMY 1 + kM/r' ,,
YTkWF dt {1+kM/r)dx {—) T^M7? dx =°>
and obtain, again to first order in r g ,
dt = dx* (l+^y
Now the total transit time is
T' = x' 2 - x[ + r„ In .
(0.27)
Since we confine to effects of first order in r g , in the logarithmic term here the
difference between the harmonic and Schwarzschild coordinates is neglected,
i.e. primes in this term are omitted.
No wonder that in different coordinates, Schwarzschild and harmonic ones,
the results for retardation,
AT =
„ Id -
.LI - _ ZT\ an d AT'--
„ In
X2 +r 2
xx +n '
respectively, are also different. Indeed, one can obtain formula (0.27) directly
from (0.20) with the change of variables (0.12). Under it, the nonlogarithmic
term in AT is cancelled by a correction ~ r g arising in xi — x\.
6.5 Retardation of Light in the Field of the Sun 53
In other words, formulae (6.26) and (6.27) differ since r and r' therein
correspond to different physical distances (see (6.12)). For instance, let us
consider two circular orbits, such that the numerical value of radius r for one
of them is equal to the numerical value of r' for another. These orbits have in
particular different periods, and the latter are directly observable. 2
Still, the natural question arises: how should one compare the theory with
experiment? The answer is as follows. There is no way to measure x 2 — %i
directly since measuring rods are of no use, and light signals do not differ
from the radar ones. But x^ — x\ can be expressed in terms of r\, r 2 , and
4> (see Fig. 6.1). On the other hand, neither of the last three parameters
can be directly measured with required accuracy. However, n and r 2 can be
expressed via the observable orbital periods, eccentricities, and times elapsed
since the perihelions. To determine the angle <j> one needs also the time elapsed
since conjunction of the planets. It goes without saying that the predictions
for the experiment, obtained in this way from formulae (6.26) and (6.27), are
identical.
The results of measurements of the signal retardation in the gravitational
field of the Sun agree for Venus with the prediction of GR within their accuracy
that constitutes 3 to 4 ( X. The best experiments performed with satellites with
the active reflection confirm this result of GR with the accuracy of 0.2%.
6.8. Derive formula (6.27) from (6.26) with the change of variables (6.12).
6.9. Prove that the third Kepler law is valid for circular orbits in the
Schwarzschild coordinates, but not in the isotropic ones.
6.10. Estimate the correction to the retardation time due to the signal de-
flection by the Sun.
6.11. A particle has an initial velocity vq at infinity and falls radially to a black
hole. How does its velocity change with the distance? Under the assumption
of a weak gravitational field, find the value of v for which the particle ve-
locity remains constant, (it/. Carrixli. 1972; S.I. Blinnikov, M.I. Vysotsky,
L.B. Okun', 2001).
One cannot but recall here the well-known comment by V.A. Fock: "Physics is
essentially a simple science. The main problem in it is to understand which letter
means what."
54 6 Variational Principle. Exact Solutions
6.6 Motion in Strong Gravitational Field
Let us consider now the motion of a point-like particle in a strong gravitational
field. The problem is solved conveniently with the Hamilton-Jacobi equation
g^d„Sd u S-m 2 =0.
For the motion in the plane 9 = n/2 this equation appears in the Schwarzschild
coordinates as follows:
(l - T -f) ~* (d t Sf " (l " y ) (d r Sf - i (d^Sf - m 2 = . (6.28)
Its solution can be presented in the form
S = -Et + L^ + s{r).
We are interested here in the radial motion of the particle, when its orbital
angular momentum vanishes, L = 0. Then
*'(r) = - (l - ^y 1 ^ 2 -- 2 (l-^).
The dependence r = r(t) is found with the usual equation dS/dE = const:
_ _ r dr
4 ~ ^ " ~ L (1 - r ff AVl " (1 " r g /r) m 2 /£ 2 '
Our choice of the sign for the radical corresponds to the motion of the particle
to the center, r decreases with the increase of t. As the initial condition for
t = we choose r = r , r = 0. Now,
For simplicity sake, assume also that rg 3> r g . Then we obtain
r dr^J7- g
f r drJr 3
l r drr g
Tims, from the point of view of a distant observer, the part iclc approaches the
gravitational radius asymptotically, reaching it only for t — > oo. In the course
of the approach the particle velocity dr/dt tends asymptotically to zero. In
the last chapter of the book we will come back to this problem.
Let us consider now the radial propagation of light from a point r to a
point r > r. Here ds 2 = 0, so that dt = dry^\g rr \/^/g 00 , and the time of
light propagation,
At= I '" dr(l- r -^j 1 = r -r + r g ln r °~ r9 , (6.29)
tends to infinity with the initial point r approaching r g . The signal from the
surface r = r g travels for infinite time. Moreover, the frequency of light as
observed by a distant observer, also decreases when the source approaches r g ,
changing according to
w ~ 1 - TjL ■ (6.30)
One factor \J\ — r g /r in this relation arises as usual from ytgoo, and the
second one from the Doppler effect due to the motion of the source to the
center.
However the fall of a particle to the center looks absolutely different for
an observer freely falling together with this particle. The interval of its proper
time is
(r r V 1/2
dr= ^g Q0 dt 2 + g rr dr 2 = ^g 00 (dt/dr) 2 + g rr dr = - - - dr.
\r r J
Clearly, the part icle reaches t Lie Schwa rzscliild sphere dm/in; 1 , liuite proper time
By the way, near the gravitational radius the velocity of this particle, according
to its proper time, tends to c.
After the particle crosses I he Schwarzschild sphere, it moves to the center,
r = 0, and reaches it also during finite time. Here, for r < r g , g 00 becomes
negative, and g rr becomes positive. In other words, inside the Schwarzschild
sphere t becomes a space-like coordinate, and r becomes a time-like one! The
motion of a particle for r < r g shows how the "time" r flows in this region: it
flows to the origin r = 0. But it means that even if one would try to reverse
the direction of motion in the region r < r g , say, by switching on a powerful
rocket, the attempt would fail, regardless of how powerful the rocket is. Inside
the sphere r = r g the motion is possible to the center only.
I Im Mi ehwarzsi hild phcre is the horizon of events, a one-way gate, it
does not let out to a remote observer any signal. Hence the name of such an
object — black hole. 3
It means in particular that the reference frame of a remote observer, iner-
tial at infinity, is incomplete: it does not describe the motion inside the sphere
We will see in the last chapter that this name is not quite
56 6 Variational Principle. Exact Solutio
, in it 5qo turns to and g rr turns to infinity at r = r g . How-
ever, this singularity is special for the Schwarzschild system of coordinates.
The invariants of the metric are regular on the surface r = r g . This is obvious
:or the determinant of the metric tensor, g = — r 4 sin 2 8, and can be proven by
direct calculation for the invariant R^ lJpT R tlv(>T . However, the last invariant
urns to infinity at r = 0. At this point the metric has a true singularity.
To construct a reference frame free of the singularity at r = r g , one can
;ake a set of freely falling pari icles of dust, enumerate them with radial marks.
and choose the proper time of a particle as the time coordinate (G. Lemaitre,
938). Indeed, there is no singularity in this reference frame. But neither there
particles at rest inside the horizon. By the way, in this comoving reference
rame not only the invariant R^ vpT W vpT remains finite at r = r g , but all
components of the Riemann tensor are finite as well. In other words, in this
rame the tidal forces acting upon an extended, non-pointlike body are finite
(see section 3.7).
6.12. Find the radii of circular orbits in the field of a black hole (S.A. Kaplan,
1949).
6.13. Find the cross-section of the gravitational capture by a black hole of a
nonrelativistic (at infinity) particle, and the correction of first order in v/c to
this cross-section (Ya.B. Zel'dovich, I.D. Novikov, 1964).
6.14. Find the cross-section of the gravitational capture by a black hole of an
ultrarelativistic (at infinity) particle, and the correction of first order in I/7
to this cross-section (Ya.B. Zel'dovich, I.D. Novikov, 1964).
6.15. A particle with the velocity Wqo <C 1 at infinity and with the impact
parameter p = 2r g (\ + 5)/v oc , 6 <C 1, is scattered by a black hole and goes
again to infinity. Describe qualitatively the motion of this particle (Ya.B.
Zel'dovich, I.D. Novikov, 1964). What is its velocity near the black hole?
6.16. Ultrarelativistic particle with the impact parameter
p= (3y/3/2)r g (l + S), 5 < 1,
is scattered by a black hole and goes again to infinity. Describe qualitatively
the motion of this particle ( Ya.B. Zel'dovich, I.D. Novikov, 1964). What is its
velocity near the black hole?
6.17. A black hole (serving as a gravitational lens), a point source of light, and
an observer are perfectly aligned, just in this order. Describe qualitatively the
picture seen by the observer around the black hole (D.E. Holz, J. A. Wheeler,
2002).
6.18. Derive relation (6.22).
6.7 Gravitational Field of Charged Point-Like Mass 57
6.7 Gravitational Field of Charged Point-Like Mass
Since any initially charged astrophysical object would lie neutralized rapidly
by the surrounding matter, the case of a charged star is unrealistic by itself.
However, the considered problem is undoubtedly of a methodological interest
as a sufficiently simple, but nontrivial generalization of the Schwarzschild
solut ion.
Even if a point-like source is charged, its metric still has the structure
(6.10). To find in this case the functions a(r) and c(r), let us consider at first
the field of the charge. Obviously, it has no magnetic field, as well as in the
case when there is no gravity. To find the electric field, we use the covariant
Maxwell equation:
F ,lv . li = 4irf. (6.31)
The left-hand side of its zeroth component has the following explicit form:
F"° = -^d m {^)F m0 ).
^ V-g
As to the right-hand side of this component, the invariant charge density
therein is
ft(rH e^f, (6.32)
just as the invariant mass density is given by formula (6.4). Correspondingly,
Arising in this way, equation
d m {^-gF m0 )=ATTe5{v-v{t))
is solved with the Gauss theorem immediately:
V~9F r0 = ^.
Hence the radial electric field is
F Qr = -F r0 = - 9 -^L^=ad^. (6.33)
V~g r r
The action for the electromagnetic field is in this case as follows:
S em = - I J dV^JTOr^ = 1 J dtdrr * _L Fl .
Now, the total action is (in the same variables u= r(l — l/d 2 ) and w = ad)
r ^0r \
-Ifdtdrfr
")8 (i Variational Principle. Exact Solutions
The variation of this action with respect to the metric should be performed
at fixed covariant field components, F 0r in the present case, since just for
them the definition in curvilinear coordinates looks the same as in cartesian
ones: F^ = d^A v — d v A^ ; it contains neither metric, nor Christoffel symbols.
Let us note also that the variation of the total action (including — e / A^dx^
in line with S g and S em ) just with respect to the covariant components A^
results in the Maxwell equation (6.31) in the Riemann space.
The variation of the obtained action (6.34) with respect to u gives w' = 0,
w = C\. As well as in the case of the Schv irz child solution, we put c\ = 1, i.e.
w = ad = 1. Then the variation with respect to w results in v! = kr 2 FQ r /w 2 =
ke 2 /r 2 , or u = r(l - 1/d 2 ) = c 2 - ke 2 /r. It follows now that
„2_ ,-2_ 1 c 2 fee 2
Recalling again that at large distances from the gravitating mass
IkM
5oo = 1 ,
we obtain c 2 = 2kM = r g . In this way we arrive at the metric created by a
charged point- like mass (H. Reissner, 1916; G. Nordtrom, 1918):
, / 2kM ke 2 \ ,
ds 2 = f 1 + — J dt 2
-fl-^i + ^pj 'dr 2 - r 2 (d8 2 + sin 2 9d^ 2 ) . (6.35)
The horizon radius here is the root
r rn = kM + \Jk 2 M 2 - fee 2 (6.36)
2kM ke 2
1 + — =0.
Of course, of the two roots of this equation we have chosen that one which
goes over into r g = 2kM for e = 0. The Reissner - Nordstrom solution has a
physical meaning only for e 2 < kM 2 . The charged black hole with e 2 = kM 2
is called extremal.
It is useful to present another derivation of the Reissner - Nordstrom
solution, a less rigorous one, but one that demonstrates explicitly the origin
of the term ke 2 /r 2 in (6.35). Let us start with the Schwarzschild solution
(6.11). When in line with the point-like mass M , there is a distributed mass
m(r), it is natural to perform in expression (6.11) the substitution
M -> M + m{r) .
6.7 Gravitational Field of Charged Point-Like Mass 59
In the present case m(r) is nothing but the part of the electrostatic energy of
the charge e that is confined inside the sphere of the radius r:
t ^ a f a 2 For e 2 r dr e 2 ( 1 1 \
m (r) = Ait I dr r z -^- = — — = — .
V ' J 8tt 2 J ro r 2 2 \ r r /
As usual, the electrostatic energy of a classical point-like charge diverges lin-
early, and to obtain a finite result one has to introduce a minimum distance
r . The term e 2 /(2r ), arising in this way, corresponds to the classical mass
renormalization, and together with the "bare" mass M combines into the
"observable" mass
And the term -e 2 /(2r) in m(r) leads to the shift
2
M -> M
2r
in the Schwarzschild metric (6.11), thus resulting in the Reissner - Nordstrom
metric (6.35).
Not only do these considerations lead to the correct result, but they are es-
sentially correct by themselves, differing in fact from the first, rigorous deriva-
tion in the following respect only: here we assume from the very beginning
that ad = 1.
Interaction of Spin with Gravitational Field
In the present chapter, we use the term spin for brevity to mean the proper
internal angular momentum of a classical particle, unrelated to its motion as
a whole. In this sense one can talk for instance about the spin of a gyroscope
installed on an Earth satellite (see section 7.2 below).
7.1 Spin-Orbit Interaction
We discuss here the interaction of spin s of a particle with its orbital angular
momentum 1, related to the motion of this particle in a centrally symmet-
ric gravitational field. We assume that the field is weak, i.e. is described by
the potential <f> = —kM/r, where, as usual, M is the mass of a source of a
gravitational field. We are interested here in the interaction linear in spin s.
Since the orbital angular momentum 1 of the particle is orthogonal both to its
radius vector r and momentum p, the spin-orbit interaction, being a scalar,
should be proportional to (Is). It is important that the scalar product (Is) of
two axial vectors is a true scalar (but not a pseudoscalar), which is necessary
in virtue of the invariance under the reflection of coordinates. Then, in a weak
external field the spin-orbit interaction should be proportional to the magni-
tude of this field, i.e. to kM. After it, simple dimensional arguments dictate
the form of the discussed interaction:
where m is the mass of the particle.
We note the correspondence between (7.1) and the operator of spin-orbit
interaction in a hydrogen-like ion with the charge of the nucleus Ze:
^=^(ls) (7.2)
(here the electron spin s and its orbital angular momentum 1 include the
Planck constant h and have the dimension of ad ion, as well as in our classical
02 1 Interaction of Spin with Gravitational Field
problem). Indeed, from the comparison of the Newton interaction kMm/r
with the Coulomb one Ze 2 /r (for the charges Ze and — e), the correspon-
dence is obvious between kMm and Z< 2 , and then between formulae (7.1)
and (7.2). Moreover, the positive sign of the numerical constant in formula
(7.2), originating in fact from the attracting Coulomb interaction, allows one
to suppose that in the gravitational spin-orbit interaction (7.1), originating
from the Newton attraction, a still unfound numerical factor will be positive
as well. This is the case indeed.
Unfortunately, the explicit calculation of this factor is quite tedious. 1
Therefore we present here without derivation the complete formula for the
gravitational spin-orbit interaction (A.D. Fokker, 1921):
(7-3)
The equations of motion for spin are written via the Poisson brackets:
Using the Poisson brackets for the spin components {s,, Sj} = —EijkSk (they
should have the same structure as those for the components of the orbital
angular momentum), we obtain
ds _ 3 kM
~dt~ 2 mc 2 r 3 *■ XS *'
Thus, the spin precesses with the angular velocity
Problem
7.1. In the gravitational field of a central body, a particle describes an ellipse
with semi-major axis a and eccentricity e. Calculate the frequency of the spin
precession averaged over the period. It is convenient to go from averaging over
time to averaging over the angle (j> by means of relations
dt _ d<j> (1-e 2 ) 3 / 2 1 _ 1 + ecos^
a(l-ei) '
s described in the end of this chapter
7.2 Spin-Spin Interaction 63
7.2 Spin-Spin Interaction
Now we discuss the interaction of the spin s of a probe particle with the spin
s of a source of gravitational field. To linear approximation, s influences
only the components /i 0n of the gravitational field of a source. We note at
once that go n = Von + h nn = h 0n - One can easily demonstrate that h 0n = h 0n .
Since we are interested here only in the effects due to the proper rotation of
the source of the field, then all other components of /i M „ can be neglected.
At first we try to guess the general structure of the vector
g= (hoi,h 2,ho 3 ) = (ffoi,5o2,ffo3)-
It enters the interval ds 2 in the combination g 0n dtdx n . Since the interval does
not change sign under time reversal dt — > — dt and is a true scalar (not a
pseudoscalar), the vector g should change sign under time reversal together
with dt, and should be a polar (not axial) vector together with dx n . Due
to the first requirement, g is proportional to So; indeed, spin, like orbital
angular momentum, changes sign under time reversal. However, spin is an
axial vector, therefore, it should enter the expression for the polar vector g
in the combination r x s , where r is the radius- vector of the probe particle.
Then, in the weak-field approximation g should be proportional to the Newton
constant k. And finally, simple dimensional arguments prompt that
g- fc ^?- (7-6)
The direct calculation is not much more complicated. In the stationary
case the equation for ho n is (see (4.7)):
Ah 0n = 16 n k T 0n . (7.7)
We assume that the internal motion in the source is nonrelativistic and rewrite
the right-hand side of this equation as I6irkpv n = —I6nkpv n , where p is the
mass density of the source, and v n are the common, contravariant, components
of the local velocity vector v. It is clear now that equation (7.7) for the vector g
coincides up to notations with the stationary equation for the vector-potential
A in electrodynamics. Using the well-known solution of this last equation, 2
we find easily
g=- 2fcI ^?- ( 7 - 8 )
Now we consider the motion of the vector of spin s of a probe particle
in the gravitational field (7.8). We start with the covariant equation of mo-
tion for spin. The covariant vector of spin of a particle S^ is defined in the
flat space-time as follows. In the rest frame of the particle it lias only space
2 See, for instance, L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields,
§44.
64 7 Interaction of Spin with Gravitational Field
components, i.e. in this frame S M = (0, s), and in any other frame its compo-
nents are found by means of the Lorentz transformation from the rest frame.
The conservation of angular momentum in flat space-time means that the free
covariant equation for spin is
(7.9)
dS*
(in the present chapter we denote the proper time by r). Due to the principle
of equivalence, in the gravitational field equation (7.9) goes over into
Now we rewrite equatic
i (7.10)
dS''
r» T s v u T = o ,
and note that for the present problem of a nonrelativistic probe particle it is
sufficient to put its velocity v = 0. Then, with 5 M = (0, s) and u^ = (1,0),
equation (7.11) simplifies to
^ = -r™ s n = r m , n0S n = c - (\7 n h 0m - v m h 0n )s n ,
f/s
<[V>
Thus, spin
(L. Schiff, 1960)
i such a gravitational field with the angular velocity
, 3r(rs ) -r 2 s
i [V x g] =
= («
(7.12)
The corrcspoudin; 1 , Hamiltonian of the spin-spin interaction appears as lollows:
3(rs )(rs)-r 2 (s s)
(7.13)
Let us note that the spin precesses in such a way as if it were considered in
a reference frame rotating with the angular velocity — u) with respect to the
inertial frame where spin is at rest. In this sense one can talk about "dragging"
the inertial frame with the angular velocity — u) caused by the proper angular
momentum of the source of gravitational field.
7.2 Spin-Spin Infciactiu]] (io
Problems
7.2. Prove relation h 0n = h 0n .
7.3. A thin spherical shell of radius R rotates with an angular velocity fl. Its
total mass is distributed uniformly. Find the metric outside and inside the
shell, assuming that the deviation of the metric from the flat one is small.
Find the angular velocity uj of dragging inertial frames inside the shell.
7.4. Find the contribution to the deviation of beam of light due to the rota-
tion of gravitating center. Assume that the plane of motion of the beam is
orthogonal to the axis of rotation of the center.
7.5. A satellite with gyroscope is on an orbit around the Earth. Estimate the
frequency of the gyroscope precession 1) due to the spin-orbit interaction, 2)
due to the spin-spin interaction with the proper angular momentum So of the
Earth rotation. How should one orient the gyroscope axis with respect to the
plane of the satellite orbit, and the plane of orbit with respect to s , to amplify
in a maximum way the relative contribution of the second effect with respect
to the first one?
Fig. 7.1. Shift of interference fringe
7.6. Beams of light emitted by a source situated at point A, propagate along
the paths ABC and ADC, and interfere on the screen situated at point C (see
Fig. 7.1). At the center of the square ABCD there is a rotating body with the
rotation axis orthogonal to the plane of the square. Estimate numerically I lie
shift of the interference fringes due to the rotation, if the rotating body is the
Earth, and the side of the square equals the Earth diameter {LB. Khriplovich,
O.L. Zhizhimov, 1980).
(i(> 7 Interaction of Spin witli Gravitational Field
7.3 Orbit Precession Due to Rotation of Central Body
Rotation of a central body causes the precession not only of the spin of a
particle, but the orbit of this particle as well. Not only the perihelion, i.e. the
Runge - Lenz vector
A-Ilpx,-*^,
precesses now. as this is the case in the central field due to the nonlinear cor-
rection to the potential. In the present case, due to the noncentral correction
to the field, the orbital angular momentum 1 is not conserved also. It pre-
cesses, and with it the plane of the orbit precesses as well since the normal to
it is directed along 1 (J. Lenac. H. Th/rring, 1918).
The correction to the Lagrangian L = — mds/dt of a particle with mass
m, due to nonvanishing vector g, is
SL = -mc(gv) = - -^3 ([r x v] s ).
The corresponding correction to the particle Hamiltonian is
^i=-j^(sol). (7.15)
Let us draw attention to the analogy between the gravitational effects,
discussed in the present chapter, and effects from atomic physics. For the spin-
orbit interaction (7.3) this analogy has been already mentioned. In the present
case, the spin-spin interaction (7.13) is an obvious analogue of the hyperfine
spin-spin interaction, 3 and (7.15) corresponds to the hyperfine interaction of
the electron orbital angular momentum with the nuclear spin.
Equation of motion for the orbital angular momentum of the particle ap-
pears as follows:
dl n/ n 2k r n
- = {V lsl ,\}= — [s xl],
i.e. the orbital angular momentum of the particle, together with the plane of
its orbit, precesses with the angular velocity
2fc
The angular velocity averaged over the period is
^) = c 2 fl 3 (1 _ e 2)3/2 S °" ( 7J6 )
The time derivative of the Runge - Lenz vector (7.14) is calculated in an
analogous way. Its averaged angular velocity is
3 Of course, we do not mean here atomic s-states where the last
ure, i.e. is proportional to 5(r) instead of 1/r 3 .
7.4 Equations of Motion of Spin in Electromagnetic Field
2fc [ ,1(1 s ) i
(«2>
z 2
Obviously, it can be said that the plane of the orbit, together with 1, also
precesses with the averaged angular velocity (a; 2 )- In other words, (0*2) is the
angular velocity of the precession in space of the ellipse of the orbit as a whole.
Problems
7.7. Prove formulae (7.16) and (7.17).
7.8. What is the form of the gravitational spin-orbit and spin-spin interaction
in the two-body problem for particles with different masses mi, m,2 and spins
Si, s 2 ?
7.4 Equations of Motion of Spin in Electromagnetic Field
In the next section the general problem of the spin precession in an external
gravitational field will be reduced to the analogous problem for the case of an
external eled romagnel ic field. The equal ions of motion for spin of a relativistic
particle in electromagnetic field are not directly related to GR, and besides,
they are well known. 4 However, at least to make the presentation coherent, we
will consider in this section just the problem referring to the electromagnetic
field.
We start with the spin precession for a nonrelativistic charged particle.
The equation that describes this precession is well known:
b=||[sxB]. (7.18)
Here B is an external magnetic field, e and m are the charge and mass of
the particle, g is its gyromagnetic ratio (for electron g sa 2). In other words,
the spin precesses around the direction of magnetic field with the frequency
— (eg/2m)B. In the same nonrelativistic limit the velocity precesses around
the direction of B with the frequency — (e/m)B:
v=±[vxB].
Thus, for .9 = 2 spin and velocity precess with the same frequency, so that the
angle between them is conserved.
Now we are going over to the relativistic generalization of equation (7.18).
We will use here at first the four-dimensional vector of spin 5 M , already dis-
cussed in section 7.2. In the reference frame where the particle moves with
4 See, for instance, V.B. Berestetsky, E.M. Lifshitz, and L.P. Pitaevsky, Quantum
Electrodynamics, § 41.
(iS 7 Interaction of Spin witli Gravitational Field
velocity v, the vector 5 M is constructed from (0,s) by means of the Lorentz
transformation, so that here
S = jvs, S = s + 7 V ^ VS) ■ (7-19)
7+1
Then, just by definition of 5 M , the following identities take place:
S^ = -s 2 (= const), S, i u ti =0; (7.20)
as usual, here u M is the four- velocity.
The right-hand side of the equation for dS^/dr should be linear and ho-
mogeneous both in the electromagnetic field strength F^, and in the same
four- vector 5 M , and may depend also on u^. In virtue of the first identity
(7.20), the right-hand side should be four-dimensionally orthogonal to 5 M .
Therefore, the general structure of the equation we are looking for, is
^V =aF lu ,S v +Pti lt F vX u v Sx. (7.21)
Comparing the nonrelativistic limit of this equation with (7.18), we find
Now we take into account the second identity (7.20), which af1 er differentiation
dS^ du^
^ dr M dr
and recall the classical equation of motion for a charge:
m *pi = eF, v u v . (7.22)
dr
Then, multiplying equation (7.21) by u M , we obtain
Thus, the covariant equation of motion for spin is
^ = ^-F vat S v - -£- (g - 2)u^F vX u„S x (7.23)
dr 2m 2m
(Ya.I. Frenkel, 1926; V. Bargman, L. Michel, V. Telegdi, 1959).
Let us discuss the limits of applicability for this equation.
Of course, typical distances at which the trajectory changes (for instance,
the Larmor radius in a magnetic field) should be large as compared to the
dc Broglic wave length h/p of the elementary particle. Then, the external
field itself should not change essentially at the distances on the order of both
7.4 Equations of Motion of Spin in Electromagnetic Field 69
the de Broglie wave length h/p and the Compton wave length h/(mc) of
the particle. In particular, if the last condition does not hold, the scatter of
velocities in the rest frame is not small as compared to c, and one cannot use
in this frame the nonrelativistic formulae.
Besides, if the external field changes rapidly, the motion of spin will be
influenced by interaction of higher electromagnetic multipoles of the particle
with field gradients. For a particle of spin 1/2 higher multipoles are absent,
and the gradient-dependent effects are due to finite form factors of the particle.
These effects start here at least in second order in field gradients and usually
are negligible.
At last, in equation (7.23) we confine to effects of first order in the external
field. This approximation relies in fact on the implicit assumption that the
first-order interaction with the external field is less than the excitation energy
of the spinning system. Usually this assumption is true and the first-order
equation (7.23) is valid. Still, one can easily point out situations when this
is not the case. To be definite, let us consider the hydrogen-like ion 3 He + in
the ground s-state with the total spin F = 1. It can be easily demonstrated
that an already quite moderate external magnetic field is sufficient to break
the hyperfine interaction between the electron and nuclear magnetic moments
(a sort of Paschen - Back effect). Then, instead of a precession of the total
spin F of the ion, which should be described by equations (7.18) or (7.23)
with a corresponding ion g-factor, we will have a separate precession of the
decoupled electron and nuclear spins.
Let us go back now to equation (7.23). We note that for g = 2 and in the
absence of electric field, its zeroth component reduces to
dr
Taking into account definition (7.19) for So and the fact that in a magnetic
field a particle energy remains constant, we find immediately that the projec-
tion of spin s onto velocity, so-called helicity, is conserved.
We will obtain now the relativistic equation for the three-dimensional vec-
tor of spin s, that directly describes the internal angular momentum of a par-
ticle in its "momentary" rest frame. This equation can be derived from (7.23)
using relations (7.19), together with the equations of motion for a charge in
external field. It will require, however, quite tedious calculations. Therefore,
we choose another way, somewhat more simple and much more instructive.
First, we transform equation (7.18) from the comoving inertial frame,
where the particle is at rest, into the laboratory one. The magnetic field B' in
the rest frame is expressed via the electric and magnetic fields E and B given
in the laboratory frame, as follows:
-V 2
B' = 7 B -
70 7 Interaction of Spin with Gravitational Field
This expression can be easily checked by comparing it component by com-
ponent with the transformation of magnetic field for two cases: when this
field is parallel to the velocity and orthogonal to it, respectively. Then
one should take into account that the frequency in the laboratory time t
is 7 times smaller than the frequency in the laboratory time r (indeed,
d/dt = dr/dt ■ d/dr = "f~ l d/dr). Found in this way contribution to the pre-
cession frequency is
-.~£[»-^»>-H-
However it is clear from equation (7.23) that spin precesses even if
g = 0. To elucidate the origin of this effect, the so-called Thomas preces-
sion (L. Thomas, 1926), we consider two successive Lorentz transformations:
at first from the laboratory frame S into the frame S' that moves with the
velocity v with respect to S, and then from S' into the frame S" that moves
with respect to S' with the infinitesimal velocity dv. Let us recall in this
connection the following fact related to usual three-dimensional rotations: the
result of two successive rotations with respect to noncollinear axes ni and ri2
contains in particular a rotation around the axis directed along their vector
product ni x n2. Now it is only natural to assume that the result of the above
successive Lorentz transformations will contain in particular a usual rotation
around the axis directed along dv x v. In result, spin in the rest frame will
rotate in the opposite direction by an angle which we denote by k [ dv x v ] .
Here a is some numerical factor to be determined below. It depends generally
speaking on the particle energy.
This is in fact the Thomas precession. Its frequency in the proper time r
U,' T = x[dv/dTXv] = K^lE'xv}.
Now we transform the electric field E' from the proper frame into the labo-
ratory one, as it was done above for the magnetic lield B', and go over also
from the proper time r to t. In result, the frequency of the Thomas precession
in the laboratory frame is
= -x— v x E - v 2 B + v(vB) .
m l J
To find the coefficient x, we recall that in a magnetic field, for g = 2 the
projection of spin onto the velocity is conserved. In other words, in this case
the total frequency of the spin precession lo = u) g + Uj- coincides with the
frequency of the velocity precession which is well known to be
u v = -— B.
7.5 Equations of Motion of Spin in Gravitational Field 71
From this we find easily that h = 7/(7 + 1) . Correspondingly, the relativis-
tic equation of motion for the three-dimensional vector of spin s in external
electromagnetic field is
*=( W . + ^)x.= £{(,-2 + ^)[.xB]
- (g - 2) -L- [s x v](vB) - f ff - -^ [s x [v x E]] 1 . (7.24)
7.9. Derive equation (7.24) directly from (7.23).
7.10. Obtain the Hamiltonian of spin-orbit interaction in hydrogen atom from
equation (7.24).
7.11. Derive equation of motion of the quadrupole moment of a relativistic
particle in homogeneous electric and magnetic fields. In the rest frame, the
operator of quadrupole moment is
7.5 Equations of Motion of Spin in Gravitational Field
It has been pointed out in section 7.2 that the covariant equation of motion
for spin is
^=0. (7.25)
However, the notion of spin is directly related to the group of rotations. It is
only natural, therefore, to describe spin in the local Lorentz coordinate frame
using the tetrad formalism (see section 3.1). The tetrad components of spin
(by the first letters of the Latin alphabet, a,b,c,d, we label here and below
four-dimensional tetrad indices) behave as vectors under Lorentz transforma-
tions of the locally inertial frame. However, they do not change under generally
covariant transformations x^ = f tl {x r ). In other words, the four components
S a are world scalars. Therefore, in virtue of relation (7.10), the equations for
them appear as follows:
^=^=S»e«u»= V " b lbcd u d Sc. (7.26)
72 7 Interaction of Spin witli Gravitational Field
The covariant derivative of a tetrad is by definition
e li-.v = 9^e^ — r*„e% ,
and the quanl ii y
7abc= e aiiiV e^e v c (7.27)
is called the Ricci rotation coefficient. By means of covariant differentiation
of the identity e a/J e b ' = ]]„),. one can easily demonstrate that these coefficients
arc antisymmetric in the first pair of indices:
Of course, the equations for the tetrad components of a 4-velocity look
exactly in the same way as those for spin:
^ = V ab 7 bcd u d u c . (7.29)
The meaning of equations (7.26) and (7.29) is clear: the tetrad components
of both vectors vary in the same way since their variation is due only to the
rotation of the local Lorentz frame.
There is a remarkable similarity between the discussed problem and the
special case of g = 2 in electrodynamics. According to equations (7.23) and
(7.22), the four-dimensional spin and four-dimensional velocity of a charged
particle with the gyromagnetic ratio g = 2 precess with the same angular
velocity:
dS a e b du a e b
—T- = — i'abO , -j— = — i'abU ■
dT m dr m
In other words, the obvious correspondence takes place:
It allows us to derive the precession frequency lj of a three-dimensional vector
of spin s in an external gravitational field from expression (7.24) by means of
the simple substitution
— B, — > - - e iknklc u c - ^Ei-^ l0lc u c . (7.31)
Thus, this frequency is {LB. Khriplovich, A. A. Pomeransky, 1998)
The factor 1/u^ in expression (7.32) is due to the transition in the left-hand
side of equation (7.26) to differentiating over the world time t:
7.5 Equations of Motion of Spin in Gravitational Field 73
d dt d d
dr dr dt w dt'
We supply here w° with the subscript w to indicate that this is the world, but
not the tetrad, component of 4- velocity. All other indices in (7.32) are tetrad
ones, c = 0,1,2,3, i,k,l= 1,2,3.
However, in some respect the first-order spin interaction with a gravita-
tional field differs essentially from that with an electromagnetic field. In the
case of an electromagnetic field, the interaction depends, generally speaking,
on a free phenomenological parameter, ^-factor. Moreover, if one allows for
the violation of invariance both under the reflection of space coordinates and
under time reversal, one more parameter arises in the case of electromag-
netic interaction, the value of the electric dipole moment of the particle. The
point is that both magnetic and electric dipole moments interact with the
electromagnetic field strength, so that this interaction is gauge-invariant for
any value of these moments. Only the spin-independent interaction with the
electromagnetic vector potential is fixed by the charge conservation and ;',aiL;',c
invariance. On the contrary, the Ricci rotation coefficients *„;„ entering the
gravitational first-order spin interaction (7.26), as distinct from the Riemann
tensor, are noncovariant. Therefore, the discussed interaction of spin with
gravitational field is fixed in unique way by the law of angular momentum
conservation in flat space-time taken together with the equivalence principle,
and thus it contains no free parameters (L.D. Landau). On the other hand, it
is no surprise that the precession frequency u> depends not on the Riemann
tensor, but on the rotation coefficients. Of course, this frequency should not
be a tensor: it is sufficient to recall that a spin, which is at rest in an inertial
reference frame, precesses in a rotating one.
One can check easily that in the weak-field approximation where
9y,v = Vfiv + V" |V,| < 1,
there is no difference between the tetrad and world indices in e a ^, and the
tetrad appears as follows:
Relation between the tetrads and metric
e a ^eb^V ab = 9\u>
in the weak-field approximation reduces to
e^y + e v/J , = hpv .
Under the demand that tetrads are expressed via metric only, (
the so-called symmetric gauge for the tetrads where
7 1 7 Interact ion of Spin wirli Gravitational Field
Then in the weak-field approximation the Ricci coefficients are:
labc = -{hbc.,a- h ac . b ). (7.33)
Now, with relations (7.32) and (7.33) one can solve, for instance, in an
elementary way (lit 1 problems of spin-orbit and spin-spin interactions for ar-
bitrary particle velocities. The combination of a high velocity for a spinning
particle with a weak gravitational field refers obviously to a scattering prob-
lem. Another possible application is to a spinning particle bound by other
forces, for instance, by electromagnetic ones, when we are looking for the cor-
rection to the precession frequency due to the gravitational interaction. So,
let us consider the spin-orbit and spin-spin problems.
We start with the spin-orbit interaction. In the centrally symmetric field
created by a mass M, the metric is
^=- T i=~, h mn =- r fS mn =- 2 ^S mn . (7.34)
Here the non mi bin Ri< i coefficients are
kM ,r r ^ kM /, o^
lijk = —5- {Ojkn - 6 tk r ) , 7(M) = 5- r % . (7.35)
Pin; 1 ,;',]!!;:; I best- expressions into formula (7.32) yields the following result for
the precession frequency:
In the limit of low velocities, 7 — > 1, the answer goes over into the classical
result (7.4).
And now the spin-orbit interaction. Using expression (7.8) for the compo-
nents of the metric due to the spin So of central body, we find the nonvanisliiu;',
Ricci coefficients:
^o=*(vJ^-vj*^), 70.= -^!^. (7.37)
The frequency of the spin-spin precession is
u, ss =fc( 2 -i)(soV)vi
- k -^— [v(s V) - s (vV) + (vso)V] (W) - . (7.38)
7+1 r
In the low-velocity limit this formula also goes over into the classical result
(7.12).
7.5 Equations of Motion of Spin in Gravitational Field 75
Problems
7.12. Prove identity
e« w a " J abX el = d x e a , - r^ x e ap - 7o6A e* =
for the tetrad. Compare it with identity
SV; a = dxg^ - r p ^ x g p „ - T p Xv g w =
for the metric tensor.
7.13. Is e" v a tensor in the Riemann space?
7.14. Find the frequency of spin precession in the Schwarzschild field for cir-
cular orbits (beyond the weak-field approximation) (T.A. Apostolatos, 1996).
Gravitational Waves
8.1 Free Gravitational Wave
In this chapter (as well as in the previous one) we do not go beyond the linear
approximation to the Einstein equations. 1 In the linear approximation, under
the auxiliary harmonic condition (4.6), the gravitational field is described by
equation (4.7). The solutions of the corresponding free equation are gravita-
tional waves.
We demonstrate first of all that for an arbitrary weak field h^ v {x) one can
always choose such a coordinate transformation
x 'v = x n _|_ e^^x),
after which the transformed field /i^„(x') will satisfy condition (4.6). Indeed,
in virtue of the transformation law (3.6) as applied to the metric tensor g /i;/ =
Vfj.i' + hpv, the following relation takes place
h'^ix) = V(x) - d,e u {x) - d v e^{x). (8.1)
Let us note that since both /i M „ and e M are small, there is no reason to distin-
guish in the arguments of these functions x and x' . Now
d^ix)- l -d v ti^{x) = d^h^{x)- )-d v h^{x)-ne v (x).
Thus, for an arbitrary initial h^(x), by choosing the vector parameters r„(.*')
in such a way that they satisfy the equation
1 Only in section (8.6) we discuss a weak gravitational wave radiated during the
motion in a strong external gra\ itational field.
78 8 Gravitational Waves
Ue v {x) = df,h^(x) - l - d u h^{x) , (8.2)
one can always make h'(x) satisfying the harmonic condition
d^ v (x)- \d u h^(x)=0.
However, this condition still does not fix the reference frame uniquely. Obvi-
ously, one can perform over the field h^ v (x), for which the harmonic condil ion
is valid, a new coordinate transformation (8.1) with parameters £„(x) sal isl'y-
ing the condil ion
Ue v {x)=0.
The harmonic condition, combined with the possibility of this last addi-
tional coordinate transformation, allows one to fix the tensor structure of a
plane wave. So, let h /iu (x) = e fiv e~ lkx . In virtue of wave equation (4.7), the
4- vector fc M satisfies the condition fc 2 = 0. The harmonic condition for the
polarization tensor e^ v appears as follows:
k^e^u k u e^ = . (8.3)
We choose the wave vector as k^ = uj(1, 0,0,1). Then the components v =
a = 1,2 of equation (8.3) give
The sum of the components with v = and v = 3 results in
e aa = eii + e 22 = 0.
Then it follows from the component with v = that
e 03 = « ( e 00 + e 33) •
Now we perform the additional transformation with parameters e v (x)
ic u cr' k ' v :
we turn to zero
respectively. In the result, the polarization tensor has only two independent
components:
8.1 Free Gravitational Wave
en = -e 22 , e 12 = e 2 i
(we omit the primes now).
Let us consider how the 2x2 matrix
transforms under the rotation by angle around the z axis. This transforma-
tion e' ab = O ac Obd,e c d, where
y — sin cos J
is conveniently rewritten as e' = O e T . After it we find easily
e' n = cos 20 en + sin 20 e i2 , e' 12 = — sin 20 en + cos 20 e 12 .
Now we go over from the linear polarizations en, ei 2 to the circular c
For e± this transformation appears as follows:
4=e* 2i *e ± . (8.4)
By the analogy with the quantum of electromagnetic field, the photon,
one introduces the notion of graviton. the quantum of the gravitational Held.
The transformation law (8.4) means that the projection of the total angular
momentum of a graviton onto the direction of its momentum, i.e. onto z axis,
equals ± 2. And since the projection of the orbital angular momentum onto the
momentum vanishes identically, it means that the projection of the graviton
spin onto the direction of its motion, i.e. its helicity, is ±2. Let us recall
that the photon helicity is ±1. The more general assertion is valid: for any
(nonzero) spin s of a massless particle, this particle has only two polarizat ion
states, with helicities ± s.
One more remark concerning the graviton. In chapter 4 it was demon-
strated that the requirement of general covariance fixes strictly the form of
the second-order equation for the gravitational field. In this way, as strictly
fixed is the linear approximation to this equation, and it follows explicitly
from the linear approximation that the gravitational field is massless. The
only additional assumption made, that of the absence of the cosmological
constant, is in fact inessential for this conclusion. It can be easily seen that
in the weak-field limit the cosmological term in the wave equation reduces to
a constant, but not to the additive term —m 2 h^„, which would correspond
to a finite mass. Thus, in the generally covariant second-order equation there
is no place for the nonvanishing graviton mass. There are no experimental
indications of the finite mass of the gravitational field. Its detection would
mean a cardinal going beyond the framework of GR.
80 8 Gravitational Waves
Problems
8.1. Find the components of the Riemann tensor for a plane gravitational
wave propagating along the z axis.
8.2. Find how the relative distance between two particles changes with time
under the action of a plane gravitational wave propagating along the z axis.
Assume that the particles were initially in the plane xy.
8.3. Find the frequency of spin precession in the field of a plane gravitational
wave propagating along the z axis.
8.2 Radiation of Gravitational Waves
We come back now to equation (4.7). Taking its trace and thus expivssiu;',
T\\ through h\\, we rewrite this equation as
-□Vv = 1671-fcT^, (8.5)
where t/v„ = /i M „ — \r) iiv h\\. It is clear from equation (8.5) that the har-
monic condition d^ip^ = agrees in a natural way with the conservation
law d^Tf^, = 0. It should be noted that though equation (8.5) is linear in the
field of a gravitational wave, it is not necessarily linear in a gravitational field
in general. Indeed, it is sufficient to consider a simple case when the source
is a system of nonrelativistic particles bound by gravitational forces. Obvi-
ously, without the account for the stress tensor quadratic in the gravitational
potential </>, T M „ will not be conserved.
The retarded solution of equation (8.5) is
^-*l d ^^
r'|)
As it was shown in section 8.1, in the wave zone h\\ — > 0, so that in this region
ippv ~^ ftju/ and, moreover, h^ u has pure space components only. Besides, here
one can change in the denominator for R 3> r' , as usual, |R — r'| — > R. Thus,
at large distances we have
4k f ,
hmn(R,t) = -— / dr'T mn (r',t-\R-
"'!)•
For a system of nonrelativistic particles, the integrand in the right-hand
side is conveniently transformed in such a way that instead of T mn , which
depends on details of the motion and interaction of these particles, only their
mass distribution enters this integrand. To this end, we at first integrate with
the weight x^ the conservation law d^T^ = 0:
I dr x k (d T 0n + d m T mn ) = d t J dr x k T 0n - f dr T kn = .
With the account for the symmetry T kn = T nk , we rewrite the obtained
relation as
J dr T kn = \dtfdr (x k T 0n + x n T ok ).
In the analogous way, integrating with the weight x k xi the conservation law
d^T^o = 0, we obtain
/ dr x k xi (d T 0Q + d m T m0 ) = d t dr x k xi T 00 - dr (x k T ol + xiT ok )=0.
We recall now that for a nonrelativistic system Too coincides with the mass
density p, and that h nn = 0. Then the result is expressed through the
quadrupole moment q mn of the mass distribution:
= f dr ( 3x m x n - r 2 5 mn ) p . (8.6)
37? Hmn '
The result is quite natural for the following r
If the multipole expansion of the vector-type electromagnetic radiation,
i.e. of the field in the wave zone, starts with the dipole term, one could expect
from the very beginning that for the tensor-type gravitational interaction such
an expansion should start with the quadrupole.
On the other hand, it is well known that for a system of particles with
the same ratio e/m there is neither electric dipole radiation, nor magnetic
dipole 2 . And for the gravitational field the role of e/m is played by the ratio
of the gravitational mass to the inert one, which according to the principle of
equivalence, is the same for all particles. Therefore, there should be no dipole
gravitational radiation.
Now we go over to the calculation of the gravitational radiation intensity.
To this end we need the energy flux density of the gravitational field, i.e. / ( "
component of its energy- momentum tensor (EMT). As to the EMT of matter
Tjjf, it satisfies the condition
TJ!„ = T» u + r*T£ - r p uv T£ = — L dJy/=gT£) - r p uv T£ = .
,1* ,M PM MP JZ^ v p
Due to the term —r p T£ in this relation, the matter EMT Tjf is not conserved,
which is quite natural in the presence of the gravitational field. But then one
See, for instance 1 . L.I). Landau and E.1N L Lifsliitz. Tin ( 'I ass it til Theory of Fields,
§67,71.
82 8 Gravitational Waves
should build from g^ (or /i M „) such a structure t% that would guarantee the
relation
Then we obtain in the standard way the conservation law for the 4-momentum:
/ rfrV^ff (T° + O = const . (8.7)
-/'
However, there is no true tensor t£- Indeed, in virtue of the equivalence prin-
ciple, one can always choose a reference frame in such a way that at any given
point the metric will be flat and its first derivatives will turn to zero. But
then the structure t£, which is built from the first derivatives, turns to zero
as well. For a true tensor it means that the tensor vanishes identically. Still, a
corresponding pseudotensor t%, which behaves as a tensor under linear coor-
dinate transformations, can be constructed, and even not in a unique way. For
an asymptotically flat system the corresponding total energy and momentum
are conserved and defined uniquely by the integrals (8.7).
In the case of interest to us, that of a weak gravitational wave, the con-
struction of the pseudotensor i 1 ', is sufficiently simple. Let us start with the
action (6.8). We note first of all that in our case the term
in the action turns to zero in accordance with the harmonic condition (6.15)
and therefore can be immediately omitted. By the way, it can be easily demon-
strated that the second factor in this term, r T pT , also vanishes for the gravita-
tional wave. Other terms of second order in h w in the action, after integrating
by parts and omitting total derivatives, reduce to
y 647rfc J
We have taken into account here the harmonic condition and the fact that
h^n = 0. Recalling also that /io M = in the wave, we present the integrand,
which is the Lagrangean density for the free gravitational wave, as follows:
Lf= -^-rh mn , p h mn , p . (8.8)
y 647Tfc
Now the energy flux density is calculated in the usual way: 3
BT {2) 1
4 = h mn , o -^— = ^~ h K m h mn . (8.9)
We have taken into account here that in a plane wave h m n,3 = —h m n,o ■
3 See, for instance, L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields,
8.2 Radiation of Gravitational Waves 83
In a plane wave propagating along the axis 3 the contraction h mn h mn
reduces to h\ x + h\ 2 + ^h\ 2 • However, the calculation of the total intensity
requires integrating over the angles, i.e. over the directions n of the wave
propagation. Therefore, the expression h\ x + h\ 2 + 2/i 12 should be rewritten
in the form valid for an arbitrary n, but not only for that directed along the
axis 3. First, the tensor, the square of which enters the result, should belong
to the plane orthogonal to n, i.e. it should satisfy the condition n m h mn = 0.
Such transverse tensor is
But still this is not all. The needed tensor should also be traceless. Therefore,
its correct form for an arbitrary direction n is
hmn = J»m„ - \ ( S ™n ~ n m 7l n ) hj- .
Tims, the contraction entering the answer, for an arbitrary n equals h mn h mn .
Simple transformations give
h mn hmn = h mn h mn - 2n i n j h ml h jn + - {n m n n h mn f .
It is convenient to average at once this expression over the directions of n. By
means of formulae
(mnj) = I 5 l3 , ( ni n 3 n m n n ) = 1 (6 tJ S mn + 8 im S jn + 5 in S jm ) , (8.10)
(h mn h mn )= -h mn h mn . (8.11)
Now, with relations (8.6), (8.9), and (8.11), we obtain the final result for
the total intensity of the gravitational quadrupole radiation (^4. Einstein,
1918; M. von Laue, 1918):
I = ^ 2 ^ mn k mn )=^ mnimn . (8.12)
We have restored in the last expression the explicit dependence on the velocity
of light c.
As it should be expected, the obtained result (8.12) is very close in struc-
ture to the corresponding formula for the electromagnetic quadrupole radia-
tion. 4 In particular, it is also of fifth order in 1/c.
The discussed effect is extremely small, the registration of gravitational
radiation from any conceivable source on the Earth is absolutely unrealistic.
See, for instance, L.D. Landau and E.R i. Lifshitz. Tht ( 'lassii al Theory of Fields,
§71.
84 8 Gravitational Waves
Problem
8.4. Derive relations (8.10).
8.3 Gravitational Radiation of Binary Stars
As to the detection of gravitational waves from cosmic sources, and from
binary stars in particular, the situation is different. Let us consider, therefore,
in more detail the problem of gravitational radiation of two bodies bound by
gravitational interaction. If the distance between the bodies is much larger
than the size of both, these bodies can be considered as point-like. Then
p(r) = mi 6{r - n(t)) + m 2 6{r - r 2 (i)) ;
here mi. 2 are the masses of the bodies, ri,2(£) are their trajectories. The
quadrupole moment of the system is
where fi = m\mil \m\ + 777,2) is the reduced mass, and r = ri(£) — r2(£) i;
relative coordinate. Using the equation of motion
and the result of its differentiation over time,
km [ 3r(rv)l
/*rr^\ [urh
( km\mi \
(8.13)
where 1 = /x [r x v] is the orbital angular momentum of the system.
In the simple case of a circular orbit (when (rv) = 0), we get easily the
total intensity of radiation, i.e. the energy loss in unit time:
ill
(8.14)
This energy loss results in particular in a decrease in orbital period T. This
decrease can be expressed via the world constants k and c. and via masses
mi, 7772, and period T itself, which are measurable. The result for a circular
orbit is
8.3 Gravitational Radiation of Binary Stars So
192 7rfc 5 / 3 / T \ ~ 5/3
T= — -g (o - ) mim 2 (mi + m 2 ) 7 . (8.15)
The gravitational radiation even of binary stars has not been directly mea-
sured up to now. There is, however, convincing quantitative proof that it
indeed exists. The detailed measurements of the pulses of radio waves from
the binary pulsar B1913+16 (sec shoil informal ion on it in section 6.4) have
demonstrated that the orbital period of this binary star decreases with the
rate -(2.4056 ± 0.0051) x 10~ 12 s/s. The effect is exactly the same quanti-
tatively as it should be due to the energy loss caused by the gravitational
radiation. The ratio of the measured rate T m to the calculated one T c (of
course, with the account for the orbit eccentricity, see problem 8.7) is
Y~ = 1.0013 ±0.0021.
Let us note that the energy of gravitational waves is huge in the present case,
it is quite comparable to the total energy of the Sun's radiation.
It is expected that in the next few years the gravitational radiation of bi-
nary stars will be directly registrated by detectors using laser interferometers.
8.5. Derive relation (8.13).
8.6. Derive formula (8.15).
8.7. Find the relation between the energy loss and the change of the orbital
angular momentum, caused by the gravitational radiation, for the circular
orbits of components of a binary star.
8.8. Find the intensity of the gravitational radiation, averaged over the rota-
tion period, for the elliptic orbits of components of a binary star (P.C. Peters,
J. Matthews, 1963).
8.9. Find the change of the orbital angular momentum, averaged over the
rotation period, for the elliptic orbits of components of a binary star (P. C.
Peters, 1964).
8.10. Find the change of the eccentricity of elliptic orbits of the components
of a binary star (P.C. Peters, 1964).
8.11. Find the change of rotation period for elliptic orbits of the components
of a binary star (P.C. Peters, 1964).
86 8 Gravitational Waves
8.12. A particle with the velocity v^ -C 1 at infinity and the impact param-
eter p = 2r g (l + S)/v 00 , 5 <C 1 scatters on a black hole (see problem 6.14).
Estimate the total energy loss due to the gravitational radiation if the particle
goes to infinity again (Ya.B. Zel'dovich, I.D. Novikov, 1964). How does the
total cross-section change due to the gravitational radiation?
8.13. Let us assume that there is a massless scalar field interacting with the
energy-momentum tensor of the usual matter. Estimate the intensity of the
radiation of this scalar field by a binary star.
8.14. Estimate the total energy loss due to the gravitational radiation when
two bodies of comparable masses scatter in such a way that the minimum
distance is on the order of their gravitational radii.
8.4 Resonance Transformation of Electromagnetic Wave
to Gravitational One
Let a free wave with the electric and magnetic field strengths e and b, respec-
tively, propagate in a constant external field with field strengths E and B.
Then it is the total field strengths that contribute to the stress tensor T mn ,
which is the source of the gravitational wave h mn , so that
-Uh mn = 16irkT mn = 4k[{E + e) m {E + e)„ + (B + b) m {B + b) n ] . (8.16)
In the present case it is just T mn (but not T mn + 1/2 5 mn T£), which enters
the right-hand side of the wave equation, since for the electromagnetic field
t a = °-
Obviously, the constant part of T mn does not generate the gravitational
wave, so that E, m E n + B m B n in the right-hand side can be at once omitted.
Then, if the electromagnetic field is weak, one can certainly neglect as well
the contribution of c ul e„ + b w b n into T mn .
In fact, even if the free wave is strong, its stress tensor
T mn ~ e m e n + b m b n
in principle cannot generate a gravitational wave. To prove it, let us choose
the direction n of the wave propagation as the z axis. Since in a free elec-
tromagnetic wave b = n x e, in our case b\ = — e 2 and 6 2 = e>\- Therefore,
T\\ = r 22 ~ eiei + e 2 e 2 , and n 2 = r 2 i = 0. Obviously, such a stress tensor
cannot serve as a source of a gravitational wave which should also propagate
along the z axis: in this gravitational wave at least one of two polarizations
should be distinct from zero, either /i n = — /i 22 , or h 12 = h 2 \-
Thus, it is sufficient to keep in the right-hand side of (8.16) only the
interference terms E m e n + E n e m + B m b n + B n b m .
The characteristic features of the discussed phenomenon can be elucidated
with the follow in;>, spcciUc example. Let the external field be purely magnetic
8.5 Synchrotron Radiation Without Special Functions 87
and directed along the x axis: B = (5,0,0), and the magnetic field of the
wave be as follows: b = (0, 6e^( z ~'', 0). Then equation (8.16) reduces to
{-dl + dl)h l2 = AkBbe lu{7 - t) .
Its solution is
hi2 = -2ikBb - e M*-*) . ( 8 .17)
The energy flux in the gravitational wave is
3 1 /t, i ^ kB 2 b 2 z 2 2 .
We are interested here only in the contribution to £q that grows with z as
z 2 . Therefore, as well as in the case of a common plane wave, we differentiate
over z only the exponential in expression (8.17). As to the energy flux in the
electromagnetic wave, it is obviously
T 3 = £ cos 2 „(*-*).
Thus, a resonance transition of an electromagnetic wave into the gravitational
one takes place in an external field (M.E. Gertsenstein, 1961), with the trans-
formation factor
^ 3 1
K = % = -kB 2 z 2 .
Despite the resonant character of the transition, i.e. the linear growth with z
of the gravitational wave amplitude, the effect is so weak that one can hardly
hope to observe it in the conceivable future.
Still, the discussion of this phenomenon is not only of methodological in-
terest. Searches for other, nongravitational, hypothetical fields with zero or
very small rest mass are based on analogous effects.
8.15. How does the quadratic growth of the energy flux of gravitational v
change at very large distances?
8.5 Synchrotron Radiation of Ultrarelativistic Particles
Without Special Functions
The synchrotron radiation, i.e. the radiation of a charged particle in an ex-
ternal magnetic field, is considered in numerous textbooks. 5 However, the
5 See, for instance, L.D. Landau and E.M. Lifshitz, The Classical Theory of
Fields, §74.
88 8 Gravitational Waves
consideration is based usually on the analysis of the exact solution of the
problem. For the radiation of relativistic particles in a gravitational field such
an analysis and exact solution by itself are incomparably more complicated.
In the case of radiation in a gravitational field the qualitative analysis is not
only more transparent, more instructive from the physical point of view, but
certainly more practical also.
The detailed qualitative analysis of the common synchrotron radiation
performed below serves as an introduction to the next section where the ra-
diation of ultrarelativistic particles in a gravitational field is considered. One
may think that the arguments presented here will be useful by themselves,
irrespective to the problems considered in the next section.
Let us start with the total radiation intensity. In the locally inertial frame
(LIF) comoving with an electron, it is
/'~eV) 2 ~ ^{E'f- (8-18)
Here e and m are the electron charge and mass, a is its acceleration, E is
the electric field strength; /, a and E are supplied with primes to point out
that they refer to the LIF. E' is obtained from the magnetic field B in the
laboratory frame (LF) by the Lorentz transformation
E'~Bj, 7 = 1 . (8.19)
We recall now that / is an invariant. Indeed, the radiation intensity is
expressed through the probability of the photon emission W and its energy
fko as follows: I = Whui. Then, the probability W in the LF is related to the
probability in the LIF W by the relation W = W/7 ust recall that the
lifetime of an unstable particle in LF is 7 times larger than that in LIF). On
the other hand, it is well known that to = 0/7. Finally, I' = /.
Now, substituting into (8.18) expression (8.19) for the electric field E' in
the LIF, we obtain the well-known result
If instead of B one fixes the radius of the electron trajectory r , related to B
via eB ~ mj/r , the expression for the total intensity becomes
/~-f. (8.21)
r o
Let us go over now to the angular distribution of the radiation. In the LIF
it has a common dipole form, it is just trigonometry. In other words, in LIF
9' = k' t /k'i ~ 1. Here kLn is the transverse (longitudinal) component of the
wave vector of the photon. In the LF these components are: k t = k' t , fc; = k[^.
8.5 Synchrotron Radiation Without Special Functions 89
Therefore, in the LF an ultrarelativistic electron radiates into a cone with a
typical angle
6 C ~ kt/ki ~ 7- 1 . (8.22)
An observer receives the signal only staying inside this cone which moves
together with the electron. An elementary consideration demonstrates that
the electron beams at the observer only from the piece of the trajectory arc
that has the same angular size as the cone itself. In the present case it means
that the angular size of this piece of the arc is 9 C ~ 7 -1 . In other words, the
formation length for radiation, which in our ultrarelativistic case (u«c= 1)
coincides with the formation time for it, is
At ~ r 9 c ~ ro^' 1 .
Then the duration of signal receiving, with the account for the longitudinal
Doppler effect, is
where n = k/k. For 9 ~ 8 C ~ 7 _1 we obtain St c ~ r 7~ 3 . It means that the
characteristic frequency of the received radiation is 7 3 times larger than the
rotation frequency ui :
uj c ~ St' 1 ~ 7 3 r^ 1 ~ 7 3 co . (8.24)
We turn now to the spectral distribution of the synchrotron radiation.
Its intensity decreases rapidly for <x> 3> uj c - Let us assume that for oj-^lOc it
changes according to a power law: I{lo) ~ ia v . Then, by comparing the total
intensity given by the integral
dw/(w) -
-■'-'
with formula (8.21), we obtain v = 1/3. In other words,
I(u>) ~ w 1/3 for uj<uj c , (8.25)
or for the discrete spectrum
In-n 1 ' 3 for n^ 7 3 . (8.26)
And at last, let us find the angular distribution of the radiation for the
frequency range
It is natural to expect that here the characteristic angles 9 are larger than
7 _1 . As previously, while the angle of the radiation cone is small, 9 <§; 1, the
90 8 Gravitational Waves
electron beams at the observer only from the piece of the trajectory arc which
has the same angular size 6. But then, instead of relation (8.23), we obtain
Thus, in this frequency region
0~
(^) ~n- 1/3 - (8-27)
In the conclusion of this section, it should be emphasized that the obtained
qualitative results are not special for the considered problem of Unite motion
of an ultrarelativistic particle in a magnetic field. They are applicable as well
to a more general case, that of scattering in external electromagnetic fields if
characteristic scattering angles exceed 7 -1 .
Problem
8.16. An ultrarelativistic electron scatters by a large angle in an external
electromagnetic field. Find the momentary intensity of the gravitational ra-
diation. In the present case the basic mechanism of its generation is the res-
onance transformation of the electromagnetic synchrotron radiation into the
gravitational one {LB. Khriplovich, O.P. Sushkov, 1974).
8.6 Radiation of Ultrarelativistic Particles
in Gravitational Field
In this section, as well as in the previous one, we are not confined to the
case of a circular motion, that for the Schwarzschild field is of methodological
interest only, due to instability of ultrarelativistic circular orbits. We discuss
also the radiation under infinite motion. The presented approach is due to
LB. Khriplovich, E.V. Shuryak (1973).
It can be easily seen that in this case as well, the radiation of a particle
is concentrated in the region of angles ~ 1/7 (see (8.22)). Still, there is a
serious distinction from the radiation in an external electromagnetic field. It is
as follows. In an external gravitational field, just in virtue of the equivalence
principle, the trajectory of an emitted particle, photon or graviton, is very
close to the trajectory of its ultrarelativistic emitter. Thus, here the formation
length of the radiation, both electromagnetic and gravitational, for the circular
motion coincides in the order of magnitude with the radius of the trajectory
ro, but is not contracted as compared to it by 7 times, as was the case in an
external electromagnetic field. Due to the nonlocal formation of the radial ion
it does not make sense in the present case to talk about its total intensity in
the LIF.
8.6 Radiation of Ultrarelativistic Particles in Gravitational Field 91
Therefore, we will estimate the differential intensity dl in the element dfi
of solid angle with the standard formula
dl ~ oj 2 u 2 R 2 dQ ^- .
dt'
Here u is the characteristic amplitude of the field of the wave; uj 2 u 2 is the
estimate for the energy flux density, i.e. for Tq and tfj in the cases of elec-
tromagnetic and gravitational waves, respectively. The factor dt/dt' is due to
the fact that the intensity is being measured with respect to the time t of the
observer, but not to the time t' of the emitter. Since t = t' + |r — r(£')|, for an
ultrarelativistic particle we have
With the radiation concentrated in the angular interval 6 ~ 1/7, its charac-
teristic frequencies are
uj c ~ 7 2 ^ ~ 7 2 - • (8-29)
They exceed the rotation frequency not by 7 3 times, as was the case with the
synchrotron radiation (see (8.24)), but only by 7 2 times. In this respect the
situation here resembles that taking place for a scattering by a small angle,
less than 1/7, in an external electromagnetic field.
For ui<.uj c the radiation propagates inside characteristic angles 0;>l/7
with respect to the direction of motion of the emitter, so that
9 ~ (^r )- 1/2 ~ (-^J (8.30)
(but not (luq/uj) 1 / 3 , as was the case with the synchrotron radiation, see (8.27)).
Let us turn now from these general relations to concrete expressions for
the electromagnetic and gravitational radiation. In the formulae below they
are described by the first and second expression, respectively.
The three-dimensionally transverse (with respect to n) field amplitudes u
in the wave zone are:
ev±_ eO y/keiv^) 2 y/ked 2
± ~ 1-nv ~ 6 2 + I/7 2 ' ±A - 1-nv ~ 6 2 + I/7 2 '
We recall here that \/ke, where e is the particle energy, plays the same role
in gravity as e in electrodynamics.
92 8 Gravitational Waves
The differential intensities of radiation for the angles #;>l/7 are
dl em e 2 1 dl gr ke 2 1 _ km 2 -/ 2 1
~de rf¥ 7 ~df ~ ~r[~6 ~ r 2 0'
And at last, the total intensities are
e 2 f 1 dd e 2 7 2 T ke 2 f 1 dd ke 2 , km 2 -, 2 ,
7 em ~ 3 / , ^ ~ T 2 " ' 5r ~ 7^ / , T ~ T 2 " 7 = "T 5- 7 '
The corresponding frequency spectra for a; <; cj c ~ 7 2 /r are
I em ~ cj° = const , J 9r ~ lnw . (8.31)
Now we briefly discuss a more realistic problem, that of the radiation
of ultrarelativistic particles moving with the impact parameter p in the
Schwarzschild field. In this case the duration of the signal is At ~ p/7 2 ,
so that the characteristic frequencies are lo c ~ J 2 /p. The total intensity of
radial ion can be obtained from the corresponding formulae for the circular
motion by the substitution 1/r 2 — > r 2 /p 4 . Indeed, while for the circular mo-
tion the acceleration equals dv/dt' ~ l/r , in the scattering problem it is
dv/dt' ~ r g l p 2 . Then, we multiply the intensity by the characteristic time of
flight, and obtain in this way the following estimates for the total energy loss:
,22 ,22
<H a, K£ r f>
2.2 2
Problems
8.17. Derive relations (8.31).
8.18. An ultrarelativistic particle with the impact parameter
p= (3V3/2)r fl (l + 5), <J<1,
scatters on a black hole (see problem 6.16). Estimate the total energy loss due
to the gravitational radiation if the particle goes to infinity again. How does
the total cross-section change due to the gravitational radiation?
General Relativity and Cosmology
9.1 Geometry of Isotropic Space
The modern cosmology is based on the solution of the Einstein equations
found by A. A. Friedmann (1922). This solution, in its turn, is based on the
assumption that the distribution of matter in the Universe is homogeneous
and isotropic.
In the real world, the matter (or at least a large part of the matter) is
condensed into stars, stars are condensed into galaxies, and galaxies are con-
densed into clusters. But on this last stage the inhomogeneities are apparently
over: astronomic observations at least are not in conflict with the assumption
that the "gas" of the clusters of galaxies is homogeneous and isotropic.
If an n-dimensional space is completely isotropic, its Riemann tensor R^ki
should be characterized by a scalar, namely by the scalar curvature R. There-
fore, with the account for the symmetry properties (3.46) and (3.47), the
curvature tensor in the locally Euclidean space should appear as follows:
Rijki = K(SikSji - SuSjk) ,
with the coefficient K being proportional to R. The natural generalization of
this equality for arbitrary coordinates is
Rijki = K {g lk 9ji - gu9jk) ■ (9-1)
The coefficient K in this formula is independent of coordinates. This can be
easily proven by plugging relation (9.1) into the contracted Bianchi identity
(3.53). Thus, an isotropic space is simultaneously a homogeneous one. How-
ever, the constant K may depend on time.
In a three-dimensional space, contracting formula (9.1) in ik and in jl
relates the coefficient K to the scalar curvature R as follows:
R=6K.
01 General Helativitx and Cosmology
Depending on the sign of the scalar curvature, three essentially different cases
are possible for the space metric of an isotropic space: 1) constant positive
curvature, K > 0; 2) constant negative curvature, K < 0; 3) zero curvature,
K = 0. Clearly, the last case corresponds to the flat, Euclidean space.
It is convenient to investigate the geometry of a space of constant positive
curvature by treating it as the geometry on a three-dimensional hypersphere
in some auxilian ionr-dimcnsiona] Euclidean space (of course, unrelated to
the four-diiiK'iisioual space-time). The equation for a hypersphere of radius a
in this space is
2 , 2 , 2 , 2 _ 2
%i + x 2 + x 3 + x 4 — a ,
and the element of length on it is
dl 2 = dx\ + dx\ + dx\ + dx\.
By expressing the auxiliary coordinate X4 via the physical ones x\, X2, £3,
and eliminating dx\ from dl 2 , we find
dl 2 = dx\ + dx\ + dx\ + (^dxjyjdx 2 jx 3 dx 3 ) 2 (9 2)
To relate the constants a 2 and K, we put x 3 = 0. It is clear that the surface
obtained in this way is a two-dimensional sphere with the Gauss curvature
Now we go from x\, x 2 , £3 to the spherical coordinates r. 6, (p. Instead of
direct calculation, one can note that under the shift along the radius, i.e. for
dr||r, the longitudinal interval is
dl 2 = dr 2 1 + -
On the other hand, for the shift rfr _L r the transverse interval i:
Then it is clear that in the spherical coordinates
Of course, any point of the space can be chosen as the origin. The length of
a circle in these coordinates is 2irr, and the surface area of a sphere is 4nr 2 .
The length of the radius of a circle and sphere
Jo v/1 - r 2 /a-
9.1 Geometry of Isotropic Space 95
Then, it is convenient to introduce four-dimensional spherical coordinates
a, 0<x<tt, 0<6»<tt, < < 2tt
in the auxiliary space, so that
xi = a sin \ sin 9 cos <j) , x 2 = a sin \ sin 9 sin <fi , a; 3 = a sin x cos # ,
x 4 = a cos x •
Obviously, now r = a sin x and the interval becomes
dl 2 = a 2 [d X 2 + sin 2 X {d9 2 + sin 2 6># 2 )] . (9.5)
In the new variables the distance of a point from the origin is ax- With the
increase of this distance, the surface area of a sphere S = 4na 2 sin 2 x at first
increases and reaches at the distance ira/2 its maximum value equal to Ana 2 .
Then it starts to decrease and turns to zero at the maximum possible distance
■na. The volume of a four-dimensional space with positive curvature is finite:
V = f dcf> [ sm8d8 f sin 2 X dxa 3
Jo Jo Jo
!9.0)
However, this space has no boundaries. Hence it follows in particular that
the total electric charge in such a space should be equal to zero. Indeed, any
closed surface in a finite space splits this space into two finite domains. The
flux of electric field through this surface is equal to the total charge of a
domain situated on one side of the surface. But the same flux is equal to the
total charge of another domain situated on the opposite side of the surface
taken with the opposite sign. It is clear that the sum of the charges from both
sides of the surface should be equal to zero. By the analogous reason, the total
4-momentum of a closed space should also vanish.
Let us discuss the spaces of constant negative curvature. It follows from
(9.3) that formally this corresponds to the substitution a — > ia. Therefore,
the geometry of a space of constant negative curvature corresponds to the
geometry on a four-dimensional pseudosphere of imaginary radius. Now
1
and the element of length in the coordinates r, 9, (f> is
rlr 2
dl 2 = ar + r 2 (d9 2 + sin 2 6dtf) ,
with < r < (X) . The change of variables r = a sinh \ ;dves
dl 2 = a 2 [d X 2 + sinh 2 X (d9 2 + sin 2 9d<p 2 )} .
The volume of a space of negative curvature is infinite.
Of course, the case of a flat, Euclidean space, with K = , is also
96 9 General Relativity and Cosmology
Problems
9.1. Prove that K in expression (9.1) is independent of coordinates.
9.2. Prove equality (9.3) by direct calculation of the scalar curvature of space.
The calculation is conveniently performed in the vicinity of the origin, i.e. for
small xi, X2, X3 .
9.3. Transform interval (9.4) to the form where it is proportional to the Eu-
clidean expression.
9.4. Prove relation (9.6).
9.2 Isotropic Model of the Universe
In the case of a closed Universe, the visual two-dimensional analogue of the
solution we are looking for is an inflating sphere, a soap bubble. In the co-
moving reference frame the matter on the sphere is at rest, i.e. the angular
coordinates of each particle of the dust do not change, and only the radius of
the sphere grows with time. In our problem of a three-dimensional space the
coordinates x, 9, <j) of each cluster of galaxies remain constant, only the scale
of the distances a(t) grows.
Since there is no singled out direction in the space, the components go m
(m = 1,2,3) of the metric tensor, which constitute a three-dimensional vector,
should vanish. The component <7oo depends only on t, so that by a suitable
choice of the time coordinate one can turn goodt 2 into dt 2 . Thus, the four-
dimensional interval transforms to
ds 2 = dt 2 - dl 2 = dt 2 - a 2 {t)[d X 2 + sin 2 X {d0 2 + sin 2 6d<j) 2 )} .
It is convenient to change from the time f to a new variable r\ defined by
relation dt = a{t)drj. In result, the interval is written as
ds 2 = a 2 {r]) [dr] 2 - d X 2 - sin 2 X {d6 2 + sin 2 9d(f> 2 )} . (9.8)
To write down the Einstein equations, one should calculate the Ricci tensor.
First of all, it is the curvature of the thrcc-diincusioua] space that contributes
to it. This contribution is found at once from formula (9.1) (taking into ac-
count that the four-dimensional space is pseudoeuclidean):
pU)
(<).<))
Another contribution to the Ricci tensor is due to the dependence of the
metric on r\. The components of the ChristoUc] symbol with the derivative
over r\ (denoted below by prime) are
9.2 Isotropic Model of the Universe 97
r oo = ^ , r% = - ^ Sy , r*,. = ^5) . (9.io)
The components -T^ an d -^oo vanish since there is no singled out direction in
our three-dimensional space. By the same reason, turn to zero the components
Roi of the Ricci tensor. Its purely time component is
At last, the corresponding contribution to the purely space components is
R% = -m (^ + ^r) • ( 9 - 12 )
The total expression for the space components of the Ricci tensor is
Rij = B g) + R f = -g.. ^ + ^ + j^ . (9.13)
The scalar curvature is
so that the 00 component of the Einstein equation is written as
Roo- £Soofl = 3f^ +lj =8^fcT 00 .
Quite analogous calculations in the case of the open Universe result in the
equation that differs from this one only by the sign at 1 in the bracket. And for
the Universe where the three-dimensional space by itself is flat, Euclidean, the
bracket simplifies to a' 2 /a 2 . As to the right-hand side, since u° = drj/ds = 1 /a,
we obtain in all three cases, for the closed, open, and flat Universe,
Too = googoopu°u° = pa 2 .
Thus, in the general case of isotropic Universe the discussed equation is
Here and below q = 1 for the closed Universe, q = — 1 for the open Universe,
q = for the flat Universe.
In the employed comoving reference frame the space components of the
four-dimensional velocity are equal to zero, i.e. the coordinates x, 9, and <f>
of each particle of the dust are independent of r\. Therefore , in this frame all
98 9 General Relativity and Cosmology
components of the energy-momentum tensor, but T 00 , vanish. The On com-
ponents of the Einstein equations turn into the identity = 0, and their mn
components appear as follows:
Rmn ~ \g mn R = ( 2 j - ^ + ^) 9mn = 0,
or simply
2^3-^ T + -|=0. (9.15)
We note that while the transition from the three-dimensional geometry of
the closed Universe to the geometry of the open one is accompanied by the
change a — > ia, the corresponding transition in the dynamic equation (9.15)
demands one more change: rj — > irj.
The volume of the closed Universe grows ~ a 3 in the process of expansion,
and the total mass of the dust remains constant. So the dust density changes
according to the law p = c/a 3 . With this dependence, we arrive for the closed
Universe at the equation
,1 2 4?r
a +a = 2a^,a , where a = — kc,
a /2 +(a-a ) 2 = a 2 .
This is obviously the energy integral for the oscillator with equilibrium at the
point a = ao- Under corresponding choice of the initial condition, the solution
for a(r]) is
a = a (l-cosr;). (9.16)
Since by definition dt = adrj, then
t = ao(r?-sin77). (9.17)
Equations (9.16) and (9.17) describe in a parametric form the evolution of the
closed Universe. This evolution is of cyclic character: the expansion from the
point (a = for t = 0) to a max = 2ao changes with contraction to the point,
and then everything starts anew.
We now examine the case of the open Universe. Looking at (9.16) and
recalling that this transition is linked to the change rj —¥ irj, and also that
a > 0, it is natural to assume that in this case
a = a (coshri- 1). (9.18)
One can check easily that function (9.18) indeed satisfies equation (9.15) (for
;/ = — I .). Correspondingly, in this case
t = a {smhri-ri). (9.19)
9.3 Isotropic Model and Observations 99
Here the expansion from the point (a = for t = 0) goes on infinitely. The
dust density, defined here by equation (9.14), falls down with the increase of
7], the regime of the expansion approaches the free one, so that asymptotically
the radius a grows linearly with time.
And at last, in the case of the flat Universe, in line with the trivial solution
a= a , t = a ri, there is the nontrivial one:
a = a oV 2 , *=|V, or a(t) ~ t 2 ' 3 . (9.20)
This solution corresponds to the interval of the form
ds 2 = dt 2 - ai t i/3 (dx 2 + dy 2 + dz 2 ) .
In fact, the dependence of a(t) ~ t 2 ' 3 takes place also in two other cases as
well, but for small times only. One can easily check it by considering the cor-
responding formulae, (9.16) and (9.17), (9.18) and (9.19), in the limit r) -> 0.
In this limit
~dl ~ ^°°'
We note that for the closed Universe the same regime sets in also under the
subsequent contraction into the point.
However, near the singularity where the density p turns to infinity, the
discussed description is inapplicable. First, the "dust" approximation, used
for the description of matter is not valid here. But there is an even deeper
reason: we deal here with superstrong fields and therefore need the quantum
theory of gravity.
9.5. Derive relations (9.10) through (9.13) for the closed Universe
analogous formulae for the open Universe.
9.6. Find the asymptotic behavior of density p in the open L
9.7. Derive relations (9.20) for the flat Universe.
9.3 Isotropic Model and Observations
Let us come back into the present epoch. We choose the position of an observer
as the origin in the isotropic Universe. Then his distance to the galaxy with
coordinates x> $■ 4> is I = a\ , and due to the expansion of the Un:
galaxy moves away from the observer with the velocity
9 Genera) Helativitv and Cosmology
dt A a dt y
Tims, the model results in the remarkable qualitative prediction: the velocity
v with which galaxies move away one from another (at a given moment i) is
proportional to the distance I between them. This prediction agrees with the
observations of the red shift in the spectra of galaxies that is interpreted as
the Doppler effect. The numerical value of the proportionality factor, the so-
called Hubble constant, obtained from the modern astronomical observations,
is
H = - -^ = 73 ± 3 km s _1 Mpc" 1 (9.21)
(1 Mpc (Megaparsec) = 10 6 parsec, 1 parsec = 3.26 light years).
We come back now to equation (9.14). It can be rewritten as follows:
We introduce here the so-called critical density
p c =^!«10.1Cn 30 g/cm 3 ;
this number for p c corresponds to the value (9.21) of the Hubble constant. It
is clear from formula (9.22) that the type of the geometry of the universe is
determined by the relation between the true density p and the parameter p c
that depends on the Hubble constant. If the density exceeds the critical one,
the Universe is closed; if (he density is less than the critical one, it is open; if
the density is equal to the critical one, it is flat.
Usually the diincusionlcss parameter Q = p/p c is considered. Modern as-
tronomical data give for the density of the common luminous matter the
following result:
Qi « 0.04 .
However, the density of some in\ isiblc dark matter .<?,; is perhaps in neb higher.
as indicated in particular by the estimates of the masses of galaxy clusters,
based on the velocity distribution of separate galaxies. Quantitatively, the
total matter density is
Q m = Qi + Q d « 0.25 . (9.23)
On the other hand, a global analysis of the modern data of observational
astronomy indicates that the Universe is flat, that
Qtot = 1.02 ±0.02. (9.24)
9.3 Isotropic Model and Observations 101
The gap ~ 0.77 between (9.24) and (9.23) should be filled in by some unknown
form of matter. Moreover, this mysterious form of matter should have negative
pressure, p rs —p, which is required by the observational data indicating
that the expansion of the Universe at the present epoch is accelerated. It is
considered as a serious indication that this peculiar "matter" , filling in the gap
between (9.24) and (9.23), is in fact the nonvanishing cosmological constant
A (see equation (4.3)).
It should be noted that the required magnitude of A is extremely small
(typical value of the corresponding density is A/8nk ~ p c ~ 10 - - 9 "/cm''), so
that it hardly could be observed anywhere, but in cosmology.
We discuss now the relation between the Hubble constant and the age of
the Universe. For the flat world, with Q = 1, it follows from relation (9.20)
that H(t) = 1 J a da/dt = 2/3 t. It means that, under the assumption of the flat
Universe, its age T (i.e. the time elapsed from the moment when the density
was infinite) is related to the present value (9.21) of the Hubble constant as
follows:
r=|l (9.25)
Obviously, the obtained relation is valid also for the early stages of expansion
in the closed and open Universe, where in both cases a ~ t 2 ' 3 . Even at the
late stage of expansion of the open Universe, when the density is so small
that a grows linearly with time, the relation between T and H differs from
(9.25) by a coefficient only: 1 instead of 2/3. Therefore, since the ratio i? is at
any rate not so far from 1, the estimate (9.25) is apparently quite reasonable.
The numerical value of the age of the Universe for H as 70 km s _1 Mpc -1 ,
according to (9.25), is
T w 13 • 10 9 years. (9.26)
The age of the Earth, according to the data on the content of radioactive
isotopes in the crust of the Earth, is about 4-10 9 years. The estimates of the
ages of the oldest galaxy clusters appear as 10-10 9 years. Therefore, the age
of the Universe certainly cannot be considerably less than result (9.26).
9.8. Prove that i]
holds. Here p is the pressure of matter (T™ = — p5™)
10
Are Black Holes Really Black?
10.1 Entropy and Temperature of Black Holes
The classical description of black holes, presented earlier in section 6.6, is
incomplete in principle. J. Wheeler (1971) was the first to realize it. His line
of reasoning looked roughly as follows. Let us take a box filled with the black-
body radiation at some temperature T. Obviously it possesses a finite entropy
as well. We drop the box into a black hole. Then the entropy of the observable
part of the universe will decrease forever. But this is an explicit violation of
the second law of thermodynamics! To save the second law we are just obliged
to assume that the black hole itself has some entropy (J. Bekenstein, 1972)
which increases when the box is absorbed. But it is only natural to ascribe
some finite temperature as well to a system with a finite entropy. In spite
of being so unexpected, this conclusion is quite natural from a somewhat
different point of view. A black hole is an ideal absorber, an absolutely black
body, for which the temperature is a quite natural property.
Let us try at first to estimate this temperature just by dimensional argu-
ments. We will measure the temperature in the same units as energy, <>,ctt iu<>,
rid in this way of the Boltzmann constant in our formulae. By the way, in
these units the entropy is dimensionless, so that it cannot be estimated by
means of dimensional arguments. A black hole by itself is characterized by
the only parameter, its mass M. Besides M, we have also at our disposal two
fundamental constants, k and c. One of them, the gravitational constant /,-.
should apparently enter the result by the very meaning of the problem. The
obvious combination Mc 2 will not do as temperature: it is too large, and does
not contain k. But one cannot construct any other combination of the dimen-
sion of energy from M, k, and c. But there is one more fundamental constant
at our disposal, the Planck constant h. By means of h, it is no problem to
const L'uct the necessary combination of the dimension of energy: the black
hole temperature is on the order of magnitude he 3 /(kM), or (up to a factor
of two) hc/r g .
104 10 Are Black Holes Really Black?
The problem is formally solved, but the natural question arises: what has
the quantum of action h to do with our problem, that is completely classical
at first sight? To reply to the question, we will consider the box filled with
radiation from a somewhat different point of view (R. Geroch). Now we lower
adiabatically this box to the black hole by means of a rope. The rope is wound
on a fly-wheel situated far away from the black hole. The fly-wheel rotates,
and its energy may be utilized in principle. We recall that the energy of a
body in the gravitational field of a black hole is
'^Jr.
Therefore, at this slow, adiabatic lowering of the box filled with radiation,
which has the total mass m, the energy extracted in this way is
0-A
AE = mc 2 [i-Ji-r g /:
When the box approaches the horizon, we open a lid in its bottom. The
radiation escapes into the black hole, and then the box is brought with the
rope back into its initial position far away from the star. At first sight, the
energy extracted in this way equals the energy of all radiation that was stored
in the box. One may think that all this radiation has stuck to the horizon.
However, this is not the case. Due to the uncertainty relation, the size of
the box cannot be smaller than the characteristic wave length of radiation A.
In its turn, the characteristic energy of quanta Huj is nothing but the radial ion
temperature T\. Therefore, the size of the box, its height included, is bounded
by the condition
j hc
d > ^r •
On the other hand, it is in principle important here to be able to bring the
box back to its initial position, far away from the horizon. Therefore, the
upper wall of the box certainly stays at a distance, that exceeds d, from the
horizon. Then it is natural that not all the radiation contained in the box can
be transformed into work, but its part only, limited by relation
,~i--<i-^. do.i)
r g r g 1 i
The discussed system, consisting of the black hole and the box, filled with
radiation and attached to a rope, can be considered as a heat machine with
the working body, which is radiation, of the temperature T\. Now, r\ is nothing
but the efficiency of this machine, and hence it is well known to be bounded
by the Carnot formula
Wz = 1 " ^ ' ( 10 - 2 )
where T is the temperature of the colder body. By comparing relations (10.1)
and (10.2) we come to the conclusion that hc/r g can be identified with the
10.1 Entropy and Temperature of Black Holes 105
temperature of the colder body, i.e. of the black hole, in complete agreement
with the result obtained by dimensional arguments.
To derive the numerical factor in the relation T ~ hc/r g , we consider the
following problem (T. Padmanabhan, 1999). Let a semiclassical wave packet
of a massless field propagate from a point r = r g + e close to the horizon, to
a distant point r (e <C r g , r 3> r g ). Since the wave packet is semiclassical, its
motion can be described by relations obtained in section 6.6 for a point-like
particle. According to (6.29), the time of this packet propagation from tq to
t = r - r + r g In T ~ Vg sa r + r g In - (10.3)
r — r g e
(we note that notations r and r are interchanged here as compared to (6.29)).
If the frequency at r = r g + e is u , then at r > r 9 it becomes
Wi = lu Jgoo{ro = r g + e) a uj , — .
V r 9
Since, in virtue of (10.3),
/ t-r\
- = exp ,
r \ r g /
the frequency u>\ depends on time as
UJl = UJ °y7~ exp (~^"J '
and the phase of the wave is
dtwi = -2ujq y /r¥ g ~ exp f - -^— \ .
The spectral function of the wave packet at large distances looks as follows:
f(uj) - J dte wt exp (-2^ y^e-(*- r )/ 2r «) .
With the change of variables
y = 2u Jo ^T g e^e- t /^,
we express this integral via T-function:
f(co) ~ (2^ y^) 2 ^ e -^ »r(-2ia,r ff )
(factors, independent of o>, are omitted here). In result, the spectral density
of the wave packet at large distances is:
10 Are Black Holes Really Black?
^\r(-2iur g )\ 2
= — exp(-47Rjr 9 ) (10.4)
(we recall that this is a semiclassical wave packet, so that Lur g 3> 1). In
a remarkable way, the spectral density of a signal, that arrives at infinity
from the vicinity of the horizon, is completely universal. And if one goes over
in it from the frequency uj to the energy hu>, it gets clear that the leading
exponential factor in (10.4) corresponds to the high frequency asymptotics of
the Boltzmann distribution with the temperature
T =^ = ^M- (1 °- 5)
This expression for the black hole temperature was obtained by S. Hawking
(1974).
The inevitable result of the finite temperature T of a black hole is the
conclusion that in fact it radiates. Black hole produces not only photons and
neutrinos with energies on the order of T, but particles of non- vanishing rest
mass to as well (only if its temperature is suificicut lv hi<>,h). Thus, one of the
most amazing properties of black holes is that they shine!
V.N. Gribov was the first who made this conclusion. 1 One of his arguments
was as follows. The uncertainty relation AEAt > h allows the creation of pairs
of particles from vacuum for the time t that does not exceed h/E; here E is
the total energy of the pair (2toc 2 for massive particles). The gravitational
field near the horizon is very strong, so that the energy conservation by itself
allows one of the particles to be absorbed by the black hole, and the second
one to go to infinity. In quantum mechanics, due to the tunneling effect of such
a sort, the processes of particle creation become possible. In particular, the
creation of electron-positron pairs in strong electric fields has not only been
studied theoretically for a long time, but has been observed experimentally
in the heavy ion collisions. In fact, an analogous phenomenon can serve as an
explanation of the black hole radiation.
One more of Gribov's arguments is worth mentioning here: a black hole
certainly cannot confine radiation with wave length exceeding its gravitational
radius. The correspondence is obvious of this argument with the expression
(10.5) for the temperature, i.e. for the frequency where the exponential fall
down of the intensity starts.
1 Gribov precisely formulated the statement that black holes radiate ir
taking place in 1971 or 1972. This was told to me independently by A.D. Dolgov,
D.I. Diakonov, L.B. Okun', who had been present at those discussions. One can
only regret that Gribov did not publish this result, apparently he considered it self-
evident. In 1974 radiation ofblacli holes was predicted independently by S. Hawking.
10.1 Entropy and Temperature of Black Holes 107
In fact, for real black holes the temperature (10.5) is negligibly small.
In particular, for the mass comparable with that of the Sun it is only about
1CT 7 K. For instance, for the temperature to be sufficient to produce electron-
positron pairs, the lightest particles of nonvanishing rest mass, the black hole
mass should be smaller than that of the Sun by 17 orders of magnitude, i.e.
it should not exceed 10 17 g. However, a star with such a small mass cannot
compress to its gravil at ioual radius, it cannot turn into a black hole. Such light
black holes in principle could arise at the most early stages of the Universe
evolution when the matter density was very high.
But could such mini-holes survive since those times? Could their age ap-
proach the Universe life time r ~ 10 10 years, or 10 17 s? The obstacle here is the
black hole thermal radiation itself. Let us estimate its intensity / by dimen-
sional arguments. To this end it is sufficient to divide T by the characteristic
time, which is nothing but r g /c:
*-%-%■ < 1M >
We have introduced here the so-called Planck mass
m P =(j) =2.2xl(T 5 g. (10.7)
On the other hand, obviously, / = —c 2 dM/dt. Solving the differential equation
dM _ m^c 2
~dT = ~ hM 2 '
we find that to survive until our time a black hole should have an initial mass
/t\ 1/3
M >m p {-\ ~ I0 15 g. (10.8)
Here t„ is the so-called Planck time
t p = 2 = ( — ) =0.54x 10~ 43 s. (10.9)
Together with the energy, a black hole also loses its mass. Then, according
to relation (10.6), the intensity of its radiation grows, it shines brighter and
brighter. The gravitational radius of a black hole gets smaller and smaller.
How does this process end? Obviously, a star cannot radiate more energy
than it has. The radiation certainly stops when the black hole temperature
s comparable to its rest energy, at
m 2 c 2
Mc 2 ~ T 1— ,
108 10 Are Black Holes Really Black?
i.e. when the mass of such a mini-hole decreases to the Planck mass:
M~m p .
Here our semiclassical consideration of quantum effects in the vicinity of black
holes, and in quantum gravity in general, becomes inapplicable. Here a con-
sistent quantum theory of gravity is necessary. However, such a theory does
not exist up to now.
Let us go back to the question of whether such bright mini-holes, arising
at early stages of the Universe evolution, could survive until our time. Simple
estimates with formula (10.8) demonstrate that the mass of the brightest
among such relics looks quite modest, it is about 10 15 g. However, the last
stage of its evolution, just before reaching the Planck scale, should be very
impressive: an explosion with a power of thousands of the biggest hydrogen
bombs. These phenomena have not been observed by astronomers up until
It is instructive to look at relation (10.5) from a somewhat different point
of view. It demonstrates that the mass of a black hole, and hence its energy as
well, decreases as the temperature increases. In other words, the heat capacity
of a black hole is negative. This unusual property is in no way special to black
holes, it is quite typical for gravitating systems in general. 2 As to a black
hole, its negative heat capacity is directly related to the instability caused
by radiation. Let us recall, however, the classical instability of the orbit of
a bound electron in the Coulomb field. It is also caused by radiation, but
is finally stabilized by quantum effects. In the case of black holes as well,
it is natural to assume that on the Planck scale their semiclassical radiative
instability is stabilized by quantum effects.
In conclusion of this section, we pay attention to the following important
fact related to the radiation of black holes. For the typical time interval At ~
r g /c between the acts of radiation, the uncertainty of the energy of a black hole
is AE ~ h/ At ~ he 3 jkM. The corresponding uncertainty in the gravitational
radius is (J. York, 1983)
kAM kAE h
r§ ~&~ ~ c 4 ~ Wc '
It is only natural to believe that at least due to this uncertainty, the time of the
fall of a point-like particle to the horizon, which is logarithmically divergent
in the classical approach (see section 6.6), becomes finite:
And the fact that the arising logarithm is very large, In M 2 /rr,
present case is not of much importance.
See L.D. Landau and E.M. Lifshitz, Statistical Physics, part 1, § 21.
10.2 Entropy, Horizon Area. Holographic Bound. Quantized Black Holes
Problem
10.1. Prove relation (10.6), starting from the Stefan - Boltzmann law.
10.2 Entropy, Horizon Area, and Irreducible Mass.
Holographic Bound. Quantization of Black Holes
Now, when the temperature of a black hole has been found, the calculation of
its entropy becomes an elementary problem. The well-known thermodynami-
cal formula
dE = TdS (10.10)
relates the increase of the energy E of a body to the increase of its entropy S.
In our case T is given by formula (10.5), and E = Mc 2 . Solving the arising
differential equation
vith the natural boundary condition
5 = for M = ,
It is convenient to introduce the so-called Planck length
Then we arrive at the following remarkable relation between the entropy of a
Schwarzschild black hole and the area of its horizon A = Aitr 2 -
U ~ 11 " 4/2 • ^—'
The corresponding analysis for a charged black hole is more intricate. In
the Schwarzschild case, the horizon area A depends on the only parameter,
the mass M of a black hole. Therefore, the adiabatic invariance of A means
that M is also an adiabatic invariant. But what happens with the Reissner
- Nordstrom black hole when a small charge e is lowered adiabatically to its
horizon? What remains constant, the horizon area or the mass (if either)?
To answer this question, we resort again to a tlumrjil experiment . To sim-
plify the electrostatic part of the problem, we modify the experiment described
110 10 Are Black Holes Really Black?
in the previous section as follows. We will consider a thin spherical shell con-
sisting of small charged particles, each of them attached by its own rope to its
own fly-wheel. All the particles are lowered synchronously and adiabatically
to the black hole. When the shell reaches the horizon, the charge of the black
hole changes from the initial value q to q + e, where e is the total charge of
the shell. As this was the case in the thought experiment of section 10.1, the
rest mass of the shell adds uotliiu; 1 , b\ ii self to the mass of the black hole. The
latter changes only due to the difference between the final and initial values
of the total electrostatic energy (see section 6.7). Since the electric field of a
charged shell exists only outside of it, this difference is as follows:
(q + ef [°°dr q 2 [°°dr _ (q+e) 2 q 2 _ qe e 2
~^~~ L * ~ Yj ri V 2 - "^T~ " 2^ " n + 2^' ( ° 3)
Here r\ is the radius of the new horizon, it has changed as compared to the
initial value r rn = kM + \/k 2 M 2 — kq 2 , together with the total charge. Thus,
to first order in small e the resulting correction to the mass is
We have taken into account here that to zeroth order in e r\ = r rn .
When the mass and the charge of a Reissner Nordsl mm black hole change
by AM and Aq, respectively, the resulting total change of the horizon area
A rn = 47rr 2 „ = 4tt (kM + \/k 2 M 2 - kq 2 ) (10.15)
AA rn = 87rTV " fc M M r -Agq). (10.16)
Vfc 2 M 2 - kq 2 K ' V ^
With AM = eq/r rn and Aq = e, it vanishes for a nonextremal black hole
(with q 2 < kM 2 ). Therefore, it is the horizon area of a Reissner - Nordstrom
black hole, but not its mass, which remains constant under the adiabatic
change of the charge.
It is useful to introduce the so-called irreducible mass M ir of a black hole
(D. Christodoulou, R. Ruffini, 1970, 1971) related to its area A as follows:
A = l6Trk 2 M? r .
Of course, for a Scliwai'zscliild black bole Mj r coincides with M. For a Reiss-
ner - Nordstrom black hole
M„ = \(m+ v/M 2 - q 2 /k ) .
Solving this equation for M, we obtain
10.2 Entropy, Horizon Area. Holographic Bound. Quantized Black Holes 111
The last form of this solution has a simple physical interpretation: the total
mass (or total energy) M of a charged black hole consists of its irreducible
mass M ir and of the energy q 2 /2r rn of its electric field in the outer space
r > r rn .
Now we will briefly discuss black holes with internal angular momentum.
Unfortunately, the solution of the equations that describe these black holes
(R. Kerr, 1963) is extremely tedious. 3 Therefore, we resort to some plausible
arguments that will allow us to guess the correct result for the area and irre-
ducible mass of a rotating black hole without mentioned tedious calculations.
According to equation (10.17), the difference between the mass and the
irreducible mass of a charged black hole is due to the energy of its Coulomb
field. It is natural to assume that for a rotating black hole this difference is due
to the kinetic energy of its rotation. Then the simplest relation between M
and Mi r , with the account for the possible relativistic nature of this rotation,
is
/2
M 2 = Mf r + -j-; (10.18)
r k
here J is the internal angular momentum of the rotating black hole, ?> is the
radius of its horizon. With ri- = 2kM lr , (10.18) can be rewritten as
M2 = M - + W (10J9)
Solving this equation for Mf r , we obtain
2M.fr = M 2 + V M2 ~ J 2 / k2 ■ (10.20)
In this way, our guess (10.18) results in the correct formula for the horizon
area of a Kerr black hole:
A k = 8tt (k 2 M 2 + ^fc 4 M 2 - fc 2 J 2 ) , (10.21)
that follows from accurate calculations. Besides, these calculations demon-
strate that the horizon surface of a rotating black hole is spherical. And this
is also one of the assumptions made in fact in our initial formula (10.18).
We note that, according to formula (10.21), the internal angular momen-
tum J of a Kerr black hole is bounded by the condition J 2 < k 2 M A . The Kerr
black hole with J 2 = k 2 M 4 is called extremal.
In this case as well, though! experiments demonstrate that the horizon area
of a nonextremal Kerr black hole remains constant under adiabatic change of
the internal angular momentum.
3 Even in the book by L.D. Landau and E.M. Lifshitz The Classical Theory of
Fields (see §104 therein), instead of the corresponding solution of equations of GR,
only a footnote is given: "In literature there is no constructive analytical derivation
of the Kerr metric, adequate to its physical meaning, and even the direct check of
this solution of the Einstein equations demands tedious calculations."
112 10 Are Black Holes Really Black?
To summarize, the horizon area of nonextremal black holes does not change
under considered adiabatic processes. Therefore, in the general case as well,
the entropy of a black hole is related to the horizon area by the same formula
(10.12) (of course, with the corresponding value of the gravitational radius).
The fact of proportionality between S and A was established by J. Beken-
stein (1973).
The second law of thermodynamics imposes serious limitations on possi-
ble processes not only in the common life. It plays an important role in the
physics of black holes as well. In particular, the following statement follows
from it: for any interaction among black holes the sum of the areas of their
horizons increases or remains constant. Originally this result was obtained by
S. Hawking (1971), but with quite different arguments.
It is appropriate to mention here the so-called holographic bound (J. Beken-
stein, 1981; G. 't Hooft, 1993; L. Susskind, 1995). According to it, the entropy
S of any spherical nonrotating body confined inside a sphere of area A is
bounded as follows:
5 "4f' (10 - 22)
with the equality attained only for a body that is a black hole.
A simple intuitive argument confirming this bound is as follows. Let us
allow the discussed spherical body to collapse into a black hole. Due to the
spherical symmetry, this process is not accompanied by radiation or any other
loss of matter. Therefore, during the collapse the entropy increases from S
to Sbh, or at least remains constant. But the resulting horizon area A^ is
certainly smaller than the initial confining one A. Then, using relation (10.12)
for a black hole, we arrive, through the obvious chain of (in)equalities
4J2 - 4/2 '
at the discussed bound (10.22).
The result (10.22) can be formulated otherwise. Among the spherical sur-
faces of a given area, it is the surface of a black hole horizon that has the
largest entropy.
The holographic bound looks ral her surprising since according to the com-
mon experience the entropy of a body is proportional to the volume of this
body, but not to the area of its surface. However, usually limit (10.22) is so
mild quantitatively that no contradiction with the common experience arises.
We consider now the temperature of charged and rotating black holes. For
a rotating black hole the thermodynamic relation (10.10) generalizes to
dM = TdS + Lod.J, (10.23)
where u> plays the role of the rotation frequency. Now, differentiating with
respect to S at J = const the expression for entropy (following directly from
(10.12) and (10.21))
10.2 Entropy, Horizon Area. Holographic Bound. Quantized Black Holes 113
S = 2ir (kM 2 + \/k 2 M 4 - jA , (10.24)
we find
OS 4nkM(kM 2 + Vk 2 M 4 - J 2 ) '
For a charged black hole the analogue of relation (10.23) is
dM = TdS + <t>dq , (10.26)
where <fr is the electrostatic potential. Differentiating the expression for entropy
in this case
S = 2irk (M + VM 2 - q 2 /k"j ^ ,
we obtain
dM hJM 2 - q 2 /k
T= — — = ^ , ' 10.27
OS 2irk{M + ^M 2 - q 2 /k ) 2
The important conclusion follows from relations (10.25) and (10.27): the
temperature of extremal black holes is equal to zero.
However, the radiation of extremal black holes in no way vanishes, though
it certainly is not of a thermal nature. For an exl iciual charged black hole the
nature of this radiation is the particle production by its electric field. This
radiation is bounded by the condition AikM 2 — q 2 ) > 0. Obviously, in this
case the loss of energy should be accompanied by the loss of charge, i.e. only
charged particles can be radiated, all of them having finite rest mass. In the
natural situation when e is comparable to the electron charge, the upper limit
on the mass \x of a radiated particle looks quite liberal:
IX < Vam p , (10.28)
where a = 1/137, and m p is the Planck mass (see (10.7)). Clearly, an extremal
black hole (of course, if its mass is sufficient) can radiate any known charged
elementary particles, VF-boson and t-quark included.
The radiation of a rotating extremal black hole can be explained in an
analogous way (LB. Khriplovich, R.V. Korkin, 2002). The loss of a charge
by a charged black hole is due in fact to the Coulomb repulsion between this
black hole and particles with the same sign of charge. In the present case the
reason is the spin-spin interaction: particles (massless mainly) whose total
angular momentum is parallel to that of a black hole are repelled from it.
Generally speaking, these mechanisms are operative, in line with the ther-
mal one, for nonextremal black holes as well.
It should be noted that neither the horizon surface, nor, consequently, the
entropy of a black hole turn to zero in the extremal case. i.e. lor the vanishing
temperature. This is in contradiction with the Nernst theorem, or with the
so-called third law of thermodynamics, according to which the entropy of a
system should vanish when the temperature tends to zero. However, there
114 10 Are Black Holes Really Black?
are no special reasons for anxiety here. In fact, the Nernst theorem is valid
only under the condition that the state of a system is nondegenerate at zero
temperature. This is the case indeed for stable ground states of common ther-
modynamic systems. However, due to the mentioned nonthermal radiation,
the state of an extremal black hole is in fact metastable one.
We come back now to the adiabatic invariance of the horizon area of a
nonextremal black hole. It is well-known that the quantization of an adiabat i<
invariant is perfectly natural. And just on this argument is based the idea of
quantizing the horizon area of black holes proposed by J. Bekenstein (1974).
Once this hypothesis is accepted, the general structure of the quantization
condil ion for large quantum numbers gets obvious, up to an overall numerical
constant (3. The quantization condition for the horizon area A should be
A = (3l 2 p N, (10.29)
where N is some large quantum number. Indeed, the presence of the Planck
length squared ti = kh/e' is only natural in this quantization rule. Then, for
the horizon area A to be finite in the classical limit, the power of N here should
be the same as that of h in I 2 . This argument can be checked by considering
any expectation value in quantum mechanics, uonvauishing in the classical
limit. It is worth mentioning that there are no compelling reasons to believe
that N should be an integer. Neither are there compelling reasons to believe
that the spectrum (10.29) is equidistant.
However, at present it is not exactly clear how black holes are quantized.
We stay, at best, within the semiclassical approximation to the quantum the-
ory of gravity, which has not been built up to now.
Problems
10.2. Find maximum energy liberated under the fusion of two black holes
s mi and mi .
10.3. Derive relations (10.25) and (10.27).
10.4. Prove condition (10.28).
Index
Adiabatic invariance, 104, 109, 110, 114
Age of the Earth, 101
Age of the Universe, 101
Bianchi identity, 22, 93
Carnot formula, 104
Christoffel symbol, 8, 16, 17, 25, 41-43,
45, 58, 96
transformation law, 14, 42
Cosmological term, 27, 28, 46, 101
Critical density, 100
Current density, 5, 6, 30
Dark matter, 100
Doppler effect, 89
Eikonal, 33
equation, 34
Einstein spaces, 28
Energy-momentum tensor, 6, 27, 29, 31,
81, 86, 98
of gravitational field (pseudotensor) ,
82
corresponding t
lological term,
of dust, 30
of electromagnetic field, 43
of point-like particle, 43
Equation of motion, 5, 8, 9, 24, 29, 30,
68, 84
Equivalence principle, 6, 7, 15, 73, 81,
82, 90
Extremal black hole, 58, 111
Four-velocity, 5, 8
Frenkel - Bargman Michel - Telegdi
(FBMT) equation, 68
noncovariant FBMT equation, 71
Gauss theorem, 57
Geodesic, 8, 9, 18-20, 24, 25
Geodesic deviation, 24, 25
Gravitational radius, 9, 54, 55
of the Earth, 9
of the Sun, 9
Graviton, 79
massless, 79
Gyromagnetic ratio, 67
Gyroscope precession, 65
Hamilton-Jacobi equation, 54
Harmonic coordinates (gauge), 6, 29,
34, 46-48, 51, 52, 77, 78, 80
Helicity, 79
Horizon, 55
Hubble constant, 100
Impact parameter. ■'! 1
Integrals of motion, 8, 49
Interval, 5, 7, 8, 12, 44, 46, 47, 51, 55,
63, 94-96, 99
Irreducible mass, 110
Isotropic coordinates, 47-49, 53
Kei
black hole, 111
- Thirring effect, 66
IK)
Locally inertial frame, 7, 9, 15, 21-24,
27, 41, 42, 71, 88, 90
Lorentz gauge, 5, 21
Lorentz transformation, 68
Lorentz transformations, 6, 9, 70, 71, 8i
Maxwell equation, 5, 17, 29, 57, 58
Newton constant, 6, 27, 63
Normal coordinates on geodesic. 9. 21
Quadrupole moment precession, 71
Radiation
of black holes, 106
of extremal black holes, 113
of nit rarela.tivisl ic part iclcs
angular distribution, 89, 90, 92
formation length, 89
frequency spectrum, 89, 92
total intensity, 88, 91, 92
quadrupole, 81, 83
Rank, 6, 7, 11, 12, 14, 17, 22, 31
Reissner - Nordtrom solution, 58, 109,
110
Relativistic spin precession in gravita-
tional field, 71, 73
Ricci rotation coefficient, 72-74
Ricci tensor, 22, 24, 28, 44, 96, 97
Riemann tensor
number of components. 22, 23
on cone surface, 24
on torus surface, 24
symmetries, 21, 22
Rotation
shift of interference fringes, 65
Runge - Lenz vector, 66
Scalar curvature, 22-24, 41, 93, 94, 96,
97
Schwarzschild solution, 44
Signature, 7
Spin-orbit interaction
gravitational, 62, 74
in hydrogen atom, 71
Superposition principle, 5, 30
Thomas precession, 70
Tidal forces, 25, 56
Uncertainty relation, 104
for horizon radius, 108
Variational principle, 2, 7, 8, 16, 43
Volume element, 12
Weak gravitational field, 6, 8, 9, 33, 53,
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