Modern cosmology

Paul S Wesson .

Five -Dimensional Physics

Classical and Quantum Consequences
of Kaluza-Klein Cosmology

Five-Dimensional Physics

Classical and Quantum Consequences
of Kaluza- Klein Cosmology

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Paul S Wesson

University of Waterloo, Canada & Stanford University, USA

Five-Dimensional Ptiijsics

Classical and Quantum Consequences
of Kaluza-Klein Cosmology

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PREFACE

Five dimensions represents a unique situation in modern theo-
retical physics. It is the simplest extension of the four-dimensional
Einstein theory of general relativity, which is the basis of astrophysics
and cosmology. It is also widely regarded as the low-energy limit of
higher-dimensional theories which seek to unify gravity with the in-
teractions of particle physics. In the latter regard, we can mention
10D supersymmetry, 11D supergravity and higher-D string theory.
However, the view of our group is pragmatic: we need to understand
5D physics, to put 4D gravity into perspective and to show us where
to go in higher dimensions.

This book provides an account of the main developments in
5D physics in recent years. In a sense, it is a sequel to the omniver-
ous volume Space-Time-Matter published in 1999. However, the pre-
sent account is self-contained. So are the chapters, which each deal
with a separate topic and has its own bibliography. The major topics
are cosmology, quantum physics and embeddings. There are cur-
rently two approaches to these topics, namely those provided by in-
duced-matter theory and membrane theory. The former uses the fifth
dimension in an unrestricted manner, to provide an explanation for
the mass-energy content of the universe. The latter uses the fifth di-
mension to define a hypersurface, to which the interactions of particle
physics are confined while gravity propagates freely into the "bulk".
Physically, these two versions of 5D physics are differently moti-

vi Five-Dimensional Physics

vated, but mathematically they are equivalent (one can always insert a
membrane into the former to obtain the latter). Therefore, in order to
be general, this volume concentrates on the mathematical formalism.

Some knowledge of tensor calculus is presumed, but each
chapter starts and ends with a qualitative account of its contents.
Many of the results presented here are the result of a group effort.
Thanks are due to the senior researchers whose work is described
herein, notably H. Liu, B. Mashhoon and J. Ponce de Leon. Acknowl-
edgements should also be made to associates from various fields in-
cluding T. Fukui, P. Halpern and J.M. Overduin. Gratitude is further
owed to a cadre of enthusiastic graduate students, notably D. Bruni,
T. Liko and S.S. Seahra. Much of this book was written during a stay
with Gravity -Probe B of the Hansen Physics Laboratories at Stanford
University, at the invitation of C.W.F. Everitt. Any omissions or er-
rors are the responsibility of the author.

Theoretical physics can be an arcane and even boring subject.
However, the author is of the opinion that the fifth dimension is fasci-
nating. Where else can one discover that the universe may be flat as
viewed from higher dimensions, or that spacetime uncertainty is the
consequence of deterministic laws in a wider world? Such issues
provide a healthy shake to the bedrock of conventional physics, dis-
lodging the plastered-over parts of its edifice and providing a stronger
foundation for future work. Physics and philosophy are not, it ap-
pears, separate. This book provides technical results whose success

leads inevitably to the insight that there is more to the world than is
apparent, provided one looks. . .

Paul S. Wesson

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CONTENTS

HIGHER-DIMENSIONAL PHYSICS 1

1 Introduction 1

,2 Dimensions Then and Now 2

.3 Higher-Dimensional Theories 6

.4 Field Equations in N > 4 Dimensions 1

5 A Primer on Campbell's Theorem 24

6 Conclusion 28

2. THE BIG BANG REVISITED 34

2.1 Introduction 34

2.2 Flat 5D Universes 35

2.3 The Singularity as a Shock Wave 43

2.4 A Bounce Instead of a Bang 47

2.5 The Universe as a 5D Black Hole 54

2.6 Conclusion 60

3. PATHS IN HYPERSPACE 64

3.1 Introduction 64

3.2 Dynamics in Spacetime 65

3.3 Fifth Force from Fifth Dimension 70

3.4 Null Paths and Two Times 77

3.5 The Equivalence Principle as a Symmetry 81

3.6 Particle Masses and Vacua 90

3.7 Conclusion 95

4. QUANTUM CONSEQUENCES 100

4.1 Introduction 100

4.2 4D Uncertainty from 5D Determinism 102

4.3 Is Mass Quantized? 107

4.4 The Klein-Gordon and Dirac Equations 112

X Five-Dimensional Physics

4.5 Gauges and Spins 119

4.6 Particles and Waves: A Rapprochement 124

4.7 Conclusion 129

5 . THE COSMOLOGICAL "CONSTANT" AND VACUUM 1 34

5.1 Introduction 134

5.2 The 5D Cosmological "Constant" 138

5.3 Astrophysical Consequences 150

5.4 Vacuum Instability 158

5.5 Mach's Principle Anew 162

5.6 Conclusion 165

6. EMBEDDINGS IN N>5 DIMENSIONS 170

6.1 Introduction 170

6.2 Embeddings and Physics 171

6.3 The Algebra of Embeddings 174

6.4 The Campbell-Magaard Theorem 182

6.5 Induced-Matter Theory 187

6 . 6 Membrane Theory 1 9 1

6.7 Conclusion 198

7. PERSPECTIVES IN PHYSICS 202
Index 221

1. HIGHER-DIMENSIONAL PHYSICS

"There's more to this than meets the eye" (Old English saying)

1.1 Introduction

Theoretical physics is in the happy situation of being able to
pluck good ideas from philosophy, work them through using the ma-
chinery of algebra, and produce something which is both stimulating
and precise. It goes beyond words and equations, because when
properly done it encapsulates what many people regard as reality.

We sometimes tend to forget what a stride was made when
Newton realized that the force which causes an apple to fall to the
ground is the same one which keeps the Moon in its orbit - and which
is now known to influence the motions of even the most remote gal-
axies. Nowadays, gravity has to be considered in conjunction with
electromagnetism plus the weak and strong forces of particle physics.
Even so, it is still possible to give an account of modern physics in a
few hundred pages or so. On reflection, this is remarkable. It comes
about because of the enormous efficiency of mathematics as the natu-
ral language of physics, coupled with the tradition whereby physicists
introduce the least number of hypotheses necessary to explain the
natural world (Occam's razor of old). At present, it is commonly be-
lieved that the best way to explain all of the forces of physics is via
the idea of higher dimensions.

In this regard, five-dimensional field theory is particularly
useful, as it is the basic extension of the four-dimensional spacetime

2 Five-Dimensional Physics

of Einstein gravity and is widely regarded as the low-energy limit of
higher-dimensional theories which more fully address the particle in-
teractions. This slim volume is a concise account of recent develop-
ments in 5D theory and their implications for classical and quantum
physics.

1.2 Dimensions Then and Now

The idea of a "dimension" is primitive and at least partly in-
tuitive. Recent histories of the idea are given in the books by Wesson
(1999) and Halpern (2004). It was already established by the time of
Newton, who realized that mass was a more fundamental concept
than density, and that a proportionality between physical quantities
could be converted to an equation if the latter balanced its ingredients
of mass, length and time (i.e., was dimensionally homogeneous).
Hence the introduction of a parameter G, which we now call New-
ton's constant of gravity.

The coordinates of an object (x, y, z) in ordinary space and
that of local time (t) are, of course, the basic dimensions of geometry.
But the concept of force, at least the gravitational kind, obliges us to
introduce another dimension related to the mass of an object (m).
And modern physics recognizes other such, notably the one which
measures a body's electric charge (q). The role of the so-called fun-
damental constants of physics is primarily to transpose quantities like
mass and charge into geometrical ones, principally lengths (Wesson
1999, pp. 2-11). This is illustrated most cogently by the conversion

Higher-Dimensional Physics 3

of the time to an extra coordinate x 4 = ct via the use of the speed of
light, a ploy due to Minkowski and Einstein which forms the founda-
tion of 4D spacetime.

The idea of a dimension is, to a certain extent, malleable. It is
also important to notice that modern field theories, like general rela-
tivity, are written in terms of tensor equations which are not restricted
in their dimensionality. One can speculate that had Einstein been
formulating his theory of gravity today, he might have established this
anonymity of dimension as a principle, on a par with the others with
which we are familiar, such as that of equivalence (see Chapter 3). It
is this freedom to choose the dimensionality which underlies the nu-
merous extensions of general relativity. These include the original
5D Kaluza-Klein theory, its modern variants which are called in-
duced-matter and membrane theory, plus the higher extensions such
as 10D supersymmetry, 11D supergravity and the higher-D versions
of string theory.

Kaluza initiated field theory with more than the 4 dimensions
of spacetime in 1921, when he published a paper which showed how
to unify gravity (as described by Einstein's equations) with electro-
magnetism (as described by Maxwell's equations). It is well known
that Einstein kept Kaluza' s paper for a couple of years before finally
as referee allowing it to go forward. However, Einstein was then and
remained in his later years an advocate of extra dimensions. For ex-
ample, a letter to Kaluza from Einstein in 1919 stated "The formal
unity of your theory is astonishing" (Halpern, 2004, p.l). Indeed, the

4 Five-Dimensional Physics

natural way in which the 4 Maxwell equations fall out of the 15 field
equations of what is a kind of general relativity in 5D, has since come
to be called the Kaluza-Klein miracle. However, the mathematical
basis of the unification is simple: In 5D there are 15 independent
components of the metric tensor, of which one refers to a scalar field
which was not at the time considered significant and was so sup-
pressed. For similar reasons, to do with the presumed unobservability
of effects to do with the extra dimension, all derivatives of the other
metric coefficients with respect to the extra coordinate were set to
zero (the "cylinder" condition). This left 14 metric coefficients,
which could depend on the 4 coordinates of spacetime
(x a , a = 0,123 for t,xyz). These 14 coefficients were determined
by 14 field equations. The latter turned out to be the 10 Einstein
equations and the 4 Maxwell equations. Voila: a unification of grav-
ity and electromagnetism.

Klein pushed the 5D approach further in 1926, when he pub-
lished a paper which showed how to incorporate quantum effects into
the theory. He did this by the simple device of assuming that the to-
pology of the extra dimension was not flat and open, but curved into a
circle. In other words, while a local orbit in spacetime (x a ) would be
straight, an orbit in the extra dimension (x 4 ) would merely go around
and around. This cyclic behaviour would lead to quantum effects,
provided the extra dimension were rolled up to a microscopic size
("compactification"). The size of the extra dimension was presumed
to be related to the parameter typical of quantum phenomena, namely

Higher-Dimensional Physics 5

Planck's constant h. Among other consequences of the closed topol-
ogy of the extra dimension, it was shown that the cyclic momentum
could be related to the charge of the electron e, thus explaining its
quantization.

The brainwaves of Kaluza and Klein just summarized are the
kind which are neat and yet powerful. They continued to be held in
high regard for many years in theoretical physics, even though the
latter was redirected by the algebraically simple and effective ideas on
wave mechanics that were soon introduced by Schrodinger, Heisen-
berg and Dirac. Kaluza-Klein theory later underwent a revival, when
Einstein's theory was recognized as the best basis for cosmology. But
something has to be admitted: Kaluza-Klein theory in its original
form is almost certainly wrong.

By this, it is not meant that an experiment was performed
which in the standard but simplistic view of physics led to a disproval
of the 5D theory. Rather, it means that the original Kaluza-Klein the-
ory is now acknowledged as being at odds with a large body of mod-
ern physical lore. For example, the compactification due to Klein
leads to the prediction that the world should be dominated by particles
with the Planck mass of order 10" 5 g, which is clearly not the case.
(This mismatch is currently referred to as the hierarchy problem, to
which we will return.) Also, the suppression of the scalar field due to
Kaluza leaves little room to explain the "dark energy" currently be-
lieved to be a major component of the universe. (This is a generic
form of what is commonly referred to as the cosmological-constant

6 Five-Dimensional Physics

problem, to which attention will be given later.) Further, the cylinder
condition assumed by both fathers of 5D field theory effectively rules
out any way to explain matter as a geometrical effect, something
which Einstein espoused and is still the goal of many physicists.

It is instructive to recall at this juncture the adage which
warns us not to throw out the baby with the bath-water. In this in-
stance, the baby is the concept of a 5 (or higher) D space; whereas the
water is the smothering algebraic restrictions which were applied to
the theory in its early days as a means of making progress, but which
are now no longer needed. Hence modern Kaluza-Klein theory,
which is algebraically rich and exists in several versions.

1.3 Higher-Dimensional Theories

These may be listed in terms of their dimensionality and
physical motivation. However, all are based on Einstein's theory of
general relativity. The equations for this and its canonical extension
will be deferred to the next section.

Induced-matter theory is based on an unrestricted 5D mani-
fold, where the extra dimension and derivatives with respect to the
extra coordinate are used to explain the origin of 4D matter in terms
of geometry. (For this reason, it is sometimes called space-time-
matter theory.) As mentioned above, this goal was espoused by Ein-
stein, who wished to transpose the "base-wood" of the right-hand side
of his field equations into the "marble" of the left-hand side. That is,
he wished to find an algebraic expression for what is usually called

Higher-Dimensional Physics 7

the energy-momentum tensor [T a A, which was on the same footing
as the purely geometrical object we nowadays refer to as the Einstein
tensor \G ap j . That this is possible in practice was proved using an

algebraic reduction of the 5D field equations by Wesson and Ponce de
Leon (1992). They were, however, unaware that the technique was
guaranteed in principle by a little-known theorem on local embed-
dings of Riemannian manifolds by Campbell (1926). We will return
to the field equations and their embeddings below. Here, we note that
the field equations of 5D relativity with a scalar field and dependence
on the extra coordinate in general lead to 1 5 second-order, non-linear
relations. When the field equations are set to zero to correspond to a
5D space which is apparently empty, a subset of them gives back the
10 Einstein field equations in 4D with sources. That is, there is an
effective or induced 4D energy-momentum tensor which has the
properties of what we normally call matter, but depends on the extra
metric coefficients and derivatives with respect to the extra coordi-
nate. The other 5 field equations give back a set of 4 Maxwell-like or
conservation equations, plus 1 scalar relation which has the form of a
wave equation. Following the demonstration that matter could be
viewed as a consequence of geometry, there was a flurry of activity,
resulting in several theorems and numerous exact solutions (see Wes-
son 1999 for a catalog). The theory has a 1-body solution which sat-
isfies all of the classical tests of relativity in astrophysics, as well as
other solutions which are relevant to particle physics.

8 Five-Dimensional Physics

Membrane theory is based on a 5D manifold in which there is
a singular hypersurface which we call 4D spacetime. It is motivated
by the wish to explain the apparently weak strength of gravity as
compared to the forces of particle physics. It does this by assuming
that gravity propagates freely (into the 5D bulk), whereas particle in-
teractions are constrained to the hypersurface (the 4D brane). That
this is a practical approach to unification was realized by Randall and
Sundrum (1998, 1999) and by Arkani-Hamed, Dimopoulos and Dvali
(1998, 1999). The original theory helped to explain the apparently
small masses of elementary particles, which is also referred to as the
hierarchy problem. In addition, it helped to account for the existence
and size of the cosmological constant, since that parameter mediates
the exponential factor in the extra coordinate which is typical of dis-
tances measured away from the brane. As with induced-matter the-
ory, the membrane approach has evolved somewhat since its incep-
tion. Thus there has been discussion of thick branes, the existence of
singular or thin branes in (4+d) dimensions or d-branes, and the pos-
sible collisions of branes as a means of explaining the big bang of tra-
ditional 4D cosmology. It should also be mentioned that the field
equations of induced-matter and membrane theory have recently been
shown to be equivalent by Ponce de Leon (2001; see below also).
This means that the implications of these approaches for physics owes
more to interpretation than algebra, and exact solutions for the former
theory can be carried over to the latter.

Higher-Dimensional Physics 9

Theories in N > 5 dimensions have been around for a consid-
erable time and owe their existence to specific physical circum-
stances. Thus 10D supersymmetry arose from the wish to pair every
integral-spin boson with a half-integral-spin fermion, and thereby
cancel the enormous vacuum or zero-point fields which would ensue
otherwise. The connection to AD classical field theory involves the
fact that it is possible to embed any curved solution (with energy) of a
4D theory in a flat solution (without energy) of a higher-dimensional
theory, provided the larger manifold has a dimension of N > 10.
From the viewpoint of general relativity or a theory like it, which has
10 independent components of the metric tensor or potentials, this is
hardly surprising. The main puzzle is that while supersymmetry is a
property much to be desired from the perspective of theoretical parti-
cle physics, it must be very badly broken in a practical sense. The
reason for this apparent conflict between theory and practice may
have to do with our (perhaps unjustified) wish to reduce physics to
4D, and/or our (probably incomplete) knowledge of how to categorize
the properties of particles using internal symmetry groups. The latter
have, of course, to be taken into account when we attempt to estimate
the "size" of the space necessary to accommodate both gravity and
the particle interactions. Hence the possible unification in terms of
(4+7)D or 11D supergravity. However, a different approach is to
abandon completely the notion of a point - with its implied singular-
ity - and instead model particles as strings (Szabo 2004, Gubser and
Lykken 2004). The logic of this sounds compelling, and string theory

1 Five-Dimensional Physics

offers a broad field for development. But line-like singularities are
not unknown, and some of the models proposed have an unmanagea-
bly high dimensionality (e.g., N =26). One lesson which can be
drawn, though, from N > 5D theory is that there is no holy value of TV
which is to be searched for as if it were a shangri-la of physics.
Rather, the value of TV is to be chosen on utilitarian grounds, in accor-
dance with the physics to be studied.

1.4 Field Equations in N > 4 Dimensions

Just as Maxwell's equations provided the groundwork for
Einstein's equations, so should general relativity be the foundation for
field equations that use more than the 4 dimensions of spacetime.

Einstein's field equations are frequently presented as a match
between a geometrical object G ap and a physical object T a/} , via a
coupling constant k , in the form G ap = /cT ap {a, J3 = 0,1 23) . Here the
Einstein tensor G a/3 = R a/3 -Rg a « 12 depends on the Ricci tensor, the
Ricci scalar and the metric tensor, the last defining small intervals in
4D by a quadratic line element ds 2 = g aj3 dx a dx p . The energy-
momentum tensor T ap depends conversely on common properties of
matter such as the density p and pressure p, together with the
4-velocities u" = dx" I ds . However, even Einstein realized that this
split between geometry and matter is subjective and artificial. One
example of this concerns the cosmological constant A . This was

Higher-Dimensional Physics 1 1

originally added to the left-hand side of the field equations as a geo-
metrical term Ag a/? , whence the curvature it causes in spacetime cor-
responds to a force per unit mass (or acceleration) Arc 2 / 3 , where r
is the distance from a suitably chosen origin of coordinates. But
nowadays, it is commonly included in the right-hand side of the field
equations as an effective source for the vacuum, whose equation of
state is p v = —p v c 2 , where p v - Ac 2 / %kG corresponds to the den-
sity of a non-material medium. (Here we take the dimensions of A
as length" 2 and retain physical units for the speed of light c and gravi-
tational constant G, so the coupling constant in the field equations is
K = %7zGI c 4 .) The question about where to put A is largely one of
semantics. It makes little difference to the real issue, which is to ob-
tain the g a p or potentials from the field equations.
The latter in traditional form are

R a p-\Rga P +^g aP =^f-T ap . (1.1)

Taking the trace of this gives R = Ah-\%nG I c A \T where
T = g"^T a/3 . Using this to eliminate R in (1 . 1) makes the latter read

=^4r^| +A ^ . o.2)

1 2 Five-Dimensional Physics

Here the cosmological constant is treated as a source term for the
vacuum, along with the energy-momentum tensor of "ordinary" mat-
ter. If there is none of the latter then the field equations are

R a p=^g aP ■ (1-3)

These have 10 independent components (since g a/} is symmetrical).
They make it clear that A measures the mean radius of curvature of a
4D manifold that is empty of conventional sources, i.e. vacuum. If
there are no sources of any kind - or if the ordinary matter and vac-
uum fields cancel as required by certain symmetries - then the field
equations just read

R afi =0 . (1.4)

It is these equations which give rise to the Schwarzschild and other
solutions of general relativity and are verified by observations.

The field equations of 5D theory are taken by analogy with
(1 .4) to be given by

^=0(45 = 0,123,4) . (1.5)

Here the underlying space has coordinates x A =(t,xyz,l) where the
last is a length which is commonly taken to be orthogonal to space-
time. The associated line element is dS 2 = g AB dx A dx B , where the
5D metric tensor now has 15 independent components, as does (1.5).
However, the theory is covariant in its five coordinates, which may be

Higher-Dimensional Physics 1 3

chosen for convenience. Thus a choice of coordinate frame, or gauge,
may be made which reduces the number of g AB to be determined
from 15 to 10. This simplifies both the line element and the field
equations.

The electromagnetic gauge was used extensively in earlier
work -on 5D relativity, since it effectively separates gravity and elec-
tromagnetism. A more modern form of this expresses the 5D line
element as parts which depend on g a/3 (akin to the Einstein gravita-
tional potentials with associated interval ds 2 = g ap dx a dx p ), O (a
scalar field which may be related to the Higgs field by which particles
acquire their masses), and A M (related to the Maxwell potentials of
classical electromagnetism). The 5D line element then has the form

dS 2 =ds 2 +£0 2 (dx 4 +A M dx M f . (1.6)

Here £- = ±1 determines whether the extra dimension (g 44 =£-0 2 jis
spacelike or timelike: both are allowed by the mathematics, and we
will see elsewhere that s = -\ is associated with particle-like behav-
iour while s = +1 is associated with wave-like behaviour. Most work
has been done with the former choice, so we will often assume that
the 5D metric has signature (+ ) . Henceforth, we will also ab-
sorb the constants c and G by a suitable choice of units. Then the dy-
namics which follows from (1.6) may be investigated by minimizing

14 Five-Dimensional Physics

the 5D interval, via £[Jjs] = (Wesson 1999, pp. 129-153). In

general, the motion consists of the usual geodesic one found in Ein-
stein theory, plus a Lorentz-force term of the kind found in Maxwell
theory, and other effects due to the extended nature of the geometry
including the scalar field. Further results on the dynamics, and the
effective 4D energy-momentum tensor associated with the off-
diagonal terms in line elements like (1.6), have been worked out by
Ponce de Leon (2002). We eschew further discussion of metrics of
this form, however, to concentrate on a more illuminating case.

The gauge for neutral matter has a line element which can be
written

dS 2 =g afi (x r ,l)dx a dx p +s0 2 (x r ,l) . (1.7)

In this we have set the electromagnetic potentials ( g 4a ) to zero, but
the remaining degree of coordinate freedom has been held in reserve.
(It could in principle be used to flatten the scalar potential via
|g 44 | = 1 , but while we will do this below it is instructive to see what
effects follow from this field.) The components of the 5D Ricci ten-
sor for metric (1.7) have wide applicability. They are:

'K„=<K„-*^^*^-

O 20M O

Higher-Dimensional Physics 1 5

_§Zs M a 1

2<D 4

(1.8)

Here a comma denotes the ordinary partial derivative, a semicolon
denotes the ordinary 4D covariant derivative, n<D = g MV <t> M . v and

l 21X
r = fiO . Superscripts are used here and below for the 5D tensors

and their purely 4D parts, whenever there is a risk of confusion.
When the components (1.8) are used with the 5D field equations
(1.5), it is clear that we obtain tensor, vector and scalar equations
which have distinct applications in physics.

The tensor components of (1.8), in conjunction with the 5D
field equations R AB = (1.5), give the 10 field equations of Ein-
stein's general relativity. The method by which this occurs is by now
well known (Wesson and Ponce de Leon 1992). In summary, we
form the conventional 4D Ricci tensor, and with it and the 4D Ricci
scalar construct the 4D Einstein tensor G a p= A R a p-^Rg ap l2 . The

1 6 Five-Dimensional Physics

remaining terms in 5 R a/3 of (1.8) are then used to construct an effec-
tive or induced 4D energy-momentum tensor via G a p=%nT ap .
Several instructive results emerge during this process. For example,
the 4D scalar curvature just mentioned may be shown using all of
(1.8) to ge given by

4 * = ^[g^M+(^J 2 ] • (1.9)

This relation has been used implicitly in the literature, but explicitly
as here it shows that: (a) What we call the curvature of 4D spacetime
can be regarded as the result of embedding it in an x 4 -dependent 5D
manifold; (b) the sign of the 4D curvature depends on the signature of
the 5D metric; (c) the magnitude of the 4D curvature depends
strongly on the scalar field or the size of the extra dimension
(g 44 =£<D 2 ), so while it may be justifiable to neglect this in astro-
physics (where the 4D curvature is small) it can be crucial in cosmol-
ogy and particle physics. Another instructive result concerns the form
of the 4D energy-momentum tensor. It is given by

artv-^-5-Ji**-

O 2<D 2 O

'""gpvASaPA S„p\ „„ I uv \2~jl

. (1.10)

Higher-Dimensional Physics 1 7

This relation has been used extensively in the literature, where it has
been shown to give back all of the properties of ordinary matter (such
as the density and pressure) for standard solutions. However, it has
further implications, and shows that: (a) What we call matter in a
curved 4D spacetime can be regarded as the result of the embedding
in an x 4 -dependent (possibly flat) 5D manifold; (b) the nature of the
4D matter depends on the signature of the 5D metric; (c) the 4D
source depends on the extrinsic curvature of the embedded 4D space-
time and the scalar field associated with the extra dimension, which
while they are in general mixed correspond loosely to ordinary matter
and the stress-energy of the vacuum. In conclusion for this paragraph,
we see that a 5D manifold - which is apparently empty - contains a
4D manifold with sources, where the tensor set of the 5D field equa-
tions corresponds to the 4D Einstein equations of general relativity.

The vector components of (1.8), in conjunction with (1.5), can
be couched as a set of conservation equations which resemble those
found in Maxwellian electromagnetism and other field theories. They
read

^=0 . 0-11)

where the 4-vector concerned is defined via

These are usually easy to satisfy in the continuous fluid of induced-
matter theory, and are related to the stress in the surface (x 4 = 0) of

1 8 Five-Dimensional Physics

membrane theory with the Z2 symmetry (see below). It should be
noted that these relations do not come from some external criterion
such as the minimization of the line element, but are derived from and
are an inherent part of the field equations.

The scalar or last component of (1.8), when set to zero in ac-
cordance with the field equations (1.5), yields a wave-type equation
for the potential associated with the fifth dimension (g 44 =£<J> 2 ) in
the metric (1.7). It is

D *-^|^ + ^--^?^l • (U3)

Here as before nO = g afi ® a . p and some of the terms on the right-
hand side are present in the energy-momentum tensor of (1.10). In
fact, one can rewrite (1.13) for the static case as a Poisson-type equa-
tion with an effective source density for the O -field. In general (1.13)
is a wave equation with a source induced by the fifth dimension.

Let us now leave the gauge for neutral matter (1.7) and focus
on a special case of it, called the canonical gauge. This was the
brainchild of Mashhoon, who realized that if one factorizes the 4D
part of a 5D metric in a way which mimics the use of cosmic time in
cosmology, significant simplification follows for both the field equa-
tions and especially the equations of motion (Mashhoon, Liu and
Wesson 1994). The efficacy of this gauge is related to the fact that a
quadratic factor in / on the 4D part of a 5D model has algebraic con-

Higher-Dimensional Physics 1 9

sequences similar to those of a quadratic factor in t on the 3D part of a
4D cosmological model. The latter case, in the context of 4D Fried-
mann-Robertson- Walker (FRW) cosmologies, is known as the Milne
universe. This has several interesting properties (Rindler 1977). We
will come back later to the Milne universe as a lower-dimensional
example of highly-symmetric 5D manifolds. For now, we note the
form of the metric and summarize its properties.

The 5D canonical metric has a line element given by

dS 2 =j^g a/; (x\l)dx a dx fl -dl 2 , (1.14)

where x 4 = I is the extra coordinate and L is a constant length intro-
duced for the consistency of physical dimensions. There is an exten-
sive literature on (1.14), both with regard to solutions of the field
equations (1.5) and the equations of motion which follow from mini-
mizing the interval S in (1.14). Some of the consequences of (1.14)
can be inferred from what we have already learned, while some will
become apparent from later study. But for convenience we here
summarize all of its main properties following Wesson (2002):
(a) Mathematically (1.14) is general, insofar as the five available co-
ordinate degrees of freedom have been used to set g 4a = 0, g 44 = -1 .
Physically, this removes the potentials of electromagnetic type and
flattens the potential of scalar type, (b) The metric (1.14) has been
extensively used in the field equations, and many solutions are
known. These include solutions for the 1-body problem and cosmol-

20 Five-Dimensional Physics

ogy which have acceptable dynamics and solutions with the opposite
sign for g 44 which describe waves, (c) When dg a/3 Idl = in (1.14),
the 15 field equations R AB = of (1.5) give back the Einstein equa-
tions as described above, now in the form G a/3 = 3g a/3 1 1? . These in
general identify the scale L as the characteristic size of the 4-space.
For the universe, the last-noted relations define an Einstein space with

A = 3/L 2 , (1.15)

which identifies the cosmological constant, (d) This kind of local
embedding of a 4D Riemann space in a 5D Ricci-flat space can be
applied to any N, and is guaranteed by Campbell's theorem. We will
take this up in more detail below, (e) The factorization in (1.14) says
in effect that the 4D part of the 5D interval is (// L)ds , which defines
a momentum space rather than a coordinate space if / is related to m,
the rest mass of a particle. This has been discussed in the literature as
a way of bridging the gap between the concepts of acceleration as
used in general relativity, and force (or change of momentum) as used
in quantum theory, (f) Partial confirmation of this comes from a
study of the 5D geodesic and a comparison of the constants of the
motion in 5D and 4D. In the Minkowski limit, the energy of a parti-
cle moving with velocity v is E =/(l-v 2 J in 5D, which agrees

with the expression in 4D if / = m. (g) The five components of the
geodesic equation for (1.14) split naturally into four spacetime com-

Higher-Dimensional Physics 2 1

ponents and an extra component. For dg a/3 Idl ^ , the former con-
tain terms parallel to the 4-velocity u a , which do not exist in 4D gen-
eral relativity. We will look into this situation later. But we note now
that for dg a p Idl = , the motion is not only geodesic in 5D but geo-
desic in 4D, as usual. Indeed, for dg ap Idl = , we recover the 4D

Weak Equivalence Principle as a kind of symmetry of the 5D metric.

The preceding list of consequences of the canonical metric
(1.14) shows that it implies departures from general relativity when
its 4D part depends on the extra coordinate, but inherits many of the
properties of Einstein's theory when it does not. In the latter case, the
4D cosmological constant is inherited from the 5D scaling, and has a
value A = ±3/L 2 depending on the signature of the extra dimension
(s = +l) . This is a neat result, and elucidates the use of de Sitter and
anti-de Sitter spaces in approaches to cosmology and particle produc-
tion, which use quantum-mechanical approaches such as tunneling.
However, in general we might expect the potentials of spacetime to
depend on the extra coordinate. Both for this case as in (1.14), and
for the case where the scalar potential is significant as in (1.7), the
vacuum will have a more complicated structure than that implied by
the simple cosmological constant just noted. It was shown in (1.10)
that in general the effective 4D energy-momentum tensor for neutral
matter in 5D theory contains contributions from both ordinary matter
and the vacuum. Ordinary matter (meaning material particles and
electromagnetic fields) displays an enormous complexity of structure.

22 Five-Dimensional Physics

"Vacuum matter" (meaning the scalar field and virtual particles which
defy Heisenberg's uncertainty relation) may display a corresponding
complexity of structure. To use a cliche, 5D induced-matter theory
implies that we may have only scratched the surface of "matter".

Membrane theory uses an exponential rather than the quad-
ratic of (1 . 14) to factorize the 4D part of a 5D metric. Thus a general-
ized form of the type of metric considered by Randall and Sundrum
(1998, 1999) is

dS 2 =e F(l) g ap dx a dx p -dl 2 . (1.16)

Here F(l} is called the warp factor, and is commonly taken to de-
pend on the cosmological constant A and the extra coordinate x 4 = /
in such a way as to weaken gravity away from the brane (/ = 0). Par-
ticle interactions, by comparison, are stronger by virtue of being con-
fined to the brane, which is effectively the focus of spacetime. An
important aspect of (at least) the early versions of brane theory is the
assumption of Zi symmetry, which means in essence that the physics
is symmetric about the hypersurface / = 0. This prescription is simple
and effective, hence the popularity of membrane theory. However, a
comparison of (1.16) and (1.7) shows that the former is merely a spe-
cial case of the latter, modulo the imposition of the noted symmetry.
In fact, examination shows that membrane theory and induced-matter
theory are basically the same from a mathematical viewpoint, even if
they differ in physical motivation. The most notable difference is that
for membrane theory particles are confined to the spacetime hypersur-

Higher-Dimensional Physics 23

face by the geometry, which is constructed with this in mind; whereas
for induced-matter theory particles are only constrained by solutions
of the 5D geodesic equation, and can wander away from spacetime at
a slow rate governed by the cosmological constant or oscillate around
it. That the field equations of membrane theory and space-time-
matter theory are equivalent was shown by Ponce de Leon (2001).
His work makes implicit use of embeddings, and we defer a discus-
sion of these plus the connection between brane and STM worlds to
the next section.

Embeddings must, however, play an important role in the ex-
tension of 5D theories to those of even higher dimension. That this is
so becomes evident when we reflect on the preceding discussion. In
it, we have morphed from 4D general relativity with Einstein's equa-
tions in the forms (1.1)— (1.4), to 5D relativity with the apparently
empty field equations (1.5). These lead us to consider the electromag-
netic gauge (1.6) and the gauge for neutral matter (1.7). The latter has
associated with it the 5D Ricci components (1.8), which imply the 4D
Ricci scalar (1.9) and the effective 4D energy-momentum tensor
(1.10). The latter balances the Einstein equations, and leaves us with
the 4 vector terms which satisfy (1.11) by virtue of (1.12), plus the 1
scalar wave equation (1.13). When the 5D metric has a 4D part which
is factorized by a quadratic in the extra coordinate, we obtain the ca-
nonical metric (1.14), which leads us to view the cosmological con-
stant (1.15) as a scale inherited from 5D. When alternatively the 5D
metric has a 4D part which is factorized by an exponential in the extra

24 Five-Dimensional Physics

coordinate, we obtain the warp metric (1.16), which leads us to view
spacetime as a singular surface in 5D. All of these results are en-
trained - in the sense that they follow from the smooth embedding of
4D in 5D. Certain rules of differential geometry underly this embed-
ding. The main one of these is a theorem of Campbell (1926), which
was revitalized by Tavakol and coworkers, who pointed out that it
also constrains the reduction from general relativity in 4D to models
of gravity in 3D and 2D which may be more readily quantized (see
Rippl, Romero and Tavakol 1995). It is not difficult to see how to
extend the formalism outlined above for N > 5, so yielding theories of
supersymmetry, supergravity, strings and beyond. But in so doing
there is a danger of sinking into an algebraic morass. An appreciation
of embedding theorems can help us avoid this and focus on the
physics.

1.5 A Primer on Campbell's Theorem

Embedding theorems can be classified as local and global in
nature. We are primarily concerned with the former because our field
equations are local. (The distinction is relevant, because global theo-
rems are more difficult to establish; and since they may involve
boundary conditions, harder to satisfy.) There are several local em-
bedding theorems which are pertinent to AD field theory, of which the
main one is commonly attributed to Campbell (1926). He, however,
only outlined a proof of the theorem in a pedantic if correct treatise on
differential geometry. The theorem was studied and established by

Higher-Dimensional Physics 25

Magaard (1963), resurrected as noted above by Rippl, Romero and
Tavakol (1995), and applied comprehensively to gravitational theory
by Seahra and Wesson (2003). The importance of Campbell's theo-
rem is that it provides an algebraic method to proceed up or down the
dimensionality ladder N of field theories like general relativity which
are based on Riemannian geometry. Nowadays, it is possible to prove
Campbell's theorem in short order using the lapse-and-shift technique
of the ADM formalism. The latter also provides insight to the con-
nection between different versions of 5D gravity, such as induced-
matter and membrane theory. We will have reason to appeal to
Campbell's theorem at different places in our studies of 5D field the-
ory. In the present section, we wish to draw on results by Ponce de
Leon (2001) and Seahra and Wesson (2003), to give an ultra-brief
account of the subject.

Campbell's theorem in succinct form says: Any analytic
Riemannian space V n (s,t) can be locally embedded in a Ricci-flat
Riemannian space V n+l (5 + 1, t) or V n+1 (s, t + 1) .

We are here using the convention that the "small" space has
dimensionality n with coordinates running to n -1, while the "large"
space has dimensionality n +1 with coordinates running to n. The
total dimensionality is N = 1+ n, and the main focus is on N = 5.

To establish the veracity of this theorem (in a heuristic fash-
ion at least), and see its relevance (particularly to the theories consid-
ered in the preceding section), consider an arbitrary manifold S n in a

26 Five-Dimensional Physics

Ricci-flat space V n+l . The embedding can be visualized by drawing a
line to represent Z n in a surface, the normal vector n A to it satisfying
n-n = n A n A =s = ±1 . If e^ form an appropriate basis and the ex-
trinsic curvature of Z^ is K ap , the ADM constraints read

G AB n A n B = --{eR° + K ap K a/} -K 2 ) =

G AB e a nB = K i, P - K ,a =0 ■ ( l • 17 )

These relations provide 1 + n equations for the 2xn(n + \)l2 quanti-
ties g a „, K ap . Given an arbitrary geometry g ap for S n , the con-
straints therefore form an under-determined system for K ap , so infi-
nitely many embeddings are possible. This implies that the embed-
ding of a system of 4D equations like (1 .1)— (1.4) in a system of 5D
equations like (1.5) is always possible.

This demonstration of Campbell's theorem can easily be ex-
tended to the case where V n+i is a de Sitter space or anti-de Sitter
space with an explicit cosmological constant, as in brane theory. De-
pending on the application, the remaining «(« + l)-(« + l) = (« 2 -lJ

degrees of freedom may be removed by imposing initial conditions on
the geometry, physical conditions on the matter, or conditions on a
boundary.

Higher-Dimensional Physics 27

The last is relevant to membrane theory with the Z 2 symmetry.
To see this, let us consider a fairly general line element with
dS 2 = g a/} (x r ,l)dx a dx 13 + e df where g ap = g ap (x r ,+/) for / >
and g a p = g a p\x r -l) for /<0 in the bulk (Ponce de Leon 2001).

Non-gravitational fields are confined to the brane at / = 0, which is a
singular surface. Let the energy-momentum in the brane be repre-
sented by S(l)S AB (where S AB n A =0) and that in the bulk by T AB .

Then the field equations read G AB - k\_5(1)S ab +T ab ~] where k is a
5D coupling constant. The extrinsic curvature discussed above
changes across the brane by an amount A a/3 =K a/3 (z i>o )-K a0 (L I<o )
which is given by the Israel junction conditions. These imply

-Js^—Sg^j . (1.18)

But the / =0 plane is symmetric, so

*<*(0=-*«*foJ=-f (^ ~\ s sSj ■ < L19 )

This result can be used to evaluate the 4-tensor

-Kg a ,=--S ap . (1.20)

28 Five-Dimensional Physics

However, P ap is actually identical to the 4-tensor
\SafiA ~g a pg f ' V &MvAy'2-® of induced-matter theory, which we noted
above in (1.12). It obeys the field equations Pf. p = of (1.1 1), which
are a subset of R AB = . That is, the conserved tensor P a/3 of in-
duced-matter theory is essentially the same as the total energy-
momentum tensor in Z 2 -symmetric brane theory. Other correspon-
dences can be established in a similar fashion.

The preceding exercise confirms the inference that induced-
matter theory and membrane theory share the same algebra, and helps
us understand why matter in 4D can be understood as the conse-
quence of geometry in 5D.

1.6 Conclusion

In this chapter, we have espoused the idea that extra dimen-
sions provide a way to better understand known physics and open a
path to new physics.

The template is Einstein's general relativity, which is based
on a fusion of the primitive dimensions of space and time into 4D
spacetime. The feasibility of extending this approach to 5D was
shown in the 1920s by Kaluza and Klein, and if we discard their re-
strictive conditions of cylindricity and compactification we obtain a
formalism which many researchers believe can in principle offer a
means of unifying gravity with the forces of particle physics (Section
1 .2). 5D is not only the simplest extension of general relativity, but is

Higher-Dimensional Physics 29

also commonly regarded as the low-energy limit of higher-TV theories.
Most work has been done on two versions of 5D relativity which are
similar mathematically but different physically. Induced-matter (or
space-time-matter) theory is the older version. It views 4D mass and
energy as consequences of the extra dimension, so realizing the dream
of Einstein and others that matter is a manifestation of geometry.
Membrane theory is the newer version of 5D relativity. It views 4D
spacetime as a hypersurface or brane embedded in a 5D bulk, where
gravity effectively spreads out in all directions whereas the interac-
tions of particles are confined to the brane and so stronger, as ob-
served. These theories are popular because they allow of detailed cal-
culations, something which is not always the case with well-
motivated but more complicated theories for N> 5 (Section 1.3). The
field equations of all theories in N > 4 dimensions have basically the
same structure, and this is why we treated them together in Section
1.4. There we concentrated again on the case N = 5, paying particular
attention to the equations which allow us to obtain the 15 components
of the metric tensor. In the classical view, these are potentials, where
the 10-4-1 grouping is related to the conventional split into gravita-
tional, electromagnetic and scalar fields. In the quantum view, the
corresponding particles are the spin-2 graviton, the spin-1 photon and
the spin-0 scalaron. The extension of the metric and the field equa-
tions to N > 5 is obvious, in which case other particles come in.
However, the extension of general relativity to TV > 4 needs to be
guided by embedding theorems. The main one of these dates again

30 Five-Dimensional Physics

from the 1920s, when it was outlined by Campbell. The plausibility
of Campbell's theorem can be shown in short order using modern
techniques, as can the mathematical equivalence of induced-matter
theory and membrane theory (Section 1.5). In summary, the contents
of this chapter provide a basis for writing down the equations for
2 < N < oo and deriving a wealth of physics.

In pursuing this goal, however, some fundamental questions
arise. In studying (say) 5D relativity, we introduce an extra coordi-
nate (x 4 = / ), and an extra metric coefficient or potential (g 44 ). The
two are related, and by analogy with proper distance in the ordinary

3D space of a curved 4D manifold we can define J g 44 ( x a , / J dl\ as

the "size" of the extra dimension. Even at this stage, two issues arise
which need attention.

What is the nature of the fifth coordinate? Possible answers
are as follows: (1) It is an algebraic abstraction. This is a conserva-
tive but sterile opinion. It implies that / figures in our calculations,
but either does not appear in our final answer, or is incapable of
physical interpretation once we arrive there. (2) It is related to mass.
This is the view of induced-matter theory, where quantities like the
density and pressure of a fluid composed of particles of rest mass m
can be calculated as functions of / from the field equations. Closer
inspection shows that for the special choice of gauge known as the
pure-canonical metric, / and m are in fact the same thing. We will
return to this possibility in later chapters, but here note that in this in-
terpretation the scalar field of classical 5D relativity is related to the

Higher-Dimensional Physics 3 1

Higgs (or mass-fixing) field of quantum theory. (3) It is a length per-
pendicular to a singular hypersurface. This is the view of membrane
theory, where the hypersurface is spacetime. It is an acceptable opin-
ion, and as we have remarked it automatically localizes the 4D world.
But since we are made of particles and so confined to the hypersur-
face, our probes of the orthogonal direction have to involve quantities
related to gravity, including masses.

The other issue which arises at the outset with 5D relativity
concerns the size of the extra dimension, defined as above to include
both the extra coordinate and its associated potential. This is a sepa-
rate, if related, issue to what we discussed in the preceding paragraph.
We should recall that even in 4D relativity, drastic physical effects
can follow from the mathematical behaviour of the metric coeffi-
cients. (For example, near the horizon of an Einstein black hole in
standard Schwarzschild coordinates, the time part of the metric
shrinks to zero while the radial part diverges to infinity.) This issue is
often presented as the question: Why do we not see the fifth dimen-
sion? Klein tried to answer this, as we have seen, by arguing that the
extra dimension is compactified (or rolled up) to a microscopic size.
So observing it would be like looking at a garden hose, which appears
as a line from far away or as a tube from close up. Since distances are
related to energies in particle experiments, we would only expect the
finite size of the fifth dimension to be revealed in accelerators of
powers beyond anything currently available. This is disappointing.
But more cogently, and beside the fact that it leads to conflicts, many

32 Five-Dimensional Physics

researchers view compactification in its original form as a scientific
cop-out. The idea can be made more acceptable, if we assume that
the universe evolves in such a way that the fifth dimension collapses
as the spatial part expands. But even this is slightly suspect, and bet-
ter alternatives exist. Thus for membrane theory the problem is
avoided at the outset, by the construction of a 5D geometry in which
the world is localized on a hypersurface. For induced-matter theory,
particles are constrained with respect to the hypersurface we call
spacetime by the 5D equations of motion. In the latter theory, modest
excursions in the extra dimension are in fact all around us to see, in
the form of matter.

Let us assume, for the purpose of going from philosophy to
physics, that a fifth dimension may exist and that we wish to demon-
strate it. We already know that 4D general relativity is an excellent
theory, in that it is soundly based in logic and in good agreement with
observation. We do not desire to tinker with the logic, but merely
extend the scope of the theory. Our purpose, therefore, is to look for
effects which might indicate that there is something bigger than
spacetime.

References

Arkani-Hamed, N., Dimopoulos, S., Dvali, G. 1998, Phys. Lett. B429,

263.
Arkani-Hamed, N., Dimopoulos, S., Dvali, G. 1999, Phys. Rev. D59,

086004.

Higher-Dimensional Physics 33

Campbell, J.E. 1926, A Course of Differential Geometry (Clarendon,
Oxford).

Gubser, S.S., Lykken, J.D. 2004, Strings, Branes and Extra Dimen-
sions (World Scientific, Singapore).

Halpern, P. 2004, The Great Beyond (Wiley, Hoboken).

Magaard, L. 1963, Ph.D. Thesis (Kiel).

Mashhoon, B., Liu, H., Wesson, P.S. 1994, Phys. Lett. B33L 305.

Ponce de Leon, J. 2001, Mod. Phys. Lett. A 16, 2291.

Ponce de Leon, J., 2002, Int. J. Mod. Phys. U, 1355.

Randall, L., Sundrum, R. 1998, Mod. Phys. Lett. AL3, 2807.

Randall, L., Sundrum, R. 1999, Phys. Rev. Lett. 83, 4690.

Rindler, W. 1977, Essential Relativity (2nd. ed., Springer, Berlin).

Rippl, S., Romero, C, Tavakol, R. 1995, Class. Quant. Grav. 12,
2411.

Seahra, S.S., Wesson, P.S. 2003, Class. Quant. Grav. 20, 1321.

Szabo, R.J. 2004, An Introduction to String Theory and D-Brane Dy-
namics (World Scientific, Singapore).

Wesson, P.S., Ponce de Leon, J. 1992, J. Math. Phys. 33, 3883.

Wesson, P.S. 1999, Space-Time-Matter (World Scientific, Singapore).

Wesson, P.S. 2002, J. Math. Phys. 43, 2423.

2. THE BIG BANG REVISITED

"Seek, and you shall find" (Matthew, New Testament)

2.1 Introduction

The big bang in 4D general relativity is a singularity, through
which the field equations cannot be integrated. While the standard
big bang can be viewed as a birth event, its lack of computability is
considered by many researchers to be a drawback. Hence the numer-
ous attempts which have been made to avoid it. Some of these are
quite innovative, and include the proposals that it involved a transi-
tion from negative to positive mass (Hoyle 1975), a quantum tunnel-
ing event (Vilenkin 1982) and matter production from Minkowski
space (Wesson 1985). The extension of the manifold from 4 to 5 di-
mensions brings in new possibilities, which we will examine in what
follows. The standard class of 5D cosmological models was found by
Ponce de Leon (1988). He solved the 5D field equations (1.5), as-
suming that the 5D metric was separable, so that the conventional 4D
Friedmann-Robertson- Walker (FRW) models were recovered on hy-
persurfaces where the extra coordinate was held fixed. But while they
are appealing from the buddhistic view, in that they are flat and empty
in 5D (though curved with matter in 4D), the Ponce de Leon models
are not unique. So after contemplating the flat approach we will con-
sider others, in which the 4D big bang can be viewed as a shock
wave, a bounce and even a black hole in 5D.

Revisited 35

These possibilities are all mathematically viable, and might be
considered as too much of a richness in return for the modest act of
extending the dimensionality from 4 to 5. However, these models
have the redeeming feature of being analyzable: we can now calculate
what happens for t < as well as insisting on agreement with astro-
physical data for t > (using the standard identification of the time t
= with the big bang). This is in the tradition of physics. We prefer
to work out the properties of the early universe, instead of being
obliged to accept its creation as fiat.

2.2 Flat 5D Universes

The standard 5D cosmological models of Ponce de Leon
(1988) have been much studied. They can be given a pictorial repre-
sentation using a combination of algebra and computer work (Wesson
and Seahra2001; Seahra and Wesson 2002). Since this also allows
us to gain insight to the nature of the 4D singularity, we follow this
approach here.

The models are commonly written in coordinates x° = t,
x 123 = r6(j) and x 4 = I (we absorb the speed of light and the gravita-
tional constant through a choice of units which renders them unity).
The line element is given by

dS 2 =l 2 dt 2 -t 2la l 2l{x - a) (dr 2 +r 2 dQ 1 )—^- T dl 2 , (2.1)
1 ) (1"«) 2

36 Five-Dimensional Physics

where JQ 2 = (dO 1 + sin 2 6d</) 2 ) . The dimensionless parameter a is

related to the properties of matter.

The latter can be obtained using the technique outlined in
Chapter 1, where we used Campbell's theorem to embed 4D general
relativity with Einstein's equations G afi =%nT ap {af3 = 0,123) in an
apparently empty 5D manifold with field equations R AB =0 (A,B =
0,123,4). Here the effective or induced energy-momentum tensor
can be taken as that for a perfect fluid with density p and pressure p.
Then the class of solutions (2.1) fixes these quantities via

%7T P = -^, %7rp = ^f^-, (2.2)

a r a t

where r = lt is the proper time. The equation of state is
p = (2a/ 3-1) p. For a = 3/2, the scale factor of (2.1) varies as t 2/3 ,
the density and pressure of (2.2) are p = \l 6;zr 2 with p = 0, and we
have the standard k = dust model for the late universe. For a = 2,
the scale factor varies as t m , p = 3/327rr 2 =3p and we have the
standard k = radiation model for the early universe. Cases with
a < 1 describe models that expand faster than the standard FRW ones
and have inflationary equations of state (see below). It should be
noted that p and p of (2.2) refer to the total density and pressure, re-
spectively. These could be split into multiple components, including
visible matter, dark matter, and possible vacuum (scalar) fields. The

The Big Bang Revisited 3 7

last could include a contribution from a time-variable cosmological
"constant" (Overduin 1999), of the type indicated by data on the dy-
namics of galaxies and the age of the universe.

Physically, the Ponce de Leon cosmologies are very accept-
able. Mathematically, they are flat in three dimensions, curved in
four dimensions, and flat in five dimensions. This means that (2.1) in
coordinates (t ,r ,9, <f> , I) is equivalent to 5D Minkowski space in
some other coordinates (T, R, 9, <p,L) with line element

dS 2 = dT 2 -(dR 2 + R 2 dQ 2 )-dL 2 . (2.3)

This does not have a big bang, but the four-dimensional part of equa-
tion (2.1) does (the 4-geometry is singular for t — »0). A situation
similar to this occurs in general relativity with the Milne model. In
many books this is presented as one of the FRW class with negative
spatial curvature, but a fairly simple coordinate transformation makes
the metric a 4D Minkowski space, and accordingly it is devoid of
matter (Rindler 1977). The coordinate transformation between equa-
tion (2.1) and equation (2.3) for the corresponding five-dimensional
case is not simple. It is given by

2\ (2a-l)/aj(l-2a)/(l-a)
r__ \vaM\-a) * I

2[r« 2 J (l-2«)

R(t,r,l) = rt Va l ll{v

„\( r 2\ , , (2a-l)laj(l-2a)/(l-a) 1

3 8 Five-Dimensional Physics

We have made an extensive study of these relations, in order to better
understand the nature of the big bang.

The 4D physics occurs in the FRW-like coordinates (t, r) of
(2.1) on a hypersurface I = k (the angular variables play no physical
role and may be suppressed). The 4D models may, however, be re-
garded as embedded in a flat space with the coordinates (T, R, L) of
(2.3) and viewed therefrom. With the help of (2.4), we can thus ob-
tain pictures of the 4D models and study the structure of their singu-
larities. This can be done for a range of the assignable parameters
(cc,l ) . As an aid to visualization, we let r and R run over positive
and negative values so that the images are symmetric about R = (if it
is desired to have r, R > 0, then one of the symmetric halves may be
deleted). To the same end, we add lines of constant t that intersect the
R = plane orthogonally (in a Euclidean sense) and lines of constant r
that run parallel to the symmetry plane at R = 0. The models grow in
ordinary space as they evolve in time. We present informative cases
which are illustrated in the accompanying figures.

Model I (a = 3/2, / = 1). This is the standard k = model for
the late universe. By (2.1) it has a scale factor that varies as t m , and
by (2.2) it has p = 0. The shape is parabolic, and lines of constant r
meet at a pointlike big bang at T = R = L = 0.

Model II (a = 1/30, / = 1). This is an inflationary k =
model for the very early universe. By (2.1) it has a scale factor that
varies as t 30 , and by (2.2) it has (p + 3p) < 0, so what is sometimes
called the gravitational density is negative and powers a strong

The Big Bang Revisited 39

FIG. 2.1 - Hypersurfaces in 5D corresponding to spatially-flat FRW
cosmologies in 4D. The upper case is the standard dust model, the
lower case is an inflationary model, as discussed in the text.

40 Five-Dimensional Physics

FIG. 2.2 - Hypersurfaces in 5D corresponding to spatially-flat, infla-
tionary FRW cosmologies in 4D. The outer surface has k = 60, the
middle surface has / = 40 and the inner surface has k = 20, as dis-
cussed in the text.

The Big Bang Revisited 41

acceleration. Comoving trajectories converge to a point arrived at by
following the null ray T + L = R = into the past, towards past null
infinity.

Model III (a =1/3, k = 20, 40, 60). This is a set of inflation-
ary k = models with scale factors that vary as f 3 , (p + 3p) < and a
common big bang at past null infinity.

Figures 2.1 and 2.2 reproduce known physics for the standard
spatially-flat FRW models while adding a new perspective. Also, our
figures for classical inflationary models are strikingly similar to those
generated by computer for a stochastic theory of inflation based on
quantum field theory (Linde 1994). In theories of the latter type, the
rest masses of particles are basically zero and become finite through a
mechanism involving the Higgs field (Linde 1990). In 5D classical
theory, it has been argued that the Higgs potential is related to the g44-
component of the metric tensor (Wesson 1999). Alternatively, this
component may be related to the effective 4D cosmological constant
(Overduin 1999). In either case, we see from (2.1) that this factor is
time-dependent, which raises the possibility of testing such models
using particle masses and gravitational lensing.

In general, the dynamics of models like (2.1) may be studied
by solving the 5D geodesic equation. We will give this detailed con-
sideration elsewhere, but note here some results which follow if we
use the 5D proper time S of (2.1) to characterize the motion. Thus the
5D geodesic gives the 5 velocities U A =dx A I dS as U' =
0(i = 1,2,3), with U =+a(2a-l)~ U2 r l and U 4 =±(l-a)V 1

42 Five-Dimensional Physics

(2«-l) r 1 . There is no motion in 3 -space, and the galaxies are
static with respect to each other because the coordinates (t, r, I) in
(2.1) are designed to be spatially comoving. This is the same pre-
scription as used in most presentations of the 4D Robertson- Walker
metric (Rindler 1977). The motions detected spectroscopically by
observational cosmology refer to a noncomoving frame. In five di-
mensions, the coordinate transformation to T = It, 7 = t lla r and / =
At A l (where A is a constant introduced for the purely algebraic pur-
pose of distinguishing T from / ) results in U° = +(2or-l) a \
U 4 =0, s =U*=0 andU l = +(2a-l)~ m TIT. The last mem-
ber is just Hubble 's law.

We see that the 5D Ponce de Leon models (2.1) have the same
law for galaxy motions as the standard 4D FRW models, as well as
the same expressions for the density and pressure (2.2). However, the
line element (2.1) may be connected to the 5D Minkowski one (2.3)
by the coordinate transformations (2.4). This remarkable fact may be
confirmed by computer, using a symbolic software package such as
GRTensor (Lake 2004). There may be a singularity in the matter-
filled and curved 4D space, but one does not exist in the empty and
flat 5D space. In other words, the 4D big bang is due to an unfortu-
nate choice of coordinates in a smooth 5D manifold. To this extent, it
is something of an illusion.

The Big Bang Revisited 43

2.3 The Singularity as a Shock Wave

In this section, we will summarize the properties of another
exact solution of the 5D field equations whose Riemann-Christoffel
tensor obeys Rabcd = 0, meaning that it is flat. However, the new so-
lution depends only on the combined variable u = [t-l^, so it de-
scribes a wave. Solutions of this type in Newtonian hydrodynamics
where the density p and/or pressure p change abruptly are called
shock waves. We can evaluate these properties of matter as before,
by studying the 4D Einstein equations G ap =%KT ap {a,j5 = 0,123)
which are contained in the 5D field equations R A b = 0(A, B = 0,123,4;
see Wesson, Liu and Seahra 2000; Ponce de Leon 2003). The proper-
ties of the 5D solution imply that we can view the 4D singularity as a
kind of shock wave.

The solution has a 5D line element given by

dS 2 = b 2 dt 2 - a 2 [dr 2 + r 2 dQ 2 ) - b 2 dl 2

i
a = (hu)(w)

b = {hu) 2(2 + 3«) . (2.5)

The notation here is the same as in the preceding section, and the so-
lution may be confirmed using the algebra of Chapter 1 or by com-
puter (Lake 2004). It depends on 2 constants, h and a. The first has
the physical dimensions of an inverse length or time, and is related to
Hubble's parameter (see below). The second is dimensionless, and is

44 Five-Dimensional Physics

related to the properties of matter. There is an associated equation of
state, and after some algebra we find
p = ap

(2 + 3a) z

(2.6)

We see that a = corresponds to the late (dust) universe, and a = 1/3
corresponds to the early (radiation) universe.

To elucidate the physical properties of the solution, it is in-
structive to change from the coordinate time t to the proper time T.
This is defined by dT = b dt, so

T

- \—(hu)wa) . (2.7)

3{l + a )h K ' K }

r hich determines the dynamics of the model by

The 4D scale factor which determines the dynamics of the model by
(2.5) and (2.7) is then

For a = 0, a(T) oc T m as in the standard (Einstein-de Sitter) dust
model. For a = 1/3, a{T) <x T % as in the standard radiation model.
The value of Hubble 's parameter is given by

TT 1 da 1 da dt h „ ,-H^-]

adT a 8t dT (2 + 3«) v '

~3(l + a)T

(2.9)

The Big Bang Revisited 45

For a = and 1/3, (2.9) shows that H has its standard values in terms

of the proper time. We can also convert the density (2.6) from t to T

using (2.7) and find

4 1 1

%np = -— . (2.10)

3(1 + or) r

For a =0 we have p = U6nT 2 , and for a = 1/3 we have p = 3/32ttT 2 ,
the standard FRW values. Thus, the 5D solution (2.5) contains 4D
dynamics and 4D matter that are the same as in the standard 4D cos-
mologies for the late and early universe.

However, while the 5D approach does no violence to the 4D
one, it adds significant insight. The big bang occurs in proper time at
T = by (2.10); but it occurs in coordinate time at a = or
u = t - I = by (2.6) and (2.5). Now the field equations R AB = are
fully covariant, so any choice of coordinates is valid. Therefore, we
can interpret the physically-defined big bang either as a singularity in
4D or as a hypersurface t = / that represents a plane wave propagat-
ing in 5D.

Some comments are in order about the shock-wave solution
(2.5), the flat-universe solution (2.1) and certain other cosmological
solutions in the literature (Wesson 1999). These all have Rabcd = 0,
and it is possible to make a systematic study of these equations
(Abolghasem, Coley and McManus 1996). However, it is not usually
possible to find coordinate transformations between solutions, or
show the explicit transformation to 5D Minkowski space like (2.4),
because of the level of complexity involved. Further, a given 5D so-

46 Five-Dimensional Physics

lution may have different 4D interpretations. This is because the
group of 5D coordinate transformations x A — » x A (x B j is wider that

the 4D group x a — > x a \x p \ , so x 4 -dependent transformations are

mathematically equivalent in 5D but physically non-equivalent in 4D.
The solutions in this and the preceding section provide an example of
this. Both have R A bcd = 0, but we interpret (2.1) as a 5D space with a
4D singularity embedded in it, and (2.5) as a wave moving in a 5D
space which "pokes" through into 4D (like a 3D shock wave pene-
trates a 2D surface). The existence of multiple 4D interpretations of a
given 5D solution raises an interesting question. If the real universe
has one (or more) extra dimensions, then what coordinate system is
being used for 4D cosmology? It seems to us that this question can be
answered empirically, because the choice of coordinates in 5D affects
the physics in 4D. An analogous situation occurs in the 4D/3D case
and was touched on in the preceding section (see also Wesson 1999
pp. 100 - 102). In the 4D FRW models, the 3D spatial coordinates
can be chosen as comoving so that the galaxies are fixed with respect
to each other, or the coordinates can be chosen in such a way as to
give the galaxies Hubble-law motions. In the 5D / 4D case, there is a
similar situation which involves, among other things, the 3K micro-
wave background. In the conventional 4D view, this is thermalized in
the big-bang fireball. In the higher-dimensional view, some other
mechanism must operate, such as a variation of particle masses that
leads to efficient Thomson scattering (Hoyle 1975). We need to look

The Big Bang Revisited 47

into the detailed physics and decide by observational data which is the
best approach.

2.4 A Bounce Instead of a Bang

Let us now move away from solutions which are flat to study
one which is curved in 5D, and has the interesting property of a big
"bounce" instead of a big bang in 4D.

It has frequently been speculated that 4D FRW models with
positive 3D curvature might, after their expansion phase, recollapse to
a big "crunch", from which they might re-emerge. However, this idea
owes more to a belief in reincarnation than to physics, where it cannot
be proved because it is impossible to integrate Einstein's equations
through the second or (nth) singularity. It has also occasionally been
suggested that an FRW model with a genuine bounce, where there is a
contraction to a minimum but finite scale followed by an expansion,
might serve to describe the real universe. But this idea fails when
confronted with observational data, notably on the ages of globular
clusters and the redshifts of quasars (Leonard and Lake; 1995; Over-
duin 1999). Furthermore, recent data on supernovae show that (pro-
vided there is negligible intergalactic dust) the universe on the largest
scales is accelerating, implying a significant positive cosmological
constant or some other dark form of energy with similar consequences
(Perlmutter 2003). Indeed, the best fit on current data is to a universe
which has approximately 70% of its density in the form of vacuum
energy, approximately 30% in the form of dark but conventional mat-

48 Five-Dimensional Physics

ter, and only a smattering of luminous matter of the type we see in
galaxies (Overduin and Wesson 2003). Thus realistic 5D models of
the universe should contain a 4D cosmological "constant".

The latter, we saw in Chapter 1, is basically a measure of vac-
uum energy, and this can actually be variable. (In standard general
relativity, the equation of state of the vacuum is p v = -p v where p v =
A / 87r in units where the speed of light and the gravitational constant
are unity, with A a true constant.) Our goal, therefore, is to find a
class of 5D solutions which not only replaces the singular big bang
with a nonsingular bounce, but is rich enough to give back 4D matter
of the appropriate normal and vacuum types.

A suitable class of models, which satisfies the 5D field equa-
tions R A b = and extends the 4D FRW ones, was studied by Liu and
Wesson (2001; see also Liu and Mashhoon 1995). It has a line ele-
ment given by

dS 2 = B 2 dt 2 - A 2 f -^r + r 2 dQ. 2 I - dl 2
1-kr

A 2 =(S+k)y 2 + 2vy + ?-^-

5 = !^Ui . (2.11)

H dt m

Here y. = ju(t) and v = v{i) are arbitrary functions, k is the 3D curvature
index (k = ±1, 0) and K is a constant. After a lengthy calculation, we
find that the 5D Kretschmann invariant takes the form

The Big Bang Revisited 49

i = R ABCD R ABCD = 7 -^r , (2-12)

which shows that K determines the curvature of the 5D manifold.
From equations (2.1 1) we see that the form of B dt is invariant under
an arbitrary transformation t = t (7) . This gives us the freedom to fix
one of the two arbitrary functions /x(t) and v{t), without changing the
basic solutions. The other arbitrary function can be seen to relate to
the 4D properties of matter, which we now discuss.
The 4D line element is

ds 2 =g ap dx a dx p =B 2 dt 2 -A 2 l-^-j + r 2 dn 2 ) . (2.13)

This has the Robertson- Walker form which underlies the standard
FRW models, and allows us to calculate the non-vanishing compo-
nents of the 4D Ricci tensor:

b 2 {a ab

1 A

+ -— - \ + 2k-

X=-ir ^ + ~ —-- +2*-d ■ (2-14)

[A A{ A B) A 2
Now from (2. 1 1) we have

B = ±,

M

50 Five-Dimensional Physics

Using these in (2.14), we can eliminate B and B from them to give

4^0 = 3////
AA

% = % --X--A^-^\ • (2..5)

I X4 4'

These yield the 4D Ricci scalar

This, together with (2.15), enables us to form the 4D Einstein tensor
G a p = A R a p - 8 a p 4 R 1 2 . The nonvanishing components of this are

^o = 7,

G\=Gl=Gl=^t + ^± . (2.17)

1 2 3 AA A 2

These give the components of the induced energy-momentum tensor,

since Einstein's equations G a p = %nT p hold.

Let us suppose that the induced matter is a perfect fluid with

density p and pressure p, moving with a 4-velocity u" = dx a / ds, plus

a cosmological term whose nature is to be determined. Then we have

G aP =%n\_(p + p)u a u p +(KI%7c-p)g afi \ . (2.18)

As in the FRW models, we can take the matter to be comoving in
three dimensions, so u a =(u°, 0, 0, 0) and u°u = 1. Then (2.18) and
(2.17) yield

3U+k)

8^-A = -^-^ . (2.19)

AA A 2

These are the analogs for our solution (2.11) of the Friedmann equa-
tions for the FRW solutions. As there, we are free to choose an equa-
tion of state, which we take to be the isothermal one

P = YP ■ (2.20)

Here y is a constant, which for ordinary matter lies in the range (dust)
< y < 1/3 (radiation or ultrarelativistic particles). Using (2.20) in
(2.19), we can isolate the density of matter and the cosmological term:
2 ( M 2 +k juju]

%np- -, —

\ + y{ A 2 AA

In these relations, /u = ju(t) is still arbitrary and A = A(t ,1) is given by
(2.11). The matter density therefore has a wide range of forms, and
the cosmological constant is in general variable.

Let us now consider singularities of the manifold in (2.11).
Since this is 5D Ricci flat, we have R = and ^Rab = 0. The third

52 Five-Dimensional Physics

5D invariant is given by (2.12), from which we see that A = (with
K^O) corresponds to a 5D singularity. This is a physical singularity,
and as in general relativity, can be naturally explained as a big bang.
However, from (2.16) and (2.21), we also see that if A = , then all
of the 4D quantities 4 R, p and A diverge. But while this defines a
kind of 4D singularity, the 5D curvature invariant (2.12) does not di-
verge. This is a second kind of 4D singularity, associated with the
minimum in the 3D scale factor A, and can be naturally explained as a
big bounce. We further note from (2. 1 1) that if A = then B = (as-
suming fx ^ 0), and the time part of the 4D line element vanishes. To
sum up: the manifold (2.11) has a 5D geometrical singularity associ-
ated with A = and a 4D matter singularity associated with B = 0,
which we can explain respectively as a big bang and a big bounce.

The physics associated with the bounce, and plots of the 3D
scale factor A = A(t, I) as a function of the time t for various values of
the extra coordinate /, were studied by Liu and Wesson (2001). They
put k = in (2.11) on the basis of observational data, and chose the
functions fi(t) and v(t) for algebraic convenience. They found that,
typically, the form of the scale factor A(t, /) is not symmetric around
the minimum or time of the bounce. This implies that in 5D models
of this type, there is a 4D production of entropy and/or matter around
the bounce.

Extensive other work has been done on the class of solutions
(2.11) due to the breadth of its algebra, with some interesting results
for physics: (a) The big bounce has characteristics of an event horizon,

The Big Bang Revisited 53

at which the spatial scale factor and the mass density are finite, but
where the pressure undergoes a sudden transition from negative to
positive unbound values (Xu, Liu and Wang 2003). (b) The models
are governed by equations which resemble the Friedmann relations
for FRW cosmologies, but when the new solutions are compactified
on an S\ I Z 2 orbitold as in membrane theory, they yield 2 branes with
different physical properties (Liu 2003). (c) By choosing the 2 arbi-
trary functions noted above in accordance with supernova data, the
models before the bounce contract from a A-dominated vacuum, and
after the bounce expand and indeed accelerate, with a dark energy
contribution which is 2/3 of the total energy density for late times, in
agreement with observations (Wang, Liu and Xu 2004). (d) This as-
ymptotic behaviour can be shown by the use of a dynamical-systems
approach to be universal, and due to the existence of two phase-plane
attractors, one for the visible / dark-matter component and one for the
scalar / dark-energy component (Chang et al. 2005). (e) The class of
solutions (2.11) can, as mentioned before, be interpreted from the
viewpoint of membrane theory, when the tension of the brane as a
hypersurface in 5D and the strength of conventional gravity in 4D are
constants (Ponce de Leon 2002). (f) The imposition of the Z 2 symme-
try of membrane theory on the solutions (2.11) results in metrics
which are even functions of the extra coordinate /, and when the de-
pendency is via I 2 the bounce has the properties of a 4D phase transi-
tion (Liko and Wesson 2005). This compendium of properties does
not exhaust the implications of the class (2.1 1). We have seen that we

54 Five-Dimensional Physics

can interpret it as a bounce in a classical cosmology, a braneworld
model, or a phase transition (which could be the classical analog of a
discontinuity in a scalar Higgs-type quantum field). However, we
will see in the following section that there is at least one more appli-
cation of (2.1 1) that, while of a different kind, is just as remarkable.

2.5 The Universe as a 5D Black Hole

The concept of a 4D black hole is now so familiar that it is
automatically associated with a central singularity, surrounded by an
event horizon which depends on the mass M at the centre of spheri-
cally-symmetric 3D space, which latter is asymptotically flat. The
latter properly and others imply that the Schwarzschild solution is -
up to coordinate transformations - unique. This is embodied in Birk-
hoff s theorem, which plays a significant role in Einstein's general
theory of relativity. By comparison, the concept of a 5D "black hole"
is considerably more complicated, due to the extra degrees of freedom
introduced by the fifth coordinate. Solutions of the 5D field equations
with a spherically-symmetric 3D space are called solitons. But even
in the static case there is a class of solutions rather than a single one,
and time-dependent cases are known (Wesson 1999). Thus Birkhoff s
theorem, in its conventional form, fails in 5D. Indeed, it is unwise to
carry over preconceptions about "black holes" from 4D to 5D. We
will see below that it is more advisable to consider a topological 5D
black hole, defined by the symmetries of its metric, and work out the
properties of these without presumptions. Specifically, our aim in this

The Big Bang Revisited 5 5

section is to consider a general 5D metric of the black-hole type, and
show that it is isometric to that of the cosmologies treated in the pre-
vious section (Seahra and Wesson 2003, 2005; Fukui, Seahra and
Wesson 2001). That is, we wish to ponder the possibility that the uni-
verse may be a 5D black hole.

As in other sections in this chapter, we let upper-case English
letters run 0-4 and lower-case Greek letters run 0-3. We use time and
spherical-polar spatial coordinates plus a length /, so x 4 = (t;r 0(f) ; /).
Then dQ. 2 2 = d6 2 + sin 2 6d<f> 2 is the measure on a 2D spherical shell.
The topological black-hole class of 5D solutions is given by

dS 2 = hdT 2 - h~ l dR 2 - R 2 d£l\ . (2.22)

Here T is the time and h = h(R) is a function of the radius R, where
the latter is defined so that when the 3-measure JQ 3 = dQ 3 (&) reduces
to the 2-measure defined above, then 2nR is the circumference. The
Kretschmann scalar for (2.22) is 72K 2 R~ S , where the 5D curvature K
depends on the mass M at the centre of the 3 -geometry (Seahra and
Wesson 2003). This scalar is the same as (2.12) for the cosmological
manifold (2.11). This coincidence and other properties suggest to us
that (2.11) and (2.22) are geometrically equivalent descriptions of the
same situation in different coordinates, or are isometries.

To prove this, we need to show a coordinate transformation
which takes us from (2.22) to (2.11) or the reverse. We proceed to

56 Five-Dimensional Physics

give the result, noting that it may be confirmed by computer. The
radial transformation is specified by

R 2 =[f/+k)l 2 +2vl + (y 2 +K)[pi 2 +ky , (2.23)

where jj, and v are the functions of t in (2. 1 1). This is an unusual map-
ping, which may repay further investigation. The corresponding tem-
poral transformation turns out to have different forms, depending on
whether k = or k =±1 . To present these forms, we introduce a dummy
variable u = u(t) and the function v = v(t,l) = kUju 2 +k^l + v \

(MT) /j' 1 . Then for k = 0, ±1 we have respectively:

In these, it is to be understood that the integrals are over t and that the
integrands involve v = v(«), ju = //(«) with v' = dvl du, ju' = djul du.
It should also be noted that while (2.24) relates the time for the black
hole (2.22) back to the time for the cosmology (2.11), there is a spe-
cial case of (2.22) where kK < and k = ±1. However, for this case
there is no Killing-vector defined horizon, so this would correspond
to a naked singularity, with a negative mass. We therefore bypass this

The Big Bang Revisited 5 7

special case as unphysical, and conclude that (2.23) and (2.24) are in
general the coordinate transformations that take the metric (2.22) for a
5D topological black hole back to the metric (2.11) for a 5D FRW-
like universe.

The isometry just shown invites further analysis based on a
comparison with the usual 4D Schwarzschild solution. The latter is
commonly presented in coordinates where there is an horizon (de-
fined by the mass M) which splits the manifold into parts, the distinc-
tion being from the geometrical viewpoint somewhat artificial. This
problem is frequently addressed by introducing Kruskal-Szekeres co-
ordinates, which effectively remove the horizon and extend the ge-
ometry. We are naturally interested in seeing if this is possible for the
metric constructed from (2.23) and (2.24). There are a large number
of choices for the parameters involved in these relations, so let us fo-
cus on the case where k = +1, K > 0. This means that the 3D sub-
manifold is spherical, so we have an ordinary as opposed to a topo-
logical black hole. Then it may be shown that for our case there are
KS-type coordinates U, V which are related to the R, T coordinates of
(2.23) and (2.24) by

U = + [h(R)y /M e- T,M \(R-M)/(R + Mf 2

V = ±e R,M e TIM \(R-M)l{R + M)\ 12 . (2.25)

In these coordinates, the metric for the black hole is

58 Five-Dimensional Physics

dS 2 =M 2 (l + M 2 /R 2 )e- 2RIM dUdV-R 2 dnl . (2.26)

A detailed investigation of the extended geometry corresponding to
(2.26), including Penrose-Carter diagrams, appears elsewhere (Seahra
and Wesson 2003). This elucidates the nature in which the manifold
is covered by the coordinates of the cosmological metric (2.11), to
which we now return.

By (2.1 1), an observer unaware of the fifth dimension or con-
fined to a hypersurface in it would experience a 4D universe with line
element

ds 2

— — 1 dt 2 - A 2 \ . drl +r 2 (d6 2 + sin 2 6d<j> 2 ) . {221)

We discussed several interpretations of this in Section 2.4 preceding,
which follow from choosing the two arbitrary functions /u(t) and v(/)
and evaluating its associated matter. We noted that the main feature
of the class of solutions (2.11) is a bounce, where the spatial scale
factor of (2.27) goes through a minimum. Now that we know that the
cosmological metric is isometric to a black hole, it is easier to see
what is involved: The scale factor A(t, /) when it passes through the
minimum (dA I dt =0) induces a singularity in the metric which is of
the same type as with a conventional black hole (goo = 0). But we ar-
gued before that this singularity is not geometrical, and indeed it is
now clear that it is of the kind found at the event horizon of a black
hole. However, an observer in the 4D manifold (2.27) would interpret
A{i) as the standard scale factor of an FRW model if A{t, J) evolves to

The Big Bang Revisited 59

be independent of the fifth dimension (see above). He would then
wrongly assume that the universe starts in the state with A = as a big
bang, whereas it actually evolves from the state with dA I dt =
which is a big bounce. In the case where the bounce is associated
with matter production, as we also mentioned in Section 2.4, it is use-
ful to introduce the mass M of the fluid out to radius r as it is defined
by the density p, pressure p and the metric (2.27). This is given by
the Misner-Sharp-Podurets mass function (Misner and Sharp 1964;
Podurets 1964; Wesson 1986). For the uniform fluid of (2.27), the
relevant relations are

M = AnAr" (ju 2 +k) = [An /3)r i A 3 p

dM/dt = -A?rr 3 A 2 (dA/dt)p . (2.28)

These allow of matter production (dMl dt > 0) both before the bounce
(dA I dt < 0, p > 0) and after it (dA I dt>0,p< 0), at least on the basis
of classical theory. However, a proper investigation of this would
require quantum theory, which would also help clarify the status of
other issues with these models, such as inflation.

In this section, we have suggested that an observer living in a
universe with the 4D line element (2.27) might be unaware that it is
part of a 5D model of the form (2.11), which is geometrically
equivalent to the 5D black hole (2.22). The argument has been
mainly mathematical in nature. From a philosophical viewpoint, the

60 Five-Dimensional Physics

idea that the universe is a higher-dimensional black hole may be
harder to accept.

2.6 Conclusion

The 5D field equations R A b - lead to startling new cosmolo-
gies. However, the 5D equations contain the 4D Einstein ones G a p =
8xT a p, so we are in the comfortable situation of keeping what we
know while finding something new. In this chapter we have looked at
four new cosmologies. The first is a universe which is flat and empty
in 5D, but contains on hypersurfaces the standard FRW models which
are curved and have matter in 4D. We learn that the big bang may be
a kind of artifact produced by an unfortunate choice of coordinates.
The second example is also 5D-flat, but in it the big bang may be in-
terpreted as the result of a shock wave propagating in the extra di-
mension. Our third example is based on a rich class of solutions
where in general there is a big bounce rather than a big bang. The
bounce may be associated with a phase transition and the creation of
matter, at least in the case where the Z 2 symmetry of membrane the-
ory is imposed. This view is in agreement with our fourth example,
where we find that the bounce has some of the properties of an event
horizon. This leads us to suggest that the universe may resemble a 5D
black hole.

It is difficult to assess the plausibility of these and related
ideas. However, many researchers would argue that none is intrinsi-
cally less plausible than the big bang. The latter phrase was coined by

The Big Bang Revisited 6 1

Hoyle, who used it in a derogatory sense. To him, it appeared daft to
assume that all of the matter in the universe was created in an initial
singularity. The steady-state model of Bondi, Gold and Hoyle sought
to provide a more logical alternative. In it, the dilution of matter by
the universal expansion was compensated by its continual creation.
This was studied using a modified form of the Einstein field equations
by Hoyle and Narlikar, and in a different though related context by
Dirac. It is well known that the steady-state cosmology foundered in
the face of observational data, but its demise did not mean the end of
new attempts to account for the nature and origin of matter. Interest
in alternative theories continued, even after it was shown by the sin-
gularity theorems that in general relativity an initial singularity was
inevitable given certain assumptions about the material content of the
universe. And herein lies the gist: we are not sure of the nature of the
very early universe, and so cannot be sure about its origin.

Modern 5D relativity has to be viewed against the historical
backdrop just outlined. We may not be clear yet as to whether this
manifold is smooth or has a membrane, but it can be argued that the
5D approach to cosmology is superior to all of its 4D predecessors.
The induced-matter picture is particularly compelling. It uses the
most basic mathematical object to form an exactly-determined set of
field equations which describe not only the curvature of 4D spacetime
but also its content of matter. And if we wish we can do away with
the big bang.

62 Five-Dimensional Physics

References

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37,361.
Chang, B., Liu, H., Liu, H., Xu, L. 2005, Mod. Phys. Lett. A 20, 923.
Hoyle, F. 1975, Astrophys. J. 196, 661.

Fukui, T., Seahra, S.S., Wesson, P.S., 2001, J. Math. Phys. 42, 5195.
Lake, K. 2004, GRTensor (Queen's U., Kingston).
Leonard, S., Lake, K. 1995, Astrophys. J. 441, L55.
Liko, T., Wesson, P.S. 2005, Int. J. Mod. Phys. A 20, 2037.
Linde, A.D. 1990, Inflation and Quantum Cosmology (Academic,

Boston).
Linde, A.D. 1994, Sci Am. 271 (1 1), 48.
Liu, H., Mashhoon, B. 1995, Ann. Phys. 4, 565.
Liu, PL, Wesson, P.S. 2001, Astrophys. J. 562, 1.
Liu, H. 2003. Phys. Lett. B 560, 149.
Misner, C.W., Sharp, D.H. 1964, Phys. Rev. B 136, 571.
Overduin, J.M. 1999, Astrophys. J. 517, LI.
Overduin, J.M., Wesson, P.S. 2003, Dark Sky - Dark Matter (Institute

of Physics, Bristol).
Perlmutter, S. 2003, Phys. Today 56 (4), 53.
Podurets, MA. 1964, Sov. Astron. (A.J.) 8, 19.
Ponce de Leon, J. 1988, Gen. Rel. Grav. 20, 539.
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Ponce de Leon, J. 2003, Int. J. Mod. Phys. D 12, 1053.
Rindler, W. 1977, Essential Relativity (2nd ed., Springer, Berlin).

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Seahra, S.S., Wesson, P.S. 2002, Class. Quant. Grav. 19, 1 139.

Seahra, S.S., Wesson, P.S. 2003, J. Math. Phys. 44, 5664.

Seahra, S.S., Wesson, P.S. 2005, Gen. Rel. Grav., vol. 37, p. 1339.

Vilenkin, A. 1982, Phys. Lett. B 117, 25.

Wang, B., Liu, H., Xu, L. 2004, Mod. Phys. Lett. A 19, 449.

Wesson, P.S. 1985, Astron. Astrophys. 151, 276.

Wesson, P.S. 1986, Phys. Rev. D 34, 3925.

Wesson, P.S. 1999, Space-Time-Matter (World Scientific, Singapore).

Wesson, P.S., Liu, H., Seahra, S.S. 2000, Astron. Astrophys. 358, 425.

Wesson, P.S., Seahra, S.S. 2001, Astrophys. J. 558, L75.

Xu. L., Liu, H., Wang, B. 2003, Chin. Phys. Lett. 20, 995.

3. PATHS IN HYPERSPACE

"Beam me up, Scottie" (Modern Startrek cliche)

3.1 Introduction

By the title of this chapter, it is implied that we will consider
the possibility that a particle may move outside spacetime. In the
early developmental stages of JVD field theory, there was some dis-
cussion as to whether particles should move on the geodesies of fa-
miliar 4D space, or be allowed to wander into the higher dimensions.
Our view is that if the fifth and higher dimensions are to be taken as
"real" in some sense, then we should take the interval in the extended
manifold, minimize it in analogy with Fermat's principle and other
applications, and investigate the resulting dynamics. In this way, we
can examine the acceptability of higher dimensions, and at least con-
strain them. Our view is that paths in N > 4D hyperspace are not the
subject of theatrics, but rather provide a way of probing new physics.

There is, at the outset, an issue to be addressed which in-
volves one of the long-standing differences between classical field
theory and quantum mechanics. The equations of motion in general
relativity involve the concept of acceleration, whereas the dynamics
of particle physics uses the concept of momentum. The former con-
cept involves only the 4D measures of space and time. The latter
concept involves these plus the measure of mass. Of course, the two
approaches overlap, and are indeed equivalent, in the case where the
rest mass of an object is constant. However, there is a theoretical dif-

Paths in Hyperspace 65

ference which goes to the root of what we mean by the concept of
mass (Wesson 1999; Jammer 2000). And there is a practical differ-
ence, as can be appreciated in cases where the mass changes rapidly,
as when a rocket burns fuel and leaves the Earth or a particle gains
mass from the Higgs field in the early universe (Rindler 1977; Linde
1990). It will turn out that there are situations in which we need to
consider carefully what happens when an object changes its rest mass
as it pursues a path through a higher-dimensional manifold. Any stu-
dent who observes the high velocity that a model rocket obtains by
dint of shedding a fraction of its mass, knows that the concept of mo-
mentum is paramount. To this extent, we will need to consider how
to introduce rest mass into the interval of general relativity, in a way
which is consistent with other parts of the theory including the field
equations (Wesson 2003a), and in agreement with observations such
as those of QSOs which show a remarkable degree of uniformity in
spectroscopic properties related to particle mass (Tubbs and Wolfe
1980). Since we can retain the usual definition of force as the product
of acceleration and mass, another way of phrasing our objective is
that we wish to elucidate forces in higher dimensions.

3.2 Dynamics in Spacetime

While this subject is one which we may be forgiven for as-
suming that we understand, it is instructive to remind ourselves of
how accelerations enter general relativity and momenta enter quantum
theory.

66 Five-Dimensional Physics

In Einstein's theory, the small element by which two arbitrary
points in spacetime are separated is given by

ds 2 =g ap dx a dx p {a,P = 0,123) . (3.1)

It is usual to take the interval (s) and the coordinates (x a = t, xyz or
similar) to be lengths, while the components of the metric tensor (g a p)
or potentials are dimensionless. The g a p are given by solutions of the
field equations (1.1), which being tensor relations obey the Covari-
ance Principle, which means that they are valid in any system of co-
ordinates (gauge). A typical example of a metric coefficient is GM I
c 2 r for the gravitational field outside an object of mass Mat distance r
in 3D spherically-symmetric space. (In this section we use physical
units for the speed of light c, the gravitational constant G and
Planck's constant h.) The Geodesic Principle asserts that the path of a
particle is obtained by minimizing the interval via

[j<fc] = . (3.2)

The geodesic or path has 4 components. These are given by the geo-
desic equation, which in its most useful form reads

Here T a pr are the Christoffel symbols of the first kind, which depend
on the first partial derivatives of the g a p {x y ). In this way, (3.3) gives

Paths in Hyperspace 67

back in the weak- field limit \\g ap <^l) the standard Newtonian ac-
celeration (GM I cV) outside an object like the Sun. When this is
multiplied by the mass of a test object (m) we obtain the gravitational
force, which balanced against the centrifugal force associated with a
circular velocity (v) gives GMm / r 2 = mv 2 I r. In this, the masses on
the left-hand side are actually gravitational in nature, while that on the
right-hand side is inertial in nature (Jammer 2000). We have used the
same symbol, because the Weak Equivalence Principle - as one of the
founding bases of general relativity - asserts that they are essentially
the same (see below). Then we can cancel the m on either side, ob-
taining the acceleration. The fact that this cancellation occurs is
commonplace but of tremendous significance. It makes gravity a par-
ticularly simple interaction compared to others. For this and other
reasons, the Equivalence Principle has been much tested and contin-
ues to be so, as we will see below. Here, we note that (3.1) - (3.3)
provide a theory of dynamics based on accelerations, not forces. This
is also apparent from the alternative but more compact form of (3.3)
given by

h : X = . (3.4)

Here u" =dx" I ds is the 4-velocity, and the semicolon denotes the
covariant derivative which takes into account the departure of the
spacetime (3.1) from flatness. The rest mass m of a test particle does
not appear. Indeed, it is acknowledged that this quantity requires a
wider rationale, such as would be provided by Mach's Principle (Rin-

68 Five-Dimensional Physics

dler 1977; Wesson, Seahra and Liu 2002). This principle motivated
Einstein, but it is widely believed that it is not properly incorporated
into standard general relativity. Thus the Covariance, Geodesic and
Equivalence Principles on their own lead to a theory where masses
and matter are sources which curve spacetime, but in which the mo-
tion of a particle does not depend on the mass of the latter.

In quantum theory, the situation is different. Here attention is
focused on momentum as the product of mass and velocity, plus its
integral the energy. In fact, much of the physics of the microscopic
world can be summed up in one simple relation between the energy
(E), the momentum (p) and the mass (m):

E 2 -p 2 c 2 =m 2 c 4 . (3.5)

This relation is closely obeyed by real particles (Pospelov and Roma-
ns 2004). It is based on dividing the line element by the squared ele-
ment of the proper time (ds 2 ), to obtain a dynamical relation. Alterna-
tively, it is based on the convention that the 4-velocities are normal-
ized via u a u a =1 or 0, depending on whether the particle is massive
or massless. Multiplying this by m 2 means that the 4-momenta are
normalized via p a p a = m 2 . With the usual identifications
(E = mc 2 u° , p Ui =mcu m ) , this results in the standard relation

noted above. However, this approach contains no information about
the possible case in which the mass of an object varies along its path
via m = m(s), which as we will see can happen in extended theories of

Paths in Hyperspace 69

gravity. This shortcoming carries over from the particle to the wave
picture. The latter is commonly derived by using the operators

\i0){dx°r ^UJUJ

(3.6)

to write the energy and 3-momentum of a particle in terms of a wave
function <f>. This causes the standard energy relation to become the
Klein-Gordon equation. For the flat spacetime of special relativity,
this reads

n 2 + (c/h) 2 m 2 = O

■v-g£(<r -.-1-1.-1) • <3,,

The non-relativistic limit of the flat-space Klein-Gordon equation is
the Schrodinger equation (which is used for systems like the hydrogen
atom), and its factorized form is the Dirac equation (which is used for
particles like the electron). For the curved spacetime of general rela-
tivity, it is necessary to proceed in a manner that is covariant. The
action is

Ht* = hr* = Jt* • <38)

which is of course quantized. The action can be used to form a wave
function (f>=e lA . The first derivative of this yields

70 Five-Dimensional Physics

p a ={hli(f>)d</)ldx a , (3.9)

as before. The second derivative of the wave function needs to be
taken covariantly, however, to account for the curvature of spacetime.
Using a semicolon to denote this and a comma to denote the ordinary
partial derivative, we define as usual nV -g^V,^- Then the second

derivative of <f> when contracted with g°^ yields

a afi

(3.10)

h 2 ra h

The l.h.s. of this is real while the r.h.s. is imaginary. The former
gives n 2 ^ + (c/ ' h) m 2 (/) = 0, which is the Klein-Gordon equation for
a curved spacetime. The latter gives p p p - , which is the conserva-
tion equation for the momenta. Both of these relations are standard in
particle physics.

The considerations of the two preceding paragraphs show that
in general relativity and quantum theory the logic of standard dynam-
ics is incomplete. In fact, 4D mechanics is based largely on well-
chosen conventions.

3.3 Fifth Force from Fifth Dimension

Coordinates as well as conventions affect dynamics. In 4D
general relativity, we saw in Section 2.3 that the galaxies can be con-
sidered static in comoving coordinates, or moving in accordance with
Hubble's law in the frame frequently used in observational cosmology.

Paths in Hyperspace 71

In 5D relativity, we will see in this section that the choice of coordi-
nates (or gauge) affects not only the form of the metric but also the
dynamics which follows from it. However, it will transpire that in the
canonical coordinates of Section 1 .4, the equations of motion in 5D
become quite transparent (Wesson et al. 1999). Then they can be
couched in the form of the usual 4D geodesic, plus an extra accelera-
tion or force (per unit mass) due to the fifth dimension.

In conventional 4D dynamics, it is often stated that the
4-velocity u a and the 4-force are orthogonal as in u a f a = (the sum-
mation convention is in effect and we absorb the fundamental con-
stants in this section by a choice of units). It is certainly the case that
conventional electrodynamics and fluid motions obey such a law.
However, it is also apparent that if we use the same formalism to set
up laws of physics in (say) 5D, the relation t/f A = would result in
iffa = - u% # and a consequent departure from the 4D conservation
laws (A,B = 0,123,4 where the argument can obviously be extended to
higher dimensions). The condition of orthogonality is built into rela-
tivity. Given an AD line element dS 2 = g AB dx A dx B in terms of a met-
ric tensor and coordinates, the velocities U A = dx A I dS for a non-null
path perforce obey \ = U A U A as a normalization condition, and
U B D B U A = as the condition which minimizes S in terms of a co-
variant derivative D B which takes into account the curvature of the
manifold. The latter TV equations, when contracted, define a kind of
force (per unit mass) F A , and result in the common relation

72 Five-Dimensional Physics

U A F A = . (3.11)

We see that this orthogonality condition depends on basic assump-
tions to do with the validity of Riemannian geometry and group the-
ory. As such, it is difficult to believe that it could be contravened as a
basis for modern physics.

It then follows that if the space is (say) 5D and not 4D, per-
fect conservation in the whole space implies imperfect conservation
in the subspace (see above: u% = - u% ^ 0). Since we know that 4D
laws are closely obeyed, this implies that the dimensionality of the
world can be tested by looking for small departures from 4D
dynamics.

The N conditions U B D B U A = are the analog of the 4D rela-
tions (3.4). The AD geodesic equation is the analog of the 4D one
(3.3). With appropriately-defined Christoffel symbols, it is

^— + T A C U B U C = (A,B,C = 0...N) . (3.12)

Solutions of this can be found once the T A BC =T A C [g DE ) are known
from solutions of the field equations, which are commonly taken to be
Rab = (see Section 1.4). However, solutions of either the geodesic
equation or the field equations in practice require some assumptions
about g A B = gAB(x c ). There are N arbitrary coordinates x c , so in 5D
we can apply 5 conditions to g A s without loss of generality. Normally,
these would be chosen with regard to some physical situation. But

Paths in Hyperspace 73

here, we adopt a different approach aimed at dynamics. There have
actually been numerous attempts at solving (3.12) in 5D (Wesson
1999). Here we choose to retain contact with modern theory by fac-
torizing the 4D part of the space using x 4 = I in a way analogous to the
synchronous coordinate system of general relativity (this does not re-
strict generality if g a p is allowed to depend on / as well as x 7 ). We
also use the 5 coordinate degrees of freedom to set g 4a = 0, g 4 4 = - 1,
which in the usual interpretation of Kaluza-Klein theory suppresses
effects of the electromagnetic and scalar fields (see Chapter 1). The
interval then takes the canonical form of (1.14):

dS 1 = — ds 1 -dl 1 (5D) . (3.13)

Here L is a constant introduced for the consistency of physical dimen-
sions, assuming that the 4D interval is a length, which is given by

ds 2 =g a/! (x'',l)dx a dx fi (4D) . (3.14)

In other words, the 5D space contains the 4D space as the hypersur-
face / = / (x y ), providing a well-defined local embedding.

The utility of (3.13) and (3.14) becomes apparent when sub-
stituted into (3.12). We also note that since we make observations in
4D it is convenient to use s in place of S. The velocities are related by

U A = d L= dc_± = u a± (315)

dS ds dS dS

74 Five-Dimensional Physics

where by (3 . 1 3) we have

ds ff/Y fJ/V

cs IUJ W ' ai6)

Then some algebra shows that (3.12) splits naturally into two parts.
The 4D part reads

du a

d - + Y a py u p u"=F a , (3.17)

whereas the part for the fifth dimension reads

ds 1 Ads) + L 2 ~~ 2

L) \ds)

(3.18)

What we have done here is to split the dynamics, with the overlap
confined to the extra acceleration or force (per unit mass) F*. It is
already obvious from (3.17) that conventional 4D geodesic motion is
recovered if F a = 0, so let us consider this quantity.

The explicit form of F 1 can be found by expanding the Y A BC
noted above. It is

p a / a P , i a p\dl dx 7 dg p

This is finite if the 4D metric depends on the extra coordinate and
there is motion not only in 4D but also in the fifth dimension (dl /ds ^
0). The latter is given by a solution of (3.18), and will in general be
finite so the extra force (3.19) will also in general be finite.

Paths in Hyper space 75

Measuring (3.19) will require a combination of exact solu-
tions and observations. But we expect the new force to be small be-
cause 4D dynamics is known to be in good agreement with available
data (Will 1993). Also, F* of (3.19) contains a part (N a ) normal to
the 4-velocity u" and a part (P") parallel to it, and strictly speaking it
is the latter which violates the usual condition of orthogonality and
would show the existence of an extra dimension. This part of the
force (per unit mass) is given by

!fir /> r\ dl a
5/ )ds

l C -^BL u P u r \ZL U ° , (3.20)

2 81 ' J -

or in short by P" = fiif where /? is a scalar that depends on the solution.
There are numerous solutions known of the field equations R AB =
(5D) which depend on x 4 = / and therefore have finite /? (Wesson
1999). It is also known that apparently empty 5D spaces can contain
curved 4D subspaces of cosmological type with matter, as discussed
in Chapter 1 . It is even possible that the 5D space may be flat, as dis-
cussed in Chapter 2. We have therefore looked at a toy model where
flat 5D space is written in Minkowski form. For this, N" = and P a =
jiu a with P = (\ll ){dl/ds). Even this simple case has an extra accelera-
tion, due essentially to the fact that we are using the 4D proper time s
rather than the 5D interval S to parametize the dynamics. We will dis-
cuss astrophysical applications of the fifth force in detail in Chapter 5.
The form of the fifth force depends, as we have noted, on the
form of the metric. Studies have been made for the induced-matter

76 Five-Dimensional Physics

approach (Liu and Mashhoon 2000; Liu and Wesson 2000; Billyard
and Sajko 2001; Wesson 2002a) and for the membrane approach
(Youm 2000; Maartens 2000; Chamblin 2001; Ponce de Leon 2001).
The results are conformable. However, the former when it uses the
canonical metric (3.13) opens a unique physical vista to which we
alluded in Section 1.4. There we learned that the constants of the 5D
metric can be identified from the field equations in terms of the 4D
cosmological constant via L 2 = 3/A. Moreover, the first part of the
5D canonical line element (3.13) is identical to the action of 4D parti-
cle physics (squared) if the extra coordinate is identified in terms of
the rest mass of a particle via / = m. Considerable work has been done
on this intriguing possibility, to which we will return in Section 3.5.
Clearly, if / is related to m then the rest masses of particles will in
general vary with the 4D proper time by (3.18). However, for the
FRW cosmologies this variation will be slow, because L is large as a
consequence of A being small. It should also be mentioned that the
variability of particle rest mass can be removed by using different pa-
rametizations for the dynamics. These include replacing the proper
time by another affine parameter (Seahra and Wesson 2001), and re-
placing the geodesic approach by the Hamilton- Jacobi formalism
(Ponce de Leon 2002, 2003, 2004). We realize that while the canoni-
cal metric (3.13) is convenient, it is also special in that it is only in
this gauge that the extra coordinate / can be identified with the rest
mass m as used in other applications of dynamics.

Paths in Hyperspace 11

3.4 Null Paths and Two Times

Photons travel on paths in spacetime which are null with ds 1 =
0, and conventional causality is defined by ds 2 > 0. But the interval in
an N > 5D manifold need not necessarily be so restricted. A study of
the 5D equations of motion suggests that particles with large
charge/mass ratios can move on paths with dS 2 < (Davidson and
Owen 1986), and that particles with no electric charge can move on
paths with dS 2 = (Wesson 1999). The idea that massive particles on
timelike geodesies in 4D are on null paths in 5 D is in fact quite feasi-
ble, both for induced-matter theory (Seahra and Wesson 2001) and
membrane theory (Youm 2001). Also, null paths are the natural ones
for particles which move through fields which are solutions of the
apparently empty Ricci-flat field equations.

The physics which follows from null paths can be different
depending on the signature of the 5D metric. We kept open the sign
of g 44 in certain preceding relations (such as those for the electromag-
netic and neutral-matter gauges in Section 1.4), but assumed that gw
was negative in some others (such as those for the canonical gauge).
This is largely because astrophysical data indicate that the cosmologi-
cal constant is positive, which for the induced-matter approach means
that the last part of the metric has to be negative. However, much
work on the quantum aspects of 5D relativity uses a de Sitter mani-
fold with a negative cosmological constant (i.e. AdeS space), which
would correspond to the opposite sign. Timelike extra dimensions are
also used in certain models of string theory (e.g. Bars, Deliduman and

78 Five-Dimensional Physics

Minic 1999). It should be mentioned in this regard that the signature
of the metric is important for finding solutions of the field equations,
both for induced-matter and membrane theory. For example, the
Ponce de Leon cosmologies considered in Section 2.2 exist only for

(h ) . Conversely, the Billyard wave considered below exists

only for (+ h) . This signature defines what is sometimes called a

two-time metric. This may be a misleading name, insofar as the fifth
dimension need not have the same nature as ordinary time. But in
quantum theory, the statistical interaction of particles can actually
lead to thermodynamic arrows of time for different parts of the uni-
verse which are different or even opposed (Schulman 2000). In gen-
eral relativity, it is well known how to incorporate the phenomenol-
ogical laws of thermodynamics; and of course a fundamental corre-
spondence can be established between the two subjects via the proper-
ties of black holes. In higher-dimensional relativity, however, the
situation is less clear, both for thermodynamics and mechanics. In-
deed, some rather peculiar consequences follow for dynamics when
we consider 5D two-time metrics Wesson (2002b). This is especially
true when we couple this idea with that of null paths.

In the Minkowski gauge, a particle moving along a null path
in a two-time 5D metric has

= dS 2 = dt 2 -(dx 2 + dy 2 + dz 2 ) + dl 2 . (3.21)

The 5-velocities U A = dx A / dX, where X is an affine parameter, obey
U a Ua = 0. With X = s for the proper 4D time, the velocity in ordinary

Paths in Hyperspace 79

space (v) is related to the velocity along the axis of ordinary time (u)
and the velocity along the fifth dimension (w) by v 2 = u 2 + w 2 . This
implies superluminal speeds. But the particle which follows the path
specified by (3.21) should not be identified with the tachyon of spe-
cial relativity, because as we saw above, in 5D theory the extra coor-
dinate x 4 = / may not be an ordinary length. What we can infer, by
analogy with 4D special relativity, is that all particles in the 5D mani-
fold (3.21) are in causal contact with each other.

In 4D, causality is usually established by the exchange of
light signals, which as viewed in a (3 + 1) split propagate as waves
along paths with ds 2 = 0. While we do not know the nature of the cor-
responding mechanism in 5D with dS 2 = 0, it is instructive to consider
two-time metrics with wave-like properties. There is one such solu-
tion of R A b = which has the canonical form (3.13) and is particularly
simple (Billyard and Wesson 1996; Wesson 2001). It is given by

dS

= ^[dt 2 -e^'^dx 2 -e i{ "" +k > y) dy 2 - e i{a>t+K2) dz 2 ~\ + dl 2 . (3.22)

Here A^ z are wave numbers and the frequency is constrained by the
solution to be co = ±2 / L. We have studied (3.22) algebraically and
computationally using the program GRTensor (which may also be
used to verify it). Solution (3.22) has two "times". It also has com-
plex metric coefficients for the ordinary 3D space, but closer inspec-
tion shows that the structure of the field equations leads to physical
quantities that are real. The 3D wave is not of the sort found in gen-

80 Five-Dimensional Physics

eral relativity, but owes its existence to the choice of coordinates. A
trivial change in the latter suppresses the appearance of the wave in
3D space, in analogy to how a wave is noticed or not by an observer,
depending on whether he is fixed in the laboratory frame or moving
with the wave. A further change of coordinates can be shown to make
(3.22) look like the 5D analog of the de Sitter solution. This leads us
to conjecture that the wave is supported by the pressure and energy
density of a vacuum with the equation of state found in general rela-
tivity, namely/> + p = 0. This is confirmed to be the case, with A < 0.
It may also be confirmed that (3.22) is not only Ricci-flat {R AB = 0)
but also Riemann-flat (Rabcd = 0). It is a wave travelling in a curved
4D spacetime that is embedded in a flat 5D manifold which has no
global energy.

As mentioned above, the logical condition on the path of a
particle for such a background field is that it be null. Let us therefore
consider a general case, where we take the metric not in the Min-
kowski form (3.21) but in the canonical form (3.22), thus:

Q = dS 2 = l —ds 2 +dl 2 . (3.23)

Here we take ds 1 = g a p{x a ,l ) dx a dxP, using all of the 5 available coor-
dinate degrees of coordinate freedom to suppress the potentials of
electromagnetic and scalar type, but leaving the metric otherwise gen-
eral. The solution of (3.23) is / = 7 exp[±z (s - s ) I L], where k and
s are constants of which the latter may be absorbed. Then / = / e ±is/L

Paths in Hyperspace 8 1

describes an /-orbit which oscillates about spacetime with amplitude
/o and wavelength L. The motion is actually simple harmonic, since
d 2 l / ds 2 = - I / L 2 . Also dl/ds = ±il / L, so the physical identification
of the mass of a particle with the momentum in the extra dimension as
in brane theory, or with the extra coordinate as in induced-matter the-
ory are equivalent, modulo a constant. In both cases, the /-orbit may
intersect the s-plane a large number of times. There is only one pe-
riod in the metric (3.23), defined by L, but of course a Fourier sum of
simple harmonics can be used to construct more complicated orbits in
the I /s plane. [Alternatively, extra length scales can be introduced to
(3.23) by generalizing L.] If we identify the orbit in the / / s plane
with that of a particle, we have a realization of the old idea (often
attributed to Wheeler and/or Feynman) that instead of there being 10 80
particles in the visible universe there is in fact only one which appears
10 80 times.

3.5 The Equivalence Principle as a Symmetry

The Weak Equivalence Principle is commonly taken to mean
that in a gravitational field the acceleration of a test particle is inde-
pendent of its properties, including its rest mass. This principle lies at
the foundation of Einstein's theory of general relativity, and by impli-
cation is satisfied by other accounts of gravity which use 4D space-
time. In higher-dimensional theories, however, it is not clear if the
principle holds. In 5D, there is an extra coordinate, which has an ex-
tra velocity or momentum associated with it. Both the extra coordi-

82 Five-Dimensional Physics

nate and its rate of change have been linked in extensions of general
relativity of the Kaluza-Klein type to the properties of a particle, such
as its rest mass and electric charge. It would be facile to assume that
all particles have the same values of those properties which can be
attributed to the extra dimension. On the other hand, classic tests of
the WEP have established its accuracy on the Earth to better than 1
part in 10 12 . And new technology indicates that tests in space can
push this to 1 part in 10 18 or better (Lammerzahl, Everitt and Hehl
2001). In this section, therefore, we wish to collect previous results
and give a coherent account of how the Equivalence Principle in its
weak form relates to 5D gravity. (The strong form has not been so
rigorously tested, since it implies that the laws of physics and their
associated parameters are the same everywhere, including the remote
parts of the universe.) Our aim is to constrain 5D relativity by the
WEP, and better understand the nature of the latter.

We use the same terminology as before, with a 5D line ele-
ment chosen to suppress electromagnetic effects but still algebraically
general, that reads

dS 2 = g AB dx A dx B = g aP {x\ T)dx a dy?- <D 2 (x\ l)dl 2 . (3 .24)

Here the 4D line element defines the conventional proper time via
ds 2 = g a p dx a dyP with 4-velocities u a = dx a Ids. As with previous
studies of the fifth dimension, we prefer to use s rather than S as pa-
rameter because we wish to make contact with established physics.
(This also allows us to handle the null 5D paths of Section 3.4 without

Paths in Hyperspace 83

difficulty.) With (3.24) as an algebraic basis and the WEP as a physi-
cal constraint, we now proceed to review certain subjects with a view
to learning about the nature of the fifth coordinate.

(a) The extra force which appears when the manifold is ex-
tended from 4D to 5D has been derived in different ways for induced-
matter theory and brane theory, and has been discussed in Section 3.3
in relation to the canonical metric (3.13). However, it also applies to
the more general metric (3.24), and is in fact generic. To see this, we
recall from Section 3.2 that there is a normalization condition on the
4-velocities:

g a ^,l)u a t/=l . (3.25)

Let us consider a slight change in the 5D coordinates (including
x 4 =/), by differentiating (3.25) with respect to s. Doing this and
using symmetries under the exchange of a and /? to introduce the
Christoffel symbols T^ , there comes

M^ +r H + ^f^° ■ (3 - 26>

This reveals that in addition to its usual 4D geodesic motion (the part
inside the parenthesis), a particle feels a new acceleration (or force
per unit mass). It is due to the motion of the 4D frame with respect to
the fifth dimension, and is parallel to the 4-velocity u'". Explicitly,
the parallel acceleration is

Five-Dimensional Physics

\{ d SaB a B \dl

.. „ , „ . (3.27)

2{ 81 ds

This agrees with (3.20), which is the result of a longer if more infor-
mative derivation where the geodesic equation is applied to a canoni-
cal metric. To return to the latter, let us write g a p[x r ,l\ =

(l 2 1 L 2 )g a/5 (x r ), where L is a length which by (1.15) is related to the

4D cosmological constant by L 2 = 3 / A. The acceleration (3.27) can
now be evaluated and simplified using (3.25). Its nature becomes
clear in the Minkowski limit, when the motion is given by

du M _ pM _
ds

\dl
Ids

or — (lu tl

ds { •

) = o

(3.28)

This is just the expected law of conservation of linear momentum,
provided / = m is the rest mass of the particle.

(b) The action can be used to confirm this. With coordinates
such that g a p {x y , /) = (/ 2 / L 2 ) g a/3 (x r ) and O 2 (x r , /) = 1 , the 5D line
element (3.24) is

dS 2 = — g ap (x Y )dx a dx p -® 2 (x r ,l)dl 2 . (3.29)

This is the pure canonical form (3.13), for which by (3.19) the fifth
force is zero and the motion is geodesic in the conventional 4D sense.

Paths in Hyperspace 85

The first part of (3.29) gives back the element of action mds of parti-
cle physics provided / = m. We have mentioned this before, and re-
call two important things. Firstly, the rest mass of a particle in 5D
theory may change along its path via m = m(s), and even in 4D the
action should properly be written as | mds . Secondly, it is only in

canonical coordinates that the simple identification / = m can be
made, though even in other coordinate systems the 4D action is part
of a5D one.

(c) The 5D geodesic equation minimizes paths via
S\ \dS =0, which generalizes the equations of motion in 4D and

adds an extra component for the motion in the fifth dimension. This
procedure can be carried out for the metric (3.24), which can be made
even more general by including the electromagnetic potential. The
working is long and boring. (See Wesson 1999, pp. 132 - 138 and
pp. 161 - 167 for the cases where electromagnetism is and is not in-
cluded respectively, as well as references to other work.) But the re-
sults of the noted variation can be summed up in terms of several
fairly simple expressions, which under some circumstances are con-
stants of the motion. Of these, let us consider the one associated with
the zeroth or time component of the geodesic equation, which is nor-
mally associated with the particle energy when the metric is static.
We take this, plus the assumptions that electromagnetic terms are ab-
sent, that the 3-velocity v is projected out, and that the 4-part of the

86 Five-Dimensional Physics

metric is quadratically factorized in / as in the canonical case. Then
the constant is

E = — r • (3-30)

One does not have to be Einstein to see that this gives back the con-
ventional 4D energy provided / is identified with the particle rest
mass m.

(d) The field equations for 5D relativity are commonly taken
to be R AB = 0, which we learned in Chapter 1 contain by virtue of
Campbell's theorem the 4D Einstein equations G a p = SttT^. The ef-
fective or induced energy-momentum tensor can be evaluated for the
metric (3.24), in terms of quantities to do with the fifth dimension. It
is given by (1.10) with the appropriate signature, namely:

This is known to give back the conventional matter content of a wide
variety of 4D solutions, but in order to bolster the physical identifica-
tion of / we note a generic property of it. For g a p A = 0, (3.31) gives
%ttT = %ng ali T ap = g^O >a;/8 / O = <£"'n<D . But the extra field equa-
tion R44 = 0, which we will examine below, gives □€> = for

Paths in Hyperspace 87

S a p A = ■ Thus T = for g a ^ 4 = 0, meaning that the equation of state
is that of radiation when the source consists of photons with zero rest
mass. This is as expected.

(e) Algebraic arguments for / = m can be understood from
the physical perspective by simple dimensional analysis. The latter is
actually an elementary group-theoretic technique based on the Pi
theorem, and one could argue that a complete theory of mechanics
ought to use a manifold in which spacetime is extended so as to prop-
erly take account of the three mechanical bases M, L, T. Obviously,
this has to be done in a manner which does not violate the known
laws of mechanics and recognizes their use of the three dimensional
constants c, G and h. The canonical metric of induced-matter theory
clearly satisfies these criteria, and we believe that it deserves its name
because of the simplifications which follow from its quadratic l-
factorization. But how unique is this form? To investigate, let us
consider a 5D line element given by

= (yj fl g^(*0^^-(y)^ 2 • ( 3 - 32 )

Here a, b are constants which can be constrained by the full set of 5D
field equations Rab = 0. It turns out that there are 3 choices: a = b =
gives general relativity embedded in a flat and physically innocuous
extra dimension; a = -\, b = gives the pure canonical metric already
discussed; while a = b = 1 gives a metric which looks different but
is actually the canonical one after the coordinate transformation

88 Five-Dimensional Physics

I — » 1} II . We see that the last two cases describe the same physics
but in terms of different choices of /. Temporarily introducing the
relevant constants, these are

/.=% /,=-*- (3.33)

c mc

in what may be termed the Einstein and Planck gauges. These repre-
sent convenient choices of x 4 = /, insofar as they represent parametiza-
tions of the inertial rest mass m of a test particle which fit with known
laws of 4D physics such as the conservation of linear momentum (see
above: the fifth force conserves l E u M or lp l u M ). However, 5D relativ-
ity as based on the field equations R AB = is covariant under the 5D
group of transformations x A — > x A \x B \ , which is wider than the 4D

group x a — > x a \x p J . Therefore 4D quantities Q(x",l ) will in gen-
eral change under a change of coordinates that includes /. This im-
plies that we can only recognize m in certain gauges. The Einstein
(canonical) and Planck gauges (3.33) are good parametizations for m,
because they allow us to geometrize mass in a way consistent with the
use of physical dimensions in the rest of physics.

The import of the preceding comments (a) - (e) is major for
dynamics and the WEP, so a short recapitulation is in order. The
general metric (3.24) is not factorized in the extra coordinate x A = I;
but when the 4-velocities are normalized as in (3.25) there results the
equation of motion (3.26) which includes a new acceleration parallel

Paths in Hyperspace 89

to the motion (3.27). This is identical to (3.20), which follows from
the canonical metric (3.13), where the 4D part is factorized in / quad-
ratically. This gauge conserves momentum via (3.28) if the identifi-
cation / = m is made. Other consequences follow for this gauge, as
summarized in equations (3.29) - (3.33). It is apparent that the con-
ventional concept of momentum is most conveniently realized by put-
ting the 5D metric into the canonical form, where the extra coordinate
plays the role of particle rest mass. Further, the Weak Equivalence
Principle is then recovered from the equations of motion (3.17) -
(3.19) when the 4D metric is independent of the extra coordinate. In
other words, conventional dynamics and the WEP are the result of a
metric symmetry.

This symmetry is geometrical in nature, but like the symme-
tries of particle physics, it can be expected to break down at some
level. The identification I = m suggested above means that our sym-
metry becomes mechanical, and that the breakdown would involve
mass terms in the 4D part of the 5D metric. The WEP is usually
phrased in terms of accelerations (rather than momenta), and is com-
monly understood to mean that the motion of an object of mass m in
the gravitational field of a larger object of mass Mis exactly geodesic
in the 4D Einstein sense. A violation of the WEP would therefore
follow from the presence of / = m in the 4D part of the metric. Ide-
ally, this would be formalized via a solution of the field equations,
and we will consider such below. Here, however, we note that viola-
tions of the WEP are to be expected in situations where m / M is not

90 Five-Dimensional Physics

negligible. Traditionally, such situations have been handled in areas
like gravitational radiation by considering the "back reaction" of the
test particle on the field of the source. But this is clearly an approxi-
mation to the real physics, which would involve the fields due to both
objects. We are led to conclude that the WEP, viewed as a symmetry
of 5D gravity, should be violated at some level.

3.6 Particle Masses and Vacua

In the foregoing section, we learned that in 5D relativity there
is a convenient form of the metric which we renamed the Einstein
gauge, because in it the extra coordinate essentially measures the
mass of a particle by its Schwarzschild radius. However, there is an
inverse form which we named the Planck gauge, because the corre-
spondence involves the quantum of action, and effectively measures
the mass of a particle by its Compton wavelength. The geometriza-
tion of rest mass in this manner is on a par with how Minkowski con-
verted the time to a length using the speed of light, creating space-
time. We have merely extended this approach, creating a space-time-
matter manifold.

In this section, we wish to address two issues which arise
from the preceding account. They both concern the metric (3.24),
which involves a scalar part and a spacetime part that can both depend
on x 4 = I. Physically, these bring up questions of how to give defini-
tions for the mass of a particle and the vacuum which are more gen-
eral than those we considered above.

Paths in Hyperspace 9 1

The mass of a particle in manifolds where there is a scalar
field |g 44 | = <& 2 {x y , l) can be defined most logically by

m=^0\dl=^(dl/ds)\ds . (3.34)

This is in line with how proper distance is defined in 3D (see Ma
1990). In practice, <D would be given by a solution of the 5D field
equations as outlined below, and dl / ds would be given by a solution
of the extra component of the 5D geodesic equation (or directly from
the metric for a null 5D path). We note that a potential problem with
this approach is that <D may show horizon-like behaviour. An exam-
ple is the Gross/Perry/Davidson/Owen/Sorkin soliton, which in terms
of a radial coordinate r which makes the 3D part of the metric
isotropic has g 44 = - <E> 2 = - [(1 - a I If) I (1 + a / 2r)] 2/?/a where a is
the source strength and a, /? are dimensionless constants constrained
by the field equations to obey a 2 = fi 2 + p + 1 (see Wesson 1999, pp.
49 - 58). This problem may be avoided by restricting the physically-
relevant size of the manifold. Another potential problem is that real
particles may have <D = O (x y , l) so complicated as to preclude finding
an exact solution. This problem may be avoided by expanding O in a
Fourier series:

®(x r ,l)= J^O {n) (x r )exp(inl/L) . (3.35)

Here L is the characteristic size of the extra dimension, which is re-
lated to the radius of curvature of the embedded 4-space which the

92 Five-Dimensional Physics

particle inhabits. It should be noted that in both modern versions of
5D relativity, namely induced-matter theory and brane theory, the ex-
tra dimension is not compactified. Thus we do not expect a simple
tower of states based on the Plank mass, but a more complicated spec-
trum of masses that offers a way out of the hierarchy problem. This
will depend on the precise form of O = O (x y , /). To obtain this, we
need to solve the R44 = component of the field equations (1.13),
which for the signature (h ) being used here is

D ^_L[£Zp +g ^„_*^] . (3 . 36)

This is a wave equation, and is source-free when the 4D part of the
metric (3.24) does not depend on x 4 = /. But in general it will have
such a dependency, and then (3.36) in combination with (3.34) raises
the interesting possibility that the mass of a local particle depends on
a universal scalar field. This is a realization of Mach's Principle,
which we mentioned in Section 3.2 as a hypothesis for the origin of
mass in classical field theory. There is also an obvious connection
with the Higgs field, which is the agent by which particles obtain
mass in quantum field theory.

The vacuum in manifolds where there is a spacetime g a p =
gap (x y ,l) is more complicated to define in 5D than it is in the 4D the-
ory of Einstein. In the latter, there is a unique vacuum state which is
measured by the cosmological constant and has the equation of state

Paths in Hyperspace 93

A = ~Pv = A /8tt. In 4D, the field equations R a p = admit solutions
which are empty of ordinary matter but have vacuum matter, the
prime example being the de Sitter solution. In 5D, the field equations
Rab = admit solutions which have ordinary matter and vacuum mat-
ter, in general mixed as evidenced by the effective energy -momentum
tensor (3.31). The last relation, on inspection, shows that it is indeed
more appropriate to talk of vacua in 5D, rather than the vacuum of
4D. To illustrate this, we note here 3 exact solutions of R AB = 0:

dS 2 =^r\\\-

l 2

M'*)

-r 2 d€l 2 \- dl

L 2

aL

l-

/

Hf + T

(l-r 2 ,L 2 )

dr 2
~(l-r 2 /L 2 )

-r 2 da 2 \-dl 2

[1 + ^-1 r 2 dQ 2 \-dl 2

(3.37)

Here dQ 2 = (dQ 2 +sin 2 6d<p 2 ), so all 3 solutions are spherically sym-
metric in 3D. The first is a 5D canonical embedding of the 4D de Sit-
ter solution provided the identification L 2 = 3 / A is made (see above).
However, in general L measures the size of the potential well associ-
ated with x 4 = /. Solutions like these depend in general on two dimen-

94 Five-Dimensional Physics

sionless constants a, /?. We have examined the properties of (3.37)
extensively, and have found that they are 5D flat (Rabcd — 0; see
Wesson 2003b). But here we note only their main 4D features.
These can be appreciated by combining the above solutions in the
form

L 2 { j

nr

f — , B, L_ C = l + 2±. (3.38)

/ (l-r 2 /L 2 ) 1 ' 2 rl

The 4D subspaces defined by these solutions are curved, with a 4D
Ricci scalar 4 R which by Einstein's equations is related to the trace of
the 4D energy-momentum tensor by 4 R = - 8^7. The general expres-
sion for 4 R for any 5D metric of the form (3.24) is given by (1.9),
which for the signature being used here is

4R = -^[s M ^A+(s f,V g M J] ■ (3-39)

The special expression for (3.38) is

2 T 1 2 1 2 1

4 R = -8xT = -— \— + ^— + — + -\ . (3.40)

L 2 IAB ABC C 2 C\

This shows that stress-energy is concentrated around singular shells
where one of A, B or C is zero. The equation of state is in general

Paths in Hyperspace 95

anisotropic (7J 1 ^ T 2 ) . If one replaces 1 / L 2 in (3.40) by its de Sitter

limit A / 3, it becomes obvious that the meaning of the cosmological
"constant" requires a drastic rethink. The effective A is in general a
function of r and /, opening the way to a resolution of the cosmologi-
cal-constant problem.

3.7 Conclusion

Mechanics is often regarded as a staid subject, but its exten-
sion to TV > 5 dimensions leads to some novel results. The 4D version
is based on conventions, and in the extension to N > 5D we have to
ensure that the dynamics is dimensionally consistent. The main con-
ventions are the normalization of the velocities, and the definition of
these in terms of the proper time (Section 3.2). The rest mass of a
particle provides the link between the concept of acceleration as used
in general relativity and the concept of momentum as used in quan-
tum theory. If the manifold is extended to 5D in a meaningful way, it
is inevitable that there appears an extra acceleration or force per unit
mass (Section 3.3). This implies that our 4D laws are modified by the
bits due to the fifth dimension. There are question marks over the
size of the 5D interval and the sign of the fifth term in the metric
(Section 3.4). It is possible that particles travelling on timelike paths
in 4D are moving on null paths in 5D, so that massive particles in
spacetime are like photons in the larger manifold. It is also possible
that the fifth dimension is not spacelike as commonly assumed, but
timelike, so particles can be multiply imaged. There is, though, no

96 Five-Dimensional Physics

question that the Weak Equivalence Principle is observed to hold to
great accuracy in 4D. This may be viewed as the consequence of a
symmetry in 5D (Section 3.5). However, it would be simplistic to
assume that the extra coordinate can never appear in the 4D part of
the 5D metric. So like the symmetries of particle physics, it is ex-
pected that the WEP will be violated in some situations at some level.
This is true for any interpretation of the fifth coordinate. When the
extra coordinate is identified with particle rest mass - for which there
are several arguments - the breakdown of the WEP is related to the
ratio of the test and source masses. A more complete theory of parti-
cle mass should involve the scalar field (which acts like the classical
version of the quantum Higgs field); but even when this is flat, a par-
ticle does not move through a unique vacuum but can experience one
of several vacua (Section 3.6). Indeed, the multiple states of "empti-
ness" admitted by 5D relativity provide a fascinating topic for future
study. If the extra coordinate is mechanical, the fact that it may in
principle have either sign brings in the possibility of negative mass.
And in general, exact vacuum solutions provide a new way to differ-
entiate between 4D and 5D.

Questions to do with the nature of particle mass and the vac-
uum will be taken up below. It is clear that the results we have de-
rived to here, in combination with others in the literature, bring us to
the brink of quantum considerations. As a classical theory, it is ap-
parent that 5D relativity is viable. We should recall that it agrees with
the classical tests carried out in the solar system and with other astro-

Paths in Hyperspace 97

physical data, as well as being in conformity with less exact cosmo-
logical observations (Wesson 1 999). It achieves this by virtue of be-
ing an extension of 4D general relativity rather than a departure from
it. In fact, the only major uncertainty about 5D relativity is whether
the manifold is smooth as for induced-matter theory or has a singular
surface as for membrane theory. Unfortunately, the strength of the
5D approach is also its weakness, insofar as the departures it predicts
from 4D theory are small and difficult to measure. Work is underway
to quantify and detect classical effects which would indicate a fifth
dimension, via for example a satellite test of the Equivalence Princi-
ple. Right now, however, we choose to leave the classical domain
and turn to the quantum one.

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Davidson, A., Owen, D.A. 1986, Phys. Lett. B 177, 77.

Jammer, M. 2000, Concepts of Mass in Contemporary Physics and

Philosophy (Princeton U. Press, Princeton).
Lammerzahl, C, Everitt, C.W.F., Hehl, F.W. 2001, Gyros, Clocks,

Interferometers ... Testing Relativistic Gravity in Space

(Springer, Berlin).

98 Five-Dimensional Physics

Linde, A.D. 1990, Inflation and Quantum Cosmology (Academic,

Boston).
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4. QUANTUM CONSEQUENCES

"To see a world in a grain of sand" (Blake)

4.1 Introduction

Practioners of quantum theory and classical theory often view
the machinations of the other camp with suspicion. Certainly the sub-
jects involve separate approaches and appear to have fundamental
differences. Wave mechanics and modern quantum field theory de-
pend on Planck's constant of action, and through it have an apparent
level of non-predictability which is formalized in Heisenberg's uncer-
tainty relation. General relativity and extensions of it depend on New-
ton's constant of gravity to measure the strength of a smooth field in
which the motion of a test particle can be predicted with unlimited
precision. It is frequently stated that quantum mechanics goes to clas-
sical mechanics in the limit in which Planck's constant tends to zero.
But there is more to the issue than this, as can be appreciated by con-
sidering quantum and classical electrodynamics. The former uses the
Dirac equation, which is first order and comes from a factorization of
the metric of spacetime. The latter uses the Maxwell equations, which
are second order and invariant under transformations of the whole
spacetime. Both are highly successful theories, able to inform us re-
spectively about (say) the spin of an electron or the emission of a ra-
dio wave. Their different qualities lie largely in their different alge-
bras. Following from this, we will in this chapter study how 5D alge-

Quantum Consequences 101

bra affects quantum and classical physics, with a view to their recon-
ciliation.

This goal may appear presumptuous. However, we will be
able to show significant technical results in key areas. These include
the inheritance of 4D Heisenberg-type dynamics from 5D laws of
motion, the plausibility that 4D mass is quantized because of the
structure of the fifth dimension, and the recovery of the 4D Klein-
Gordon and Dirac equations from 5D null paths. These topics, occu-
pying Sections 4.2 - 4.4, draw for their discussion on results we es-
tablished in the preceding chapter. There, we saw that the extension
of the manifold from 4D to 5D necessarily brings in the existence of a
fifth force, raising the possibility that conservation laws in 5D are im-
perfect when viewed in 4D, and that we may observe as anomalous in
4D the bits of the dynamics left over from 5D. In the previous chap-
ter we also saw that massive particles on timelike paths in 4D can be
viewed as moving on null or photon-like paths in 5D, implying that
objects which appear to be causally separated in 4D can be in contact
in 5D (like when objects apparently separated in ordinary 3D space
are connected by photons in 4D). These two properties of 5D relativ-
ity have, we should recall, been studied for both membrane theory
(Youm 2000, 2001) and induced-matter theory (Wesson 2002, 2003).
We will use the latter approach, since it lends itself more readily to
our purpose, but the two approaches are mathematically equivalent
(Ponce de Leon 2001). The null-path hypothesis can be applied in
two natural coordinate frames, which in the previous chapter we

1 02 Five-Dimensional Physics

dubbed the Einstein and Planck gauges. We will look at how quanti-
zation can depend on the gauge in Section 4.5, and discover how it
may carry over from the microscopic to the macroscopic domain in
the form of a (broken) symmetry for the spin angular momenta of
gravitationally-dominated systems. Then in Section 4.6 we will em-
ploy the insights gained previously to revisit that most hoary of sub-
jects, the difference between a particle and a wave. Encouragingly,
we will find that the flexibility afforded by 5D coordinate transforma-
tions allows us to resolve this problem in terms of gauges. We round
off our itinerary with a short discussion in Section 4.7, where we
make some comments about unification.

4.2 4D Uncertainty from 5D Determinism

In this and the following section, to aid interpretation we use
physical units for Planck's constant h, the gravitational constant G
and speed of light c. We will also need to consider the cosmological
constant A, which we take to have units of an inverse length squared.
This parameter measures the energy density of the vacuum in 4D
general relativity, and is related to the length L which scales the ca-
nonical metric in 5D theory, via A = 3 / L 2 (see Chapter 1). However,
when the metric is transformed to suit other problems, L measures the
scale of the potential well in which a particle finds itself, and may
therefore be related to the vacuum or zero-point fields characteristic
of quantum interactions.

Quantum Consequences 103

The canonical line element which we have already examined
has the form dS 2 = (I / L) 2 ds 2 - dl 2 . The coordinates are x A = (x a , l)
and the 5D interval contains the 4D one defined by ds 2 =
g a p dx a dx 6 (a , /? = 0, 123 for time and space). The 5D metric is
mathematically general if we allow g a p = g a p (x a , /), though it is
physically special in that the electromagnetic potentials are g 4a =
and the scalar potential is g 44 = - 1. However, we saw in Section 3.5
that if g a p = g a p (x y only) then we recover the Weak Equivalence
Principle as a geometrical symmetry. This and several other proper-
ties led us to identify the extra coordinate x 4 = 1 = Gm / c 2 in terms of
the rest mass m of a test particle. Since this is the particle's
Schwarzschild radius, we renamed the canonical metric the Einstein
gauge. It has a coordinate transformation / — >• L 2 1 1, which changes
the form of the line element to one whose properties led us to identify
the coordinate in the new metric as / = h / mc. Since this is the parti-
cle's Compton wavelength, we named this form the Planck gauge. It
is specified by

dS 2 =^ds 2 -^-dl 2 , (4.1)

I 2 I 4

and it is this which we will study in the present section.

The dynamics associated with (4.1) are best brought out if we
use the 4D proper time s as parameter, since we have a large body of
data couched in terms of this, with which we wish to make contact.

1 04 Five-Dimensional Physics

The Lagrangian density £ = [dS I ds) for (4.1) has associated with
it 5 -momenta given by

d£ 21} dx p

a 8(dx a /ds) I 2 aP ds

d£

2L A dl

' d{dllds) l A ds

(4.2)

These define a 5D scalar which is the analog of the one used in 4D
quantum mechanics:

jP A dx A = \(P a dx a + Pjdl)

C2L 2
"J I 2

[-mi]

This is zero for dS 2 = 0, since then (4.1) gives

l = l e ±s/L ,

ds L

(4.3)

(4.4)

where / is a constant. The second member of this shows why some
workers have related the (inertial) rest mass of a particle to / (Wesson
2002) and some to its rate of change (Youm 2001) with consistent
results: the two parametizations are essentially the same. In both
cases, the variation is slow if s/L<$:l (see below). We prefer to
proceed with the former approach, because it makes the first part of

Quantum Consequences 105

the 5D line element in (4.1) essentially the element of the usual 4D
action mc ds, with the identification / = h/mc. It should be noted that
the test particle we are considering has finite energy in 4D, but zero
"energy" in 5D because \P A dx A = .

The corresponding quantity in 4D is \p a dx a , and using rela-
tions from the preceding paragraph it is given by

f/V^=fa^f— = ±-- ■ (4-5)

J J J cl c I

The fact that this can be positive or negative goes back to (4.4), but
since the motion is reversible we will suppress the sign in what fol-
lows for convenience. We will also put L / 1 =n, anticipating a physi-
cal interpretation which indicates that it is not only dimensionless but
may be a rational number. Then (4.5) says

) mc ds-nh . (4.6)

Thus the conventional action of particle physics in 4D follows from a
null line element (4.1) in 5D.

The other scalar quantity that is of interest in this approach is
dp a dx a . (It should be recalled that dx a transforms as a tensor but x a
does not.) Following the same procedure as above, there comes

,„ h( du n dx a 1 dl^\ds 2 .._.

} a dx =— — • ( 4 - 7 )

c I ds ds I ds j I

106 Five-Dimensional Physics

The first term inside the parenthesis here is zero if the acceleration is
zero or if the scalar product with the velocity is zero as in conven-
tional 4D dynamics (see Section 3.3). But even so, there is a contri-
bution from the second term inside the parenthesis which is due to the
change in mass of the particle. This anomalous contribution has
magnitude

i, , a . h\dl\ds 2 hds 2 h(dl) 2

\dp„dx\ = —\ — — 7- = = n—\ — , (4.8)

1 " ' c\ds\ I 2 c LI c\l )

where we have used (4.4) and n =L / I. The latter implies dn / n =
- dl / 1 = dKi / Ki where Ki = III is the wavenumber for the extra di-
mension. Clearly (4.8) is a Heisenberg-type relation, and can be writ-
ten as

\d Pa d X "\ = ^ . (4.9)

1 ' c n

This requires some interpretation, however. Looking back at the 5D
line element (4.1), it is apparent that L is a length scale not only for
the extra dimension but also for the 4D part of the manifold. (There
may be other scales associated with the sources for the potentials that
figure in g a p, and these may define a scale via the 4D Ricci scalar 4 R,
but we expect that the 5D field equations will relate 4 R to L.) As the
particle moves in spacetime, it therefore "feels" L, and this is re-
flected in the behaviour of its mass and momentum. Relations (4.6)
and (4.9) quantify this. If the particle is viewed as a wave, its 4-
momenta are defined by the de Broglie wavelengths and its mass is

Quantum Consequences 107

defined by the Compton wavelength. The relation dS 2 = for (4.1) is
equivalent to P A P A = or K A K A = 0. The question then arises of
whether the waves concerned are propagating in an open topology or
trapped in a closed topology. In the former case, the wavelength is
not constrained by the geometry, and low-mass particles can have
large Compton wavelengths / = h/mc with I > L and n= L/l <\. In
the latter case, the wavelength cannot exceed the confining size of the
geometry, and high-mass particles have small Compton wavelengths
with 1<L and n > 1 . By (4.9), the former case obeys the conventional
uncertainty principle while the latter case violates it. This subject
clearly needs an in-depth study, but with the approach adopted here
we tentatively identify the former case as applying to real particles
and the latter case as applying to virtual particles.

The fundamental mode {n = 1) deserves special comment.
This can be studied using (4.6) - (4.9), or directly from (4.1) by using
/ = h/mc with dS 2 = 0. The latter procedure gives \dm\ = m ds/L
which with (4.6) yields m = ( \mc ds\l cL = nhl cL . This defines for

n=la fundamental unit mass, m = h/cL. So apart from giving back
a Heisenberg-type relation in (4.9), the 5D null-path approach appears
to imply the existence of a quantum mass.

4.3 Is Mass Quantized?

This question cuts deeper than the formalism of any particular
theory. It is basically asking if there is a minimum length scale for
the universe, provided by as yet unproven properties of particle

108 Five-Dimensional Physics

physics. If it exists, this small scale would compliment the large one
which is indicated by the supernova and other data discussed in Chap-
ters 1 and 2. There we learned that the cosmological constant is posi-
tive and finite, implying a scale for the present cosmos of order A" 1/2
or 10 28 cm (Lineweaver 1998; Overduin 1999; Perlmutter 2003). We
also learned that A could be interpreted as the energy density of the
vacuum, which is Ac 4 / %nG and constant in general relativity, but
could be the result of a scalar field and time-variable in generaliza-
tions of that theory (Weinberg 1980; Wesson 1999; Padmanabhan
2003). It is plausible that any length scale - small or large - which
we measure during our "snapshot" view of physics (of order 10 2 yr),
could be changing over the longer periods typical of the evolution of
the universe (of order 10 10 yr). In this case the universe would be
scale-free, or in the jargon of dimensional analysis, self-similar. It is
hard to see how we could test this global hypothesis, attractive though
it may be. But what we can do is take the generic approach provided
by dimensional analysis, and see what it implies given present data
about a smallest scale or a quantum of mass. We can then compare
the result of this generic approach with the specific one provided by
5D relativity.

There are 2 masses which can be formed from a suite of 4
constants with degenerate or "overlapping" physical dimensions (De-
sloge 1984). Thus from h, G, c and A we can form a microscopic
mass

Quantum Consequences 109

= 2xl(T 65 g , (4.10)

-A J )

and a macroscopic mass

-mr~>

■Amr

(4.11)

Here the two masses are relevant to quantum and gravitational situa-
tions, and so may be designated by the names Planck and Einstein
respectively. [To avoid confusion, it can be mentioned that the mass
m PE = (hc/G) m — 5 x 10" 5 g which is sometimes called the Planck
mass does not involve A and mixes h and G, so from the viewpoint of
higher-dimensional field theory and (3.33), it arises from a mixture of
gauges and is ill-defined, possibly explaining why this mass is not
manifested in nature.] The mass (4.1 1) is straightforward to interpret:
it is the mass of the observable part of the universe, equivalent to 10 80
baryons of 10" 24 g each. The mass (4.10) is more difficult to interpret:
it appears to be the mass of a quantum perturbation in a spacetime
with very small local curvature, measured by the astrophysical value
of A as opposed to the one sometimes inferred from the vacuum fields
of particle interactions (Weinberg 1980; Padmanabhan 2003). We
will return to the cosmological-constant problem below, whose es-
sence is that to make (4.10) and (4.11) the same would require
(hG / c 3 ) = 3 / A = L 2 , requiring the cosmological constant to have a
value many orders bigger than what is inferred from astrophysics.
This is a major problem, but is moot in the present context, because

110 Five-Dimensional Physics

the astrophysical value of A is the smallest and so the mass given by
(4.10) is also the smallest. That is, dimensional analysis predicts on
general grounds a unit or quantum of mass of approximately
2xl0" 65 g.

Relativity in 5D provides a more specific means of analysis.
As we have seen, there are two coordinate frames relevant to me-
chanical problems, with different identifications for the extra coordi-
nate (3.33). These are l E = Gm / c 2 and l P = h / mc, for the Einstein
(canonical) and Planck gauges respectively. These gauges have some
common properties. For example, the null-path hypothesis results in
dl/ds = ± / / L for both. This is relevant to the old version of Kaluza-
Klein theory, in which / was compactified to a circle and its rate of
change was related to electric charge, which was thereby quantized.
However, we are using the extra dimension not as a means of under-
standing the electron charge but as a means of understanding mass
scales, and we need to distinguish between the gauges. For this we
use the labels introduced above. Thus

is the Planck gauge which we used to study uncertainty in Section 4.2,
and

dS 2 =i l -A ds 2 -dl 2 E (4.13)

Quantum Consequences 111

is the Einstein gauge which has been studied in various contexts as
summarized elsewhere (Wesson 1999). These gauges imply different
mass scales.

To see where the microscopic mass (4.10) originates, we can
take (4.12) in the null case to obtain

Hti-GO 1 " * ■ <4 - 14)

Here we know that the conventional action is quantized and equal to
nh where n is an integer. Thus L / l P = n. This means that the Comp-
ton wavelength of the particle cannot take on any value, but is re-
stricted by the typical dimension of the (in general curved) spacetime
in which it exists. Putting back the relevant parameters, the last rela-
tion says that m = (nh/c)(AJ3) m . For the ground state with n = 1,
there is a minimum mass (/z/c)(A/3) 1/2 = 2 x 10" 65 g. This is the same
as (4.10). To see where the macroscopic mass (4.11) originates, we
can take (4.13) in the null case to obtain

j-\dl E = \ds . (4.15)

Here we do not have any evidence that the line element by itself is
quantized, so the discreteness which is natural for the Planck gauge
does not carry over to the Einstein gauge. However, in the Planck
gauge the condition L / 1 = n could have been used to reverse the ar-
gument and deduce the quantization of the action from the quantiza-

112 Five-Dimensional Physics

tion of the fifth dimension, implying that the latter may be the funda-
mental assumption. Let us take this in the form L/l E = n. (This im-
plies dls/ds = \/n, which holds too in the Planck gauge, so the veloc-
ity in the extra dimension is also quantized.) Putting back the rele-
vant parameters, we obtain m = (c 2 /nG)(3/A) ia . For the ground state
with n = 1, there is a maximum mass (c 2 /G)(3/A) 1/2 - 1 x 10 56 g.
This is the same as (4. 1 1 ) above.

In summary, astrophysical data imply that we should add A to
the suite of fundamental physical parameters, which implies on di-
mensional grounds that there are two basic mass scales related to h
and G. These mass scales can be understood alternatively as the con-
sequences of discreteness in the fifth dimension. The smaller scale
defines a mass quantum of approximately 2 x 10" 65 g. This is so tiny
that mass appears in conventional experiments to be continuous.

4.4 The Klein-Gordon and Dirac Equations

We discussed the Klein-Gordon equation briefly in Section
3.2, noting that it is the wave equation which corresponds to the stan-
dard energy condition for a particle. Its low-energy form is the
Schrodinger equation, and its factorized form is the Dirac equation.
In the present section we wish to review the status of these equations
in 4D and show how they can be derived from 5D (Wesson 2003).
We use the terminology of previous sections in this chapter, except
that we absorb the constants to smooth the algebra.

Quantum Consequences 113

The Klein-Gordon equation relates the (inertial rest) mass m
of a spin-0 particle to a scalar wave function ^ via

a 2 + m 2 <p = O . (4.16)

Here a 2 ^> = r] afl 8 2 ^/ dx a dx /} for a flat 4D space rj a ^ = diag
(+1, -1, -1, -1) but the generalization to a curved 4D space g a ^ is
straightforward using the comma-goes-to-semicolon rule to introduce
the covariant derivative, whence n 2 ^ = g a/ V, a7 j ■ The 4-velocities
u a = dx a / ds yield 4-momenta// 2 = mu a which can be obtained from a
wave function via p a = (1 / ifi) d</)/ dx a where = exp / \p a dx a =

exp / \mds . The 4-velocities are conventionally normalized via
u a u a = 1 , which on multiplication by m 2 yields p a p a = m 2 or

E 2 -p 2 -m 2 =Q . (4.17)

Here E and/» are the energy and 3 -momentum of the particle. These
preliminaries may be familiar, but are necessary because it is required
that (4.16) and (4.17) in 4D be recovered from a situation which is
quite different in 5D. In such a manifold, the noted relations should
clearly be replaced by

□ 2 d> = (4.18)

P A P A =0, (4.19)

114 Five-Dimensional Physics

where the 5D parameters are defined in analogous fashion to their 4D
counterparts.

The embedding of 4D dynamics in 5D requires a choice of
gauge, as mentioned above, and three such are in use. (a) The Min-
kowski gauge is flat 5D space, and with x 4 = Im the null path is given
by dS 2 = = dt 2 - (dx 2 + dy 2 + dz 2 )- dl 2 M , so l M = ± s where a fm-

ducial value of the 4D proper time has been absorbed, (b) The Planck
gauge is (4.12) above, and with x 4 = l P the null path is given by
dS 2 = = (Lll p f ds 2 -{Lll P ) 4 dl 2 , so l P = /„ exp (± s / L) where k
is a constant, (c) The Einstein gauge is (4.13) above, and with x 4 = l E
the null path is given by dS 2 = = (l E / L) 2 ds 2 - dl\ , so l E =

1 exp (± s / L) as before. We noted previously that the Planck gauge
is obtained from the Einstein gauge by the simple coordinate trans-
formation l E —>■ L 2 / l P . In the Planck gauge, L is the scale of the po-
tential well in which the particle moves. In the Einstein gauge this
scale is large and related to the cosmological constant via A = 3 / L 2
of (1.15). The 3 gauges are suited to different kinds of problem. The
Minkowski gauge is just a flat coordinate manifold; while the Planck
and Einstein gauges with the identifications l P = Mm, l E = m of (3.33)
are (in general) curved momentum manifolds. The connection
between the null Minkowski and Einstein manifolds is via l E =

k exp (± l M / L), which with the coordinate transformation between
the Einstein and Planck gauges noted above means that we can go
between any of the three.

Quantum Consequences 115

A wave function which is extended from 4D to 5D has the
form = exp \i\(p a dx a +p M dl M )\ in the Minkowski gauge.
Changing to the Einstein gauge, it transpires that the 4D and extra
parts of the integrand are equal, yielding © = exp 2i \mds . This

resembles the usual 4D wave function, but m = m{s) in general so the
mass has to stay in the integrand. Also, it is algebraically more con-
venient to take the root of this quantity, so the wave function is just

^ = exp[iJWfc] . (4.20)

Taking the logarithm of this, writing dS 4 for the 4D part of the 5D
metric, and noting that for a null path \dS 4 \ = dl E , there comes

ln(<f>) = iL$\dS 4 \ = iLl E . (4.21)

For a superposition of states, this invites us to form the natural loga-
rithm of the product of several wave functions, as

ln(<f> l 2 -) = iL(l E +l 2 E +-) = iLl t °' al • (4.22)

That is, the wave function for a system of particles is basically the
total extension in the fifth dimension or the total mass.
Returning to (4.20), it gives

116 Five-Dimensional Physics

dx a dx

2 ds ds . du a ( ds \ ds
dx a dx p ds {dx a J dx p

dm ds ds
ds dx a dx 13

(4.23)
Here the last two terms in the bracket are related to the fifth force dis-
cussed in Section 3.3, and are imaginary. Taking the real part of
(4.23) and contracting we obtain

f p v a v p , (4.24)

6 dx a 8x fi YS aP '

where v a = ds / dx a , which implies u a v a = 1 . However, the 4-velocities
are normalized via u a u a = 1 (see above), so v a = u a . With the same
definition of the d'Alembertian as before, (4.24) then reads

n 2 + m 2 <fl = O . (4.25)

This is formally the same as the conventional Klein-Gordon equation
(4.16), but its physical interpretation is somewhat different insofar as
m = m(s) in general. In other words, 5D relativity yields the 4D
Klein-Gordon equation but with a variable mass.

The Dirac equation relates the mass mofa spin- 1/2 particle to
a bispinor field if/ via

iy a —-rmf/ = . (4.26)

8x a

Here y a are four 4x4 matrices which obey the relationship
y a yP + yPy a = 2ri a P. This decomposes the metric of spacetime, and

Quantum Consequences 117

(4.26) is commonly regarded as resulting from the decomposition of
the relativistic energy equation (4.17), or equivalently the relativistic
wave equation (4.16). As above, these preliminaries may be familiar,
but are necessary because it is required that (4.26) in 4D be recovered
from a situation which is noticeably different in 5D. The main differ-
ence lies in the fact that the quantities which enter the problem in 4D
are not covariantly defined whereas those in 5D are so. (The field if/
can be thought of as a 4-element column matrix, but neither this nor
y" are 4-vectors.) This technical problem can be overcome by realiz-
ing that if if/ is the conjugate of if/ then the combination y a (dy//dx a )yj
must by (4.26) measure the real scalar quantity m, which can be iden-
tified as a geometrical quantity in 5D after choosing one of the gauges
discussed above. A related issue concerns the distinction between the
electron (e) and the positron (e + ). In 4D it is customary to regard the
upper two elements of yi as the spin states of the e~ and the lower two
elements as the spin states of the e + . But in 5D such an assignation
cannot be made a priori, and the e" degeneracy is lifted by the pres-
ence of an external electromagnetic field (see below). A last issue
which requires comment is the definition of an origin for the spin an-
gular momentum of a system. In 4D, it is customary to set the zeroth
component of the spin vector S a to zero by referring the angular mo-
mentum to the centre of mass, the other 3 components being given by
the relation u a S a = which follows from the properties of the angular-
momentum tensor. In 5D, a similar relation holds because S A is
spacelike while dx 4 / dS is timelike, so their inner product can always

118 Five-Dimensional Physics

be made to vanish. However, S 4 cannot in general be set to zero by an
argument analogous to that for So, because insofar as angular momen-
tum for the fifth dimension can be defined it involves a moment arm
x 4 = I which is related to particle mass. The upshot of these com-
ments is that while it makes sense to seek a 5D analog of (4.26), cer-
tain differences of physical interpretation are to be expected.

Bearing this in mind, it is convenient to rewrite the spin /
velocity orthogonality relation

u A S A = u a S a + u 4 S 4 = (4.27)

in the alternative form

u a S' a ±w = , (4.28)

where S' a =S a l S 4 is defined in analogy with the electromagnetic
potentials A a = g a 4/g44 of Kaluza-Klein theory and w = \u 4 \ = \dl / ds\.
These are related in the null Einstein gauge with electromagnetism
(Section 1.4) by

dS 2 =0 = \ds 2 -(d! E + A a dx a f , (4.29)

which gives

w = ^-\l + — Au a \ . (4.30)

L\ I \

Here the constant L can be absorbed within the Einstein gauge, and
the weak-field limit of (4.30) with l E = m then causes (4.28) to read

Quantum Consequences 119

u a S' a ±m = . Redefining the scalar u"S' a in terms of a bispinor
field (see above) then gives a relation which is formally the same as
(4.26). However, a more direct route to the Dirac equation than that
provided by (4.27) is to assume the existence of a 5D field yj (x 4 )
which obeys

dx'

H'-fr'SY ■ (4 - 31)

Here the Dirac matrices are extended to y A {A = a, 4); and dy/ / dx 4 =
(dy/ / ds)(ds / dx 4 ) = ±(L/l)(dyi / ds) for both the Einstein and Planck
gauges. Within the latter, L can be absorbed and l P = Mm causes
(4.31) to become formally the same as (4.26), modulo y 4 . This has no
obvious rationale, but may be given one via the Hoyle-Narlikar
identity

\^y/\ 2 =(w>)(wr a w)-(W4W) 2 , (4-32)

which may be proved using the Pauli matrices. We see that the 4D
Dirac equation (4.26) can be understood as a consequence of the 5D
relations (4.27) and (4.31), in the Einstein and Planck gauges, respec-
tively.

4.5 Gauges and Spins

It is apparent from the contents of the preceding sections that
quantization is, from a 5D standpoint, gauge-dependent. This should
not be too surprising, if we recall that standard 4D relations to do with
quantization cannot in general be invariant under a 5D change of

120 Five-Dimensional Physics

gauge x A — >x A (x a ,l) which involves the extra coordinate. How-
ever, it can then happen that a familiar quantization rule may take on
a strange guise when the gauge is altered. In this section, we will
briefly examine a situation of this kind, where the quantization of the
spin in a microscopic state takes on a new appearance in a macro-
scopic state.

The basis for discussing spin is that Planck's constant h de-
fines a unit for both the action and the spin angular momentum of a
particle. (The latter involves a factor 2k, of course, but we are not
here concerned with that.) It is reasonable to ask what happens to the
spin when we change from the Planck gauge to the Einstein gauge.
For both gauges, the null-path hypothesis with n = L/l yields ds = ±n
dl irrespective of the nature of/. Here dl corresponds to a change in n
by dn = - (L / I 2 ) dl, so ds = ± / dn for both gauges. For the Planck
gauge with / = 1 / m, this means that mds = ± dn. This is of course
the standard rule for the action, and defines the quantum for it as h.
This is also the quantum for the spin, insofar as bosons and fermions
have integral and half-integral multiples of it. But if we use the pre-
vious relation ds = ± / dn with / = m for the Einstein gauge, the impli-
cation is that it is now ds / m which takes on discrete values (rather
than mds). The associated quantum has the physical dimensions of
G / c 2 (rather than h). By analogy, we might expect a constant with
these or similar dimensions to figure also in the spin angular momenta
of gravitationally-dominated systems.

Quantum Consequences 121

We can express the preceding argument in another way by
restoring the physical constants. Then the Planck and Einstein gauges
have quantization rules of the form

jmcds = ±nh (4.33)

reds

J = ±np , (4.34)

where p is a constant with the physical dimensions of M" L T ,
which by analogy with the dual role of h might be expected to be
relevant to the spins of astrophysical systems.

In this regard, it is interesting to note that there is evidence of
a kind of (broken) symmetry for the spin angular momenta (J) and
masses (M) of astrophysical systems (Wesson 1981, 2005). It is ex-
pressed in the rule

J = pM 2 , (4.35)

where the constant p has the required physical dimensions. (This
symbol should not be confused with that for the 3 -momentum of a
particle used earlier.) The value of this constant may be determined
from the extensive but older set of data illustrated in Figure 4.1,
which agrees with the newer but more restricted set of data illustrated
in Figure 4.2 (see Wesson 2005; Steinmetz and Navarro 1999; Bul-
lock et al. 2001). The approximate value is/? - 8 x 10" 16 g- _1 cm 2 s' x .
The relation (4.35) has been compared to the Regge trajectories of
particle physics, but as a gravitational symmetry it is broken by forces

122 Five-Dimensional Physics

"1 — 1 1

„OCAL SUPERCLUSTER .
COMA CLUSTER . "

-

SPIRAL GALAXIES j?

.

STAR CLUSTERS

H

n

DOUBLE STARS

SOLAR SYSTEM

1, 1 (

PLANETS

I i i

15 20 25 30 35 40

Log 10 M(g)

FIG. 4. 1 - The angular momenta J of various astronomical systems
versus their masses M. This plot is adapted from one by Wesson
(1981). Data like those presented here have been much discussed in
the literature, and what should be included / decluded remains contro-
versial. (Asteroids are excluded here because they are not gravita-
tionally dominated, being supported mainly by solid-state forces. Lo-
cal supercluster dynamics are still under discussion.) The data support
J=pM 2 , where p is a constant whose value is/?^ 8xl0" 16 ^ _1 cm 2 s'K

Quantum Consequences 123

3r 3-

FIG. 4.2 - The specific angular momenta j (= J/M) for the disks of
spiral galaxies versus their rotation speed v. This plot is adapted from
one by Steinmetz and Navarro (1999; h is a dimensionless measure of
Hubble's parameter). The Tully-Fisher relation as revealed here
needs modest assumptions to put it into correspondence with basic
physics, as discussed by Bullock et al. (2001). But the data support
j= pM or J = pM 2 , where p is a constant fixed by the larger class of
data in Fig. 1.

1 24 Five-Dimensional Physics

of other kinds. There has been considerable discussion of its origin,
since while it is compatible with standard gravitational theory it ap-
pears to require some additional factor to account for why it holds
over such a large mass range. (It may or may not be a coincidence
that the dimensionless combination G / pc is the same order of magni-
tude as the fine structure constant, though this supports the conjecture
that/; is the analog of h.) What we have shown here is that an angular
momentum / mass relation of the form (4.35) might be expected on
the basis of 5D theory.

4.6 Particles and Waves: A Rapprochement

It appears contradictory that something can behave as a parti-
cle in one state and a wave in another. The archetypal example is the
double-slit experiment, where electrons as discrete particles pass
through a pair of apertures and show wave-like interference patterns.
Wave-particle duality is widely regarded as a conceptual conflict be-
tween quantum and classical mechanics. However, particles and
waves can both be given geometrical descriptions, which raises the
possibility that these behaviours are merely different representations
of the same underlying geometry (i.e. isometries). We have in Chap-
ter 2 seen that a coordinate transformation can change the appearance
and application of a solution, and we have in Chapter 3 noted a solu-
tion where a flat 5D space can represent a wave in ordinary 3D space.
In this section, we will consider flat manifolds of various dimension-
alities, with a view to showing that a 4D de Broglie wave which de-

Quantum Consequences 125

scribes energy and momentum is isometric to a flat 5D space.

2D manifolds, like the one which describes the surface of the
Earth, are locally flat. A brief but instructive account of their iso-
metries is given by Rindler (1977, p. 1 14; a manifold of any TV is ap-
proximately flat in a small enough region, and changes of coordinates
that qualify as isometries should strictly speaking preserve the signa-
ture). Consider, as an example, the line element ds 2 = dt 2 - t 2 dx 2 .
Then the coordinate transformation t — »■ e ,mt / ico, x — ► e lkx causes the
metric to read ds 2 = e 2mt dt 2 - e 2 '^ 1 "' dx 2 , where co is a frequency, k
is a wave number and the phase velocity co/k has been set to unity. It
is clear from this toy example that a metric which describes a freely-
moving particle (the proper distance is proportional to the time) is
equivalent to one which describes a freely-propagating wave. For the
particle, we can define its energy and momentum via E = m {dt I ds)
and p = m(dx/ ds). For the wave, E = me mt {dtlds) and

p = mr ' [dxl ds) . In both cases, the mass m of a test particle
has to be introduced ad hoc, a shortcoming which will be addressed
below. The standard energy condition (4.17), in the form m 2 =
E 2 - p 2 , is recovered if the signature is (+ — ) . If on the other hand
we have a Euclidean signature of the kind used in certain approaches
to quantum gravity, it is instructive in the 2D case to consider the
isometry ds 2 = x 2 dt 2 + t 2 dx 2 . The transformation t—*e ,a " / ico,
x — > e lkx / ik causes this to read ds 2 = - (l/k) 2 e 2, y at+kx >
(dt 2 + dx 2 ), after the absorption of a phase velocity as above. Thus a

126 Five-Dimensional Physics

particle metric becomes one with a conformal factor which resembles
a wave function.

3D manifolds add little to what has been discussed above. It
is well known that in this case the Ricci and Riemann-Christoffel ten-
sors can be written as functions of each other, so the field equations
bring us automatically to a flat manifold as before.

4D manifolds which are isotropic and homogeneous, but non-
static, lead us to consider the FRW models. These have line elements
given by

R 2 (t)

ds 2 =dt 2 K -^(dx 2 +dy 2 +dz 2 ) , (4.36)

(\ + kr 2 IA)

where R (t) is the scale factor and k = ± 1, defines the 3D curvature.
(This should not be confused with the wave number.) In the ideal
case where the density and pressure of matter are zero, a test particle
moves away from a local origin with a proper distance proportional to
the time. (I.e., R = t above where the spatial coordinates xyz and

r = ^jx 2 +y 2 +z 2 are comoving and dimensionless.) This specifies
the Milne model, which by the field equations requires k = -\. (We
can think of this as a situation where the kinetic energy is balanced by
the gravitational energy of a negatively-curved 3D space.) However,
(4.36) with R = t and k = - 1 is isometric to Minkowski space (Rin-
dler 1977, p. 205). Indeed, the Milne model is merely a convenient
non-static representation of flat 4D space. In the local limit where
\r 2 /4| <K 1, the ^-behaviour of the 3D sections of (4.36) allows us to

Quantum Consequences 127

specify a wave via the same kind of coordinate transformation used in
the 2D case. We eschew the details of this, since the same physics is
contained in more satisfactory form if the dimensionality is extended.
5D manifolds which are canonical have simple dynamics and
lend themselves to quantization, as we have seen. It is therefore natu-
ral that we should consider a 5D canonical analog of the 4D Milne
model discussed in the preceding paragraph (Liko and Wesson 2005).
We desire that it be Ricci-flat (R AB = 0) and Riemann-flat (Rabcd = 0).
The appropriate solution is given by

dS 1 =(- ] dt 2 - /sinhf - ] da 2 -dl 2 . (4.37)

Here the 3 -space is the same as that above, namely do 2 =
(dx^dy^dz 2 ) (1 + kr 2 1 4)" 2 with k = -1 . That the time-dependence of
the 3-space in (4.37) is different from that in (4.36) is attributable to
their different dimensionalities. However, the local situation for
(4.37) is close to that for (4.36). To see this, we note that for
laboratory situations t /L <rl in (4.37), so it reads

dS 2 J-]dt 2 -(-)da 2 -dl 2 . (4.38)

In this, let us multiply by L 2 and divide by ds 2 . Also, we take the
null-path hypothesis, which for any canonical metric results in the
constraint (dl /ds) = ±l / L. Then (4.38) gives

128 Five-Dimensional Physics

= / 2

-(")■

dx)\(±
ds) [ds.

-I 2

(4.39)

This with the identification / = m (see before) and the recollection that
proper distances are defined by \tdx etc., simply reproduces the

standard condition (4.17) in the form = E 2 -p 2 -m 2 .

To convert the 5D metric (4.38) to a wave, we follow the
lower-dimensional examples noted before. Specifically, we change
t —> e mt I ico, x — » exp[ik x x) etc., where <x> is a frequency and k x etc.
are wave numbers for the x, y, z directions. After setting the phase
velocity to unity, (4.38) then reads

dS 2 J-) e 2,M dt 2 -(-) {exp[2i(^ + ^x)]^ 2 +etc}-J/ 2 .

(4.40)
This with the null condition causes the analog of (4.39) to read

= j/ e " Br — 1 -llexp[i(cot + k x x)]—\ -etc-/ 2 . (4.41)
We can again make the identification I = m and define

E = le i0 " — , p = lexp[i(cot + k x x)'] — etc. (4.42)

Then (4.41) is equivalent to

Quantum Consequences 129

0^E 2 -p 2 -m 2 . (4.43)

This is of course the wave analog of the standard relation (4.17) for a
particle.

The fact that it is possible to convert a particle solution to a
wave solution, with consistency of their energy relations, demon-
strates in a formal sense that the concepts are compatible. It should
also be pointed out that while we have done this only for (4.37), that
solution is 5D fiat. Many solutions have this property, so it must be
possible to do the same thing for them, at least in principle. [In prac-
tice, it may be difficult to find the appropriate coordinate transforma-
tions. The wave of (3.22) found by Billyard and Wesson (1996) is
also 5D flat, but has a different signature to (4.37), and a coordinate
transformation between them is not known.] Furthermore, even 5D
solutions which are globally curved are locally flat, so the correspon-
dence outlined above has generality. We are led to conclude that as
regards their description by 5D relativity, particles and waves can be
regarded as isometries. That is, they are different 4D representations
of the same flat 5D space.

4.7 Conclusion

Quantum and classical physics parted ways in the 1930s.
Then, there were good experimental data on atomic systems, which
could be adequately explained by the simple but effective theory pro-
vided by Schrodinger's wave equation and Heisenberg's account of
quantum states. By contrast, cosmological observations of galaxies

1 30 Five-Dimensional Physics

were sketchy, and Einstein's theory of general relativity was too
complicated to be widely appreciated, let alone the extensions of it
due to Kaluza and Klein. Unfortunately, knowledge has its own kind
of inertia: the more one learns, the more one wishes to learn, usually
along the same path. Therefore, more work was done on quantum
mechanics than on general relativity. Interest in gravitation only ac-
celerated in the 1 960s, due largely to Wheeler, who pointed out that a
proper understanding of condensed astrophysical objects could only
be obtained via Einstein's theory. That progress was enlivened by
Hoyle, who sprinkled astronomy with pregnant ideas; and Hawking,
who made the idea of a black hole acceptable to a wide audience.
However, it is the case that modern physics remains split into two
camps: the quantum and the classical.

The contents of the current chapter have shown how this split
might be mended by the device of a fifth dimension. We have in ef-
fect used an extra coordinate related to the rest mass of a particle to
give a semi-classical account of several problems which are usually
regarded as the purview of quantum theory. (Here "semi-classical"
means a formalism which resembles Einstein's theory but contains
Planck's constant.) The issues we have examined involve the follow-
ing: The derivation of 4D uncertainty from the laws of a 5D determi-
nistic world (Section 4.2); the possible existence of a mass quantum
of small size, which is connected to the cosmological constant as a
measure of the small curvature of the universe (Section 4.3); the
demonstration that the Klein-Gordon and Dirac equations can be un-

Quantum Consequences 131

derstood as consequences of 5D dynamics (Section 4.4); the realiza-
tion that in 5D quantization is gauge-dependent, so a relation for a
microscopic system like a particle might reappear as a relation for a
macroscopic system like a galaxy (Section 4.5); and the possibility
that a particle and a wave may be the same thing described in differ-
ent coordinate frames, or isometries (Section 4.6). These issues do
not, of course, exhaust the list of problems we might wish to solve.
Rather, we have dealt with the issues which can be readily treated
with one of the two natural gauges of the theory.

These gauges are those named after Einstein and Planck,
where respectively the mass of a particle is measured by its
Schwarzschild radius or its Compton wavelength. These gauges exist
because of the historical development of physics, which in its bipolar
concentration on gravitation and quantum mechanics has shown us
the relevant constants involved (G and h). While these are only two
out of an infinite number of coordinate frames allowed by the covari-
ance of the theory, they are ideally suited to the physics in their re-
spective domains. The situation here is similar to that in other covari-
ant theories, like general relativity. In the latter, we recognize the
Schwarzschild solution as relevant to the solar system when it is
couched in its original coordinates, whereas other coordinates (like
those due to Eddington-Finkelstein or Kruskal) may have analytical
value but do not correspond directly to our observations. In this re-
gard, it is apparent that the results derived in the present chapter have
relevance mainly for induced-matter (or space-time-matter) theory,

132 Five-Dimensional Physics

rather than membrane theory with its singular hypersurface. That said,
in both versions of 5D relativity a central role is played by the cosmo-
logical "constant", or more correctly stated, by the energy density of
the vacuum.

Indeed, both classical cosmology and quantum field theory
now ascribe great importance to that part of the universe we cannot
"see". It is to this which we now turn our attention.

References

Billyard, A., Wesson, P.S. 1996, Gen. Rel. Grav. 28, 129.

Bullock, J.S., Dekel, A., Kolatt, T.S., Kravtsov, A.V., Klypin, A.A.,

Porciani, C, Primack, J.R. 2001, Astrophys. J. 555, 240.
Desloge, E.A. 1984, Am. J. Phys. 52, 312.
Kaluza, T. 1921, Sitz. Preuss. Akad. Wiss. 33, 966.
Klein, O. 1926, Z. Phys. 37, 895.
Liko, T., Wesson, P.S. 2005, J. Math. Phys. 46, 062504.
Lineweaver, C.H. 1998, Astrophys. J. 505, L69.
Overduin, J.M. 1999, Astrophys. J. 517, LI.
Padmanabhan, T. 2003, Phys.Rep. 380, 235.
Perlmutter, S. 2003, Phys. Today 56 (4), 53.
Ponce de Leon, J. 2001, Mod. Phys. Lett. A16, 2291.
Rindler, W. 1977, Essential Relativity (2nd ed., Springer, Berlin).
Steinmetz, M., Navarro, J.F. 1999, Astrophys. J. 513, 555.
Weinberg, S. 1980, Rev. Mod. Phys. 52, 515.
Wesson, P.S. 1981, Phys. Rev. D23, 1730.

Quantum Consequences 133

Wesson, P.S. 1999, Space-Time-Matter (World Scientific, Singapore).

Wesson, P.S. 2002, Class. Quant. Grav. 19, 2825.

Wesson, P.S. 2003, Gen. Rel. Grav. 35, 111.

Wesson, P.S. 2005, Astrophys. Sp. Sci., 299, 317.

Youm, D. 2000, Phys. Rev. D62, 084002.

Youm, D. 2001, Mod. Phys. Lett. A16, 2371.

5. THE COSMOLOGICAL "CONSTANT" AND VACUUM

"Sir, the invisible man is outside - but I said you couldn't see him"
(Hollywood gag)

5.1 Introduction

In all of the preceding chapters we have mentioned the cos-
mological "constant", which is a true constant in Einstein's 4D theory
of general relativity, but a possibly variable measure of the properties
of the vacuum in N > 5D theories. In the present chapter, we wish to
focus on A and the concept of vacuum in 5D.

The role of A as a scale for the universe was clearly perceived
by Eddington, who was a sagacious student of general relativity at a
time when Einstein's theory was not widely appreciated. Einstein's
semi-technical book The Meaning of Relativity was based on the Staf-
ford Little lectures at Princeton (New Jersey) in May 1921. Edding-
ton's semi-popular volume The Expanding Universe was based on his
International Astronomical Union lecture at Cambridge (Massachu-
setts) in September 1932 (though did not come out in accessible form
til considerably later). The fact that both books are under 200 pages
long, but have had profound impacts on physics, is a lesson that one
does not need a tome of technicalities to convey the gist of a subject.
It is also remarkable that these books and their authors held very dif-
ferent views about A. Einstein's distrust of this parameter is, of
course, well known; whereas Eddington was led to state that "To drop
the cosmical constant would knock the bottom out of space" (loc cit.,

The Cosmological "Constant" and Vacuum 135

p. 104). The latter worker, based largely on his studies in cosmology,
regarded the relationship between the Einstein tensor and the metric
tensor G a p = Ag a p as the basis of gravitation. He went on to use the
fundamental length scale defined by A with Planck's constant, to ar-
gue that an uncertainty of the kind associated with Heisenberg meant
that we can never know the precise momentum (and therefore loca-
tion) of any particle in the universe. In this and other ways, Edding-
ton presaged parts of what today we derive from the application of
quantum field theory to curved spacetimes, which includes the Hawk-
ing radiation around black holes. He also predicted to order of mag-
nitude the role played by the number of particles («10 80 ) in the visible
part of the universe, an idea we now attribute mainly to Dirac who
formalized it in the Large Numbers Hypothesis; and from this Ed-
dington vaguely anticipated the constraints on evolution which nowa-
days we identify with Carter, Dicke and Hoyle in the form of the An-
thropic Principle. All this from a belief in the cosmological constant
and some very clear, succinct thinking.

We are endeavoring, in the present treatise, to emulate the
Eddington approach. However, one aspect has come to the forefront
since his time which plays a central part in modern thinking on A.
Namely, that by the field equations, it measures the energy density
and pressure of the vacuum via pc 2 = - p = Ac 4 / %kG (in conven-
tional units). That A as an explicit measure of scale coupled to the
metric tensor, can be viewed instead as an implicit part of the energy-
momentum tensor in the field equations G a p = (87rG / c A )T a p, is due

136 Five-Dimensional Physics

largely to the algebraic properties of the latter object. This was real-
ized by Zeldovitch and others in the 1960s, and the usage is now
commonplace via the relation just quoted. As elsewhere, however, an
apparent conflict appears when the classical approach is extended to
the quantum one. Modern quantum field theory involves vacuum
fields, akin to the older zero-point fields, which can be measured by
an effective value of A. Unfortunately, the values of A as inferred
from astrophysics and particle physics differ. This is the crux of the
cosmological-"constant" problem, which we do not wish to delve into
here because good reviews are available (Weinberg 1980; Padmanab-
han 2003). The offset is model-dependent, but in round figures is of
the order of 10 120 . This is a number which would have given even the
numerically-minded Eddington pause for thought.

One of the most promising ways to account for the nature and
various estimates of A is that it is a parameter which we measure in
4D but originates in N > 5D (Rubakov and Shaposhnikov 1983;
Csaki, Ehrlich and Grojean 2001; Seahra and Wesson 2001; Wesson
and Liu 2001; Padmanabhan 2002; Mansouri 2002; Shiromizu, Ko-
yama and Torii 2003; Mashhoon and Wesson 2004). There are differ-
ent versions of this idea; but in its most general form it involves a re-
duction of field equations in JV > 5D to effective ones in 4D which
contain a vacuum field which can be variable. For 4D models of cos-
mological type, like the FRW ones, this may only make the cosmo-
logical "constant" a time-variable parameter. (The physical dimen-
sions of A in Einstein's equations are those of inverse-length squared,

The Cosmological "Constant" and Vacuum 137

which means that it defines a distance of order 10 28 cm; but via the
speed of light this implies that it might be expected to decay as
inverse-time squared, with a period of order 10 10 yr.) In more compli-
cated situations, it is possible in principle that A as the 4D measure of
a 5D scalar field could vary in both time and space. This would
resolve the cosmological-"constant" problem in a most satisfying
manner.

In Section 5.2, we will look at a simple but instructive model
where A is variable in a manner which is readily calculable (Mash-
hoon and Wesson 2004). This model is based on the generic property
that physics which is covariant in 5D is gauge-dependent (via the ex-
tra coordinate) in 4D. We will use the induced-matter approach, but
only as a mathematical framework, which implies applicability to
other 5D formalisms (Ponce de Leon 2001). The model will use
boundary conditions set by accepted wisdom concerning the big bang.
But any approach which introduces effects due to the fifth dimension
into standard 4D cosmology ought to predict more than it explains, so
in Section 5.3 we will list consequences of the extra dimension for
conventional astrophysics. These will be seen to provide no serious
obstacle. So in Section 5.4 we will take up a more fundamental issue
concerned with the gauge-dependence of A, namely the question of
vacuum instability. We hasten to add that our model for this is of a
modest kind, distinct from more radical ones which raise the possibil-
ity of a catastrophic destabilization of the vacuum due to high-energy
experiments with particle accelerators. In 5D theory, there is no sharp

138 Five-Dimensional Physics

division between what we call "vacuum" and what we call "matter",
so a change in the former can lead to the creation of the latter, in ac-
cordance with the appropriate laws (Birrel and Davies 1982; Alvarez
and Gavela 1983; Kolb, Lindley and Seckel 1984; Huang 1989; Linde
1991; Liko and Wesson 2005). The subject of vacuum instability is
grave but speculative, so to balance things we return in Section 5.5 to
the traditional subject of Mach's Principle, and use an exact solution
of the 5D equations to show how it can be realized in 4D (Wesson,
Seahra and Liu 2002). This presupposes, as do other considerations
in this chapter, that we are willing to relax somewhat our conven-
tional distinction between "matter" and "vacuum".

5.2 The 5D Cosmological "Constant"

In this section we absorb G, c and h, but the terminology is
otherwise as before. Then the general 5D metric

dS 2 =g a/} (x r ,l)dx a dx /3 + £® 2 (x r ,l)dl 2 , (5.1)

describes gravity and a scalar field for both a spacelike and timelike
extra dimension (s = +l) . As we saw in Chapter 1, for (5.1) the field
equations R AB = can be expressed as sets of 10,4 and 1 relations,
which we restate here for convenience:

G a fj = &7tT a p

The Cosmological "Constant" and Vacuum 139

° 8a P _g + e AA e e

4

#„=0

p-^ CT P — S^P^ P

(5.2)

(5.3)

nd> s g^O^

(5.4)

Here O a = dO/dx a , a semicolon denotes the ordinary 4D covariant

derivative and an overstar means d I dx A . The canonical metric is ob-
tained from (5.1) by factorizing it in terms of x 4 = / and a constant
length L:

dS 2 = l ^[g ap (x\l)dx a dx P Ydl
ises tb

(5.5)

This form causes the Einstein tensor to read

+/V

ygafi-jg^S

1 40 Five-Dimensional Physh

t-^T \ 6 + 21 g f ' V S MV + ~g MV g^ + jig^ g MV | \g aP -(5-6)

In (5.5) the Weak Equivalence Principle is recovered as a symmetry
via g a p = (Section 3.5). Then (5.6) gives G a p = 3>g a p / L 2 , which
are Einstein's equations with a cosmological constant A = 3 / L 2 . We
quoted this as (1.15). The analysis just given, and others of a similar
type in the literature, show how the cosmological constant of Einstein
theory is derived from Kaluza-Klein theory.

The preceding argument, however, begs for generalization. It
is clear from (5.5) that this will involve a choice for g a p = g a p (x y , I ).
Such a choice of gauge will not in general produce a constant A, so to
this extent we expect a gauge-dependent cosmological "constant".

Let us look at a special but physically instructive case of
(5.5). That metric is general, so to make progress we need to apply
some physical filter to it. Now, the physics of the early universe is
commonly regarded as related to inflation; and the standard 4D metric
for this is that of de Sitter, where ds 2 = dt 2 -exp\ 2?VA/3 Ida 2 .

(Here da 2 = dr 2 + r 2 d9 2 + r 2 sin 2 0d0 2 in spherical polar coordi-
nates.) The physics flows essentially from the cosmological constant
A. However, it is well known that the de Sitter metric is conformally
flat. This suggests that physically-relevant results in 4D may follow
from metric (5.5) in 5D if the latter is restricted to the 4D confor-
mally-flat form:

The Cosmological "Constant" and Vacuum 141

dS 2 =^[f(x'',l)r ?a/} dx a dx^]-dl 2 . (5.7)

Here 77^ = diagonal (+1 - 1 - 1 - 1) is the metric for flat Minkowski
space. We are particularly interested in the / - dependence of fix 1 , 1 ).
To determine the latter, we need to solve the field equations.

We could take these in the form (5.2) - (5.4), but since we
have suppressed the scalar field in (5.5) it is more convenient to cal-
culate the components of the 5D Ricci tensor directly. These are:

f)A a
R A4 =- — ~A a a -A a0 A aP (5.8)

44 dl I a aP

ST"

5 R MV = 4 R MV -S My , (5.10)

where Sftv is a symmetric tensor given by

4[

-^ + \- + A a a \A uv -2A u a A va +-U3 + L4"
dl \l ") MV M m \ L 2y

(5.11)
Here 4 Rf, v and T* p are, respectively, the 4D Ricci tensor and the con-
nection coefficients constructed from g a p. Moreover

d^L , (5 . 12)

2 dl

142 Five-Dimensional Physics

where AJ? =g fiS A aS . We need to solve (5.8) - (5.10) in the form
Rab = 0, subject to putting g^x 7 ,! ) =f(x\l )% v as in (5.7), which
ensures (4D) conformal flatness. We note that g" v = rj ^/f and

A MV = ft] M J 2 , where / = df(x r ,l)l 8! . Also A aP = frf* / (2/ 2 ) ,

A a a —2f/f and A a p = frj a p l(2f) . Then the scalar component of
the field equation (5.8) becomes

, ■ , +- — =0 . (5.13)

To solve this, we define U = fl f + 2/l . Then (6) is equivalent to

2U+U 2 =0 , or d (U~ l ) I 31 = 1/2, so on introducing an arbitrary
function of integration / = h{x y ) we obtain U~ l =[/- /o(x y )] / 2. This,

in terms of the original function / means that // f + 211 = U =

2/[/-/ (x r )], or a[ln(/ 2 /)]/5/ = a{ln[/-/ (x r )] 2 }/a/. This

gives I 2 fl [I -l (x y )] 2 = k (x y ), where k = k (x y ) is another arbitrary
function of integration. We have noted this working to illustrate that
the solution of the scalar component of the field equations (5.8) or
(5.13) involves an arbitrary length / (x y ) and an arbitrary dimen-
sionless function k (x y ). The solution for the conformal factor in the
metric g MV (x y , I ) =f(x y , I ) % v is

The Cosmo logical "Constant" and Vacuum 143

/(*',/):

Jo(x r )

I

k( X r)

(5.14)

and involves both arbitrary functions.

However, one of these is actually constrained by the vector
component of the field equations (5.9). To see this we note that^v of
(5.12) is symmetric, and it is a theorem that then

1

v-g dx v '

2 8x v

(5.15)

Here g is the determinant of the 4D metric, so since g MV = frj MV we
have yj-g = f 2 . Then using (5.15), equation (5.9) becomes

f/K

f df _ d ,

(5.16)

If dxf\! J ~ v ) f dx v dl K va/
The r.h.s. of this can be expressed using the identity T" a -lyf^g
d I yj—g I / dx v , whence (5.16) becomes

_L a

2f dx 1

7\ff\-^ = 2 ^{jp\ • ( 5 - 17 >

f 2 dx v a/L/ax v

In this form, we can multiply by 2/ and rearrange to obtain

1 44 Five-Dimensional Physics

/|4 = /£ ■ (5-18)

ox <9x

Dividing by / / ^ , we find

. , - ■ ( 5 -!9)

But the term in parentheses here, by (5.14), is // / = 2l (x r )/
\l\ l-l (x r )\\. Thus (5.19) implies that k (x y ) = / and is constant.

We have noted this working to illustrate that the scalar and vector
components of the field equations (5.8) and (5.9) together yield the
conformal factor

/(*V) = (l-A)V) , (5,

20)

which involves only one arbitrary function that is easy to identify: if
the constant parameter l vanishes, then krj^ is simply our original de
Sitter metric tensor.

The tensor component of the field equations (5.10) does not
further constrain the function k(x r ). However, we need to work
through this component in order to isolate the 4D Ricci tensor A R /iv
and so obtain the effective cosmological "constant". To do this, we
need to evaluate S^ of (5. 1 1). The working for this is straightforward
but tedious. The result is simple, however:

The Cosmological "Constant" and Vacuum 145
S »v=j2 k ( Xr )%r ■ ( 5 - 21 )

By the field equations (5. 10) in the form 5 R /iv = , this means that the
4D Ricci tensor is also equal to the r.h.s. of (5.21). We recall that our
(4D) conformally-flat spaces (5.7) have g MV =flx y ,r) % v = (1 - / / I) 2
k (x y ) % v using (5.20). Thus k (x y )% v = l 2 g flv / (/ - kf and

3 I 2
4 R=— r? . (5.22)

This is equivalent to the Einstein field equation for the de Sitter met-
ric tensor krj MV , since under a constant conformal scaling of a metric
tensor, the corresponding Ricci tensor remains invariant. Nonethe-
less, (5.22) defines an Einstein space 4 R ftv = Ag MV with an effective
cosmological constant given by

4^' •

This is our main result. It reduces for / = to the standard de Sitter
value A = 3 / L 2 . The latter, as we showed above, holds when there is
no /-dependence of the 4D part of the canonical metric (5.5). By con-
trast, when there is an /-dependence of the form given by (5.20) we
obtain (5.23). The difference between the A forms is mathematically
modest, but can be physically profound, because (5.23) says that for
/ —*■ /o we have A —»■ oo. In other words, the cosmological constant is

146 Five-Dimensional Physics

not only gauge-dependent but also divergent for a certain value of the
extra coordinate.

This is a striking result, and at first sight puzzling. However,
we should recall that if we have a theory which is covariant in 5D and
from it derive a quantity which is 4D in nature, then in general a
change in the 5D coordinate frame will alter the form of the 4D quan-
tity. In our case, the field equations R AB = are clearly covariant, and
we have changed the coordinate frame away from the canonical one
and found that we have altered the form of A. We can sum up the
situation as follows. The pure-canonical metric and the conformally-
flat metric have line elements given respectively by

dS 2 =—g aP {x r )dx a dx p -dl 2 (5.24)

(I -I \ 2
ds i = \ *>_i c (xr} riapdx * dx P_ dl 2 (525 )

Clearly the two are compatible, and the first implies the second if we
shift /—>(/- / ) and write g a ^x y ) = k(x 7 )rjap. The shift along the
/-axis does not alter the last part of the metric, so both forms describe
flat scalar fields. But the shift alters the prefactor on the first part of
the metric, with the consequence that A changes from 3 / L 1 to (3 / L 2 )
/ 2 (/-/ V 2 as in (5.23) above.

To investigate this in more detail, we will adopt the strategy
of Chapter 3. There we saw that there is often an extra force per unit
mass or acceleration which acts in 4D when a path is geodesic in 5D

The Cosmological "Constant" and Vacuum 147

(the pure canonical metric is an exception). Also, we saw that the 5D
path may be null. To proceed, we return to the general form of the
metric (5.5), for which the path splits naturally into a 4D part and an
extra part:

d 2 x M M dx" dx p __
ds 2 aP ds ds

dj__2fd£\
ds 2 l{ds)

1 dx" dx a "I dl dx p

2 ds ds Jds ds

--f-T

L 2 [ds)

dx a dx p dg ap
ds ds dl

(5.26)

(5.27)

In these, following (5.25) and the preceding discussion of metrics, we
substitute

x r ,l)--

Kp{x r )

(5.28)

where k a p is any admissible 4D vacuum metric of general relativity
with a cosmological constant 3 / L 2 . Furthermore we assume a null
5D path as noted above, and rewrite the line element as

dS 2 --

(5.29)

Since a massive particle in spacetime has ds # 0, we have that the
velocity in the extra dimension is given by (dl / ds) 2 = (/ / L) 2 . Then

148 Five-Dimensional Physics

the r.h.s. of (5.27) disappears, and to obtain the /-motion we need to
solve

4_2r4 + ^ =0 (5 . 30)

ds 2 l{ds) L 2 V

and (dl / dsf = (I / L) 2 simultaneously. Substituting the latter into
(5.30), we find that / is a superposition of simple hyperbolic functions.
There will be two arbitrary constants of integration involved in this
solution, which can be written as

>(i)

/ = ^cosh — +flsinh| T | . (5.31)

Moreover, (dl / ds) = (I / L) implies that A = B . To fix the con-
stants A and B here, it is necessary to make a choice of boundary con-
ditions. It seems most natural to us to locate the big bang at the zero
point of proper time and to choose / = / (s = 0). Then A = / and B =
± / in (5.31), which thus reads

l = l Q e ±s,L . (5.32)

The sign choice here is trivial from the mathematical perspective, and
merely reflects the fact that the motion is reversible. However, it is
not trivial from the physical perspective, because it changes the be-
haviour of the cosmological constant.

This is given by (5.23), which with (5.32) yields

The Cosmological "Constant" and Vacuum 149

A = 4" 5- • (5-33)

In the first case (upper sign), A decays from an unbounded value at
the big bang (s = 0) to its asymptotic value of 3 / L 2 (s — > oo). In the
second case (lower sign), A decays from an unbounded value (s = 0)
and approaches zero (s — * oo). We infer from astrophysical data that
the first case is the one that corresponds to our universe.

To investigate the physics further, let us now leave the last
component of the 5D geodesic (5.27) and consider its spacetime part
(5.26). We are especially interested in evaluating the anomalous
force per unit mass/" of that equation, using our metric tensor (5.28).
The latter gives d gap /dl = 2(1 - /„)(/„ / I 3 ) Kp (x 7 ) = 2/„[/ (/ - /o)]" 1 g a p
in terms of itself. We can substitute this into (5.26), and note that the
4-velocities are normalized as usual via g a /s (dx a I ds)(dx? / ds) = 1 .
The result is

/*=—-!*—**£. . (5.34)

1(1-1,) ds ds

This is a remarkable result. It describes an acceleration in spacetime
which depends on the 4-velocity of the particle and whose magnitude
(with the choice of boundary conditions noted above) is infinite at the
big bang. It is typical of the non-geodesic motion found in other ap-
plications of induced-matter and membrane theory. It follows from
(5.32) that

150 Five-Dimensional Physics

r^—T^TT, ■ (535)

L ds \e- s ' L -\)

In the first case (upper sign),/ 1 " — > (-1 / s) (dx M 7 ds) for s — > and
f — > for s — > oo. In the second case (lower sign), f M —>
(-1 / s) (dx 1 " / ds) for s -»• and/'' -► (-1 / i) (^ /<*) for s -* oo.
Thus both cases have a divergent, attractive nature near the big bang.
However, at late times the acceleration disappears in the first case, but
persists (though is small if L is large) in the second case. As in our
preceding discussion of A, we infer from astrophysical data that the
first case is the one that corresponds to our universe.

In (5.33) for A = A(s) and (5.35) for/" =f(s) we have rela-
tions which form an interface between theory and observation. Indeed,
we have already chosen the signs in our relations by appeal to the
broad aspects of data on cosmological timescales and the dynamics of
galaxies (Overduin 1999; Strauss and Willick 1995). However, there
are more detailed comparisons and predictions which can be made.

5.3 Astrophysical Consequences

The model derived in the preceding section has a time-
dependent cosmological "constant" A = (3 / L 2 ) (1 - e' s/L )' 2 given by
(5.33) and a velocity-dependent fifth force f= — iyc/L) {e s/L - l)" 1
given by (5.35). Here s is the 4D proper time, v is the 3D velocity
which we take to be radial (so v « c implies s «: c t), and L is a
length we take to be 1 x 10 28 cm approximately (see above). We rein-
troduce conventional units for the speed of light c and the gravita-

The Cosmological "Constant" and Vacuum 151

tional constant G, because we wish to make some comments about the
physics of the model with a view to comparison with observational
data (Wesson 2005). In this regard, it should be recalled that all of
the standard FRW models can be written in 4D conformally-flat form
(Hoyle and Narlikar 1974), so the following 6 comments are expected
to have some generality.

(a) The 5D model is like 4D inflationary ones, insofar as it is
dominated by A at early times. Indeed, by the specified relation, A
formally diverges near the big bang. To illustrate the potency of this,
we note that over the period 10 8 to 10 10 yr the value of A decreases by
a factor of approximately 4000. From the viewpoint of general rela-
tivity, A has associated with it a force (per unit mass) Arc 2 I 3 and an
energy density Ac 4 / 8tt G. It may be possible to test for the decay of
these using high-redshift sources such as QSOs.

(b) Galaxy formation is augmented in this model, because
there is a velocity-dependent extra force which tends to pull matter
back towards a local origin. In the usual 4D scheme, galaxy forma-
tion is presumed to occur when an over-dense region attracts more
material than its surroundings, so that the density perturbation grows
with time. Unfortunately, it does not do so fast enough in most mod-
els to account for the observed galaxies or other structures (Padma-
nabhan 1993). And while there are ways out of this dilemma (for ex-
ample by using seed perturbations due to quantum effects or pregalac-
tic stars), the fifth force naturally aids galaxy formation and deserves
in-depth study.

152 Five-Dimensional Physics

(c) Peculiar velocities are naturally damped by the fifth force
noted above. It is standard in modern cosmology to break the veloci-
ties of galaxies into two components, the (regular) Hubble flow and
(random) departures from it. In theoretical work, the former compo-
nent is often removed by a choice of coordinates, defining a comov-
ing frame where the regular velocities of the galaxies are zero and to
which their peculiar velocities can be referred. A practical definition
of the comoving frame is the one in which the 3K microwave back-
ground looks completely homogeneous. At the present epoch, the
peculiar velocities of field galaxies do not exceed a few 100 km/s; and
while there are other ways to account for this, the fifth force provides
a natural mechanism. This can be appreciated by noting that the ve-
locity associated with the force is v = v (e s/L - 1) "', where v is the
value when s = L ln(2). This is asymptotic to zero, and is like the
damped motion characteristic of a toy model based on 5D Minkowski
space (Wesson 1999 pp. 169 - 172). In both cases, the motion has a
form which is due essentially to our use of the 4D proper time s - ct
as parameter. We discussed the consequences of this in Chapter 3,
and in the present context we can express the motion in an alternative
way: the comoving frame which is assumed in most 4D work on cos-
mology is a natural equilibrium state of 5D gravity.

(d) The damping mechanism outlined above can lead to a
cosmological energy field which is significant. We can calculate an
approximate upper limit to this by using previous expressions in the
following manner. The magnitude of the force on an object of mass m

The Cosmological "Constant" and Vacuum 153

is (mvc / L)(e s/L - l)" 1 . The product of this with v gives the associated
rate of change of energy or power, which is (wcv 2 IV) (e s/L - l)" 3 .
The integral of this over proper time from s\ to s 2 gives the total en-
ergy change, and if we assume si <s: s 2 and s\ «; L this is approxi-
mately (mv 2 I 2) (L / si) 2 . This is the energy lost by one object
through the damping of its kinetic energy by the fifth force, and is
sharply peaked at early epochs. If the objects concerned form a uni-
form distribution, and presently have a mean distance d from each
other, the energy density of the field produced by the damping
iss-(mvQ I2d 3 UL/^) . We can write this in a more instructive

form if we introduce the mass density (p m ), the epoch when the damp-
ing was severe (t t ) and the epoch now (t ). Then a rough estimate of
the present energy density of the field that results from the damping is
£ — (/7 m v 2 I2\[t ltt) . This is a theoretical upper limit because of

the approximations made, but of course the parameters in it are them-
selves highly uncertain. For the purpose of illustration, let us substi-
tute p m = 2 x lO" 31 g cm" 3 , v = 100 km/s, J = 1 x 10 10 yr and t, =
1 x 10 8 yr. Then s = 1 x 10" 13 erg cm" 3 . This is a significant fraction
of the energy density of the cosmic microwave background, and com-
parable to the energy densities of other components of the intergalac-
tic photon field (Henry 1991). However, the physics which leads to
the present field is different from that involved in the production of
the CMB and the electromagnetic fields at other wavelengths such as
the extragalactic background light (Wesson 1991). Indeed, the energy

1 54 Five-Dimensional Physics

field currently being discussed may not be electromagnetic in nature
(this would require that the damping act on plasma protogalaxies or
young galaxies with a significant content of ionized material). It
could have a different nature, such as thermal energy or gravitational
waves. We should recall, though, that while the nature of the energy
field is open to discussion, its existence follows necessarily from the
model being discussed. In view of this, we suggest that it would be
wise to use observations of known fields and their isotropy to con-
strain the underlying theory.

(e) The extra force associated with 5D strengthens local grav-
ity and can therefore have dynamical effects on field galaxies and
galaxies in clusters. This is because in the local limit Newton's law is
modified so that the force (per unit mass) is

(5.36)

L(S>--l)

Here M =M (r) is the mass interior to radius r for a system with ap-
proximately spherical symmetry and we have put s = ct in the fifth-
force part. In view of the many implications of (5.36), we would like
to discuss them in a form which is generic. One way to do this is to
rewrite (5.36) as if the system was Newtonian and redefine the gravi-
tational "constant" to be

gm\l

(5.37)

The Cosmological "Constant" and Vacuum 155

In this, vr 2 / GM has the dimensions of a time and would convention-
ally define a dynamical timescale (t d ), while L / c is a timescale asso-
ciated with the cosmology which we expect to be approximately equal
to the present epoch (to). Both this and the remaining factor in (5.37)
are uncertain, so to be general we write the latter relation as

•tel

(5.38)

Here a is a dimensionless factor which depends on parameters to do
with both the cosmology and the system, but which at present is of
order of magnitude unity. In adopting this approach, it should be em-
phasized that we are not suggesting that the Newtonian "constant" is
really a variable parameter, as in 4D gravitational theories of the type
proposed by Brans / Dicke, Dirac, Hoyle / Narlikar, Canuto and oth-
ers, or in certain N ( > 5)D theories of the Kaluza / Klein type. Rather,
we are taking a pragmatic approach to see how a new effect fits into a
framework of existing data. In this respect, the parametization (5.38)
also has the advantage that we can use limits set on departures from
Newtonian gravity in other contexts (Will 1993). Since the relations
(5.36) - (5.38) will require detailed future study, we content ourselves
here with noting the results for two situations of interest. First, field
galaxies which interact via their quadrupole moments at early times
do so with an effective value of G which is 4 times the conventional
one, thus largely resolving the discrepancy found in standard applica-
tions of this mechanism for the generation of the spin angular mo-

156 Five-Dimensional Physics

menta of spirals (Hoyle 1949; Wesson 1982). Second, galaxies in
clusters which interact over periods comparable to the crossing time
do so with an effective value of G which is only modestly larger than
the conventional one, thus only fractionally resolving the virial dis-
crepancy found in many clusters and implying that most harbour large
amounts of dark matter / energy, as usually assumed.

(f) The observed solar system is believed to be dynamically in
agreement with 4D gravitational theory, with the possible exception
of one situation to which we will return below. Before proceeding to
this, it is instructive to recall that while Campbell's theorem guaran-
tees the embedding of any 4D Einstein solution in a Ricci-flat 5D so-
lution (Section 1.5), it does not guarantee that the 4D Birkhoff theo-
rem carries over to 5D. Since Birkhoff's theorem, which ensures the
uniqueness of the Schwarzschild solution (up to coordinate transfor-
mations) depends not only on the assumption of (3D) spherical sym-
metry but also on boundary conditions at infinity, it is perhaps not
surprising that it breaks down when the theory is extended to 5D.
This is why there are (at least) two solutions of the 5D theory, both of
which agree with the observed dynamics of the solar system. One of
these, called the 5D canonical Schwarzschild solution, has exactly the
same dynamics as the 4D solution (Wesson 1999 pp. 177 - 179). The
other, called the 5D soliton solution, has unmeasurably small depar-
tures from the 4D solution (Kalligas, Wesson and Everitt 1995). Both
of these solutions are spherically symmetric in the three dimensions
of ordinary space, and also static. By comparison, the cosmological

The Cosmological "Constant" and Vacuum 157

model we are considering is non-static, principally by virtue of the
time-dependent cosmological "constant" (5.33). This leads to the ex-
tra force (5.35), which affects the radial motion and depends critically
on the ordinary velocity v in that direction. Clearly, if we have any
prospect of seeing a cosmological 5D effect in the solar system, we
have to look towards a situation where a test body has a large radial
velocity. The planets, in their slow elliptical orbits, do not satisfy this
criterion, and are expected to show no significant departures from
standard motion. Other objects in the solar system, such as the parti-
cles of the solar wind or comets on parabolic orbits, do meet the crite-
rion but are not well studied. By contrast, the Pioneer spacecraft are
suitable (Anderson et al. 1998, 2002). These two craft, launched
more than thirty years ago, have approximately radial orbits: Pioneer
10 is on a path just 3° out of the ecliptic, while Pioneer 11 is moving
out of the ecliptic at about 17° inclination. At a distance of over 20
AU from the Sun, they are indicating an anomalous acceleration of
negative sign of about 10~ 7 cm s~ 2 . Many possible explanations of
this have been discussed, of which several are instrument-related (see
Bertolami and Paramos 2004 for a review). Among those which are
astrophysics-related, a plausible one involves an acceleration of the
Sun due to its own asymmetric activity, but this falls short of explain-
ing the anomaly by 4 orders of magnitude (Bini, Cherubini and
Mashhoon 2004). By coincidence, the force (5.35) fails to account
for the motion of these spacecraft by approximately the same factor.
(The escape velocity from the solar system at 20 AU is close to 10 km

158 Five-Dimensional Physics

s _1 , and with this value for v and L — 1 x 1 28 cm, the result is as noted
to within order of magnitude.) Nevertheless, we see here the oppor-
tunity for a future test of the 5D extra force. We need a high- velocity,
radially-moving spacecraft. Better still would be two such craft, from
which other influences could be cancelled as in the GRACE project
(Tapley et al. 2004), leaving the cosmological effect we wish to verify.
The preceding comments (a) - (f) show that the model of
Section 5.2, with a decaying cosmological "constant" (5.33) and an
extra force (5.35), has wide-ranging consequences for astrophysics
and cosmology. Indeed, the options are almost intimidating in their
number, like the alternatives to the big bang we studied in Chapter 2.
However, unlike some other forays into extra dimensions, the basic
extension to 5D has the redeeming feature of testability.

5.4 Vacuum Instability

This possibility is raised by the considerations of the two pre-
ceding sections, where we saw that a translation along the extra axis
of a 5D manifold can cause a gauge change in the effective value of
the cosmological "constant" in 4D spacetime. This raises the ques-
tion of changes in the energy density of the vacuum, which in general
relativity is Ac 4 / 8^G. We will give an alternative derivation of the
expression for a variable A, and then outline the implications of this.
The subject is speculative, so our discussion will be brief.

To see that A can depend on the extra coordinate of 5D the-
ory (x 4 = I ), let us reconsider the canonical form of the line element.

The Cosmological "Constant" and Vacuum 159

This has dS 2 = (/ / Lfds 2 - dl 2 , where L is a length and ds 2 =
g a p (x Y ) dx a dx^ specifies the 4D proper time. The coordinate trans-
formation or gauge change /—>■(/- / ) leaves the extra part of the 5D
metric unchanged, while the prefactor on the 4D part changes from
I 2 /L 2 to (/ - kf I L 2 = {I 2 1 L 2 ) [(/ - /„) / / f . This means in effect

that the original metric tensor g a/} changes to g a/S = [(/ - 1 ) / /] g afi .

Now it is a theorem that solutions of the source-free 5D field equa-
tions R AB = with the canonical metric satisfy the source-free 4D
field equations 4 R a p = Ag a p with A = 3 / L 2 (see above and Mashhoon,
Liu and Wesson 1994). It is also true that a constant conformal trans-
formation of the metric leaves the Ricci tensor invariant, which is in
effect the situation here since the change from g a p to g ap only de-
pends on / and not x 7 . Then the 4D field equations still hold with
R ap (g afi ) = A g ap and A = (3 / L 2 ) I 2 1 (/ - l Q ) 2 . This is the same as

(5.23) above.

We see that a translation along the / -axis preserves the form
of the canonical metric, and since the 5D field equations are covariant
we obtain again the 4D field equations, but with a different cosmo-
logical constant, namely

Ai-h)

A = Jt|tVI • (5-39)

We examined the astrophysical consequences of this in Section 5.3,
where we identified the divergence at / = / with the big bang. We

1 60 Five-Dimensional Physics

now proceed to take another look at (5.39), by adding a series of
mathematical and physical conditions to it.

Firstly, let us take derivatives of (5.39) to obtain
dh = - (6 / L 2 ) (I - l y 3 llodl. We are mainly interested in the region
near / = / , where the energy density A = A (/ ) is changing rapidly
but smoothly. Putting dl = I - / for the change in the extra coordi-
nate, we obtain

dAdl 2 =-6l 2 /L 2 . (5.40)

This is an alternative form of the instability inherent in (5.39) near to
its divergence.

Secondly, let us assume that the instability has a dynamical
origin, and that the / -path involved is part of a null 5D geodesic as
before. Then with dS 2 = we have / = l e ±s/L as in (5.32). This with
the upper sign implies dl / I = ds / L, which in (5.40) yields

dAds 2 =-6 . (5.41)

This is remarkable, in that it contains no reference to x 4 = / and is
homogeneous in its physical dimensions (units), with no reference to
fundamental constants. That is, it is gauge and scale invariant. [An
alternative derivation of (5.41) may be made by using (5.33) and not-
ing that s is measured from where A diverges at the big bang.] Again
(5.41) implies instability, since dh — »■ oo for ds —*■ 0. This behaviour
can be put into better physical perspective by recalling that the action
for a particle of rest mass m in 4D dynamics is usually defined as

The Cosmological "Constant" and Vacuum 161

/ = J mcds. So (5.41) can be interpreted as a change in the energy
density of the vacuum for a particle of unit mass which changes its
action.

Thirdly, let us assume that the action is quantized. In some
higher-dimensional theories, the rest mass of a particle can change as
it pursues its 4D path, so m = m (s). But irrespective of this, we have
that dl = mcds = h, introducing Planck's constant. Then (5.41) gives

<M = -6(^J . (5.42)

This says that a change in the energy density of the vacuum (Ac 4 /
%nG ) is related to the square of the mass of a particle (m). It is a
rather strange relation, in that the l.h.s. is classical in nature and the
r.h.s. is quantum in nature. However, relations of a similar type have
appeared in the literature (Matute 1997; Liu and Wesson 1998; Seahra
and Wesson 2001; Mansouri 2002). We are reminded of the old
Dirac theory, in which an underlying sea of energy develops a hole
which is interpreted as a positron. On this basis, (5.42) can be inter-
preted to mean that a perturbation in the energy density of a global
sea of energy has a size related to the Compton wavelength of an as-
sociated particle whose mass is m. Of course, in modern particle
physics the masses are usually thought of as arising from a mecha-
nism involving the Higgs field. We do not venture into this or related
issues, because (5.42) is derived from classical as opposed to quantum
theory.

1 62 Five-Dimensional Physics

In fact, the relations (5.40), (5.41) and (5.42) are phenome-
nological, in that they are semi-classical and lack a deeper foundation
in quantum mechanics. They are akin to the relations of thermody-
namics, which are compatible with - but lack detailed knowledge of -
atomic physics. Nevertheless, (5.41) in particular is remarkably sim-
ple and deserves study.

5.5 Mach's Principle Anew

Most works on gravitation mention this subject, if only as a
motivation for Einstein's general relativity. The latter is an excellent
theory of gravity, but lacks a proper account of the source of that
field, namely mass. It is possible to approach this subject from many
different directions, which it would be inappropriate to review here.
But to most workers, Mach's Principle means that the local properties
of a massive particle are dependent on the nature and distribution of
other matter in the universe. This has a nice, philosophical ring to it.
However, it is indisputably the fact that the environment in which a
particle finds itself is the vacuum. Therefore, it is reasonable to ex-
pect that any future theory of (say) the masses of the elementary par-
ticles will involve the physics of the vacuum. In the preceding sec-
tion, we came across some unusual relations which appear to link the
properties of the global vacuum to the mass of a local particle. In the
present section, we wish to give a brief account of a similar subject
from a different direction. Specifically, we wish to go back to a
known wave-like solution in 5D (Liu and Wesson 1998), and take its

The Cosmological "Constant " and Vacuum 1 63

4D part (Wesson, Liu and Seahra 2002). This will be seen to have
Machian properties.

We use the same approach as before, where a particle with
energy E, ordinary momentum/? and rest mass m may alternatively be
regarded as a wave spread through spacetime, with the corresponding
de Broglie and Compton wavelengths (see Sections 3.2 and 4.6). We
retain conventional units as in Section 5.4, and as there we are inter-
ested in connecting the particle labels (E, p, m) to the properties of
spacetime. As the basic descriptors for the latter, we take the compo-
nents of the (4D) Ricci tensor R a p . From this we can obtain the Ein-
stein tensor G a p , and from this we can construct if we wish an energy-
momentum tensor T? in accordance with Einstein's field equations.

The procedure as outlined so far runs parallel to that used for
the induced-matter approach to fluids (see Chapter 1); except that
now we are attempting to do it purely in 4D, and for a particle. Be-
cause of these constraints, we run into two technical issues: the metric
turns out to be complex, and the energy-momentum tensor has an un-
usual form. Some workers may consider these issues to be problems.
However, the subject we are addressing is essentially a technical one,
namely: how to go from a particle, to a wave, to a spacetime. The
fact that there is a wave in the middle of the analysis should alert us to
the possibility of a complex metric; and the fact that we wish to ob-
tain a particle (rather than fluid) description should alert us to the pos-
sibility that quantum (rather than classical) parameters may appear in

1 64 Five-Dimensional Physics

the effective energy-momentum tensor. Due to these unorthodox
properties, we keep the analysis short.

Consider, therefore, the simple case of a "wavicle" moving
along the z-axis. The 4D line element is given by

ds 2 = c 2 dt 2 -exp[2i(Et-pz)/h~]dx 2

-exp[-2i(Et-pz)/h]dy 2 -dz 2 . (5.43)

Despite the complex nature of the metric coefficients here, it tran-
spires that the corresponding components of the Ricci tensor are real,
as for certain other wave-like solutions (Section 4.6). The non-
vanishing components for (5.43) are:

_2E 2 _ 2Ep 3 _ 2p 2

he he h

The trace of these is R = 2\E 2 -p 2 c 2 )/ h 2 c 2 . Here the energy and
momentum are constants, so we can introduce the mass in accordance
with the standard relation E 2 -p 2 c 2 = m 2 c 4 of (3.5). Then we obtain
R = 2(mc/h) . We see that the Ricci scalar is related to the Comp-
ton wavelength of the particle (h /mc ), while the components of the
Ricci tensor are related to its de Broglie wavelengths (he / E, h / p ).
If we use the components (5.44) to construct G^ and so obtain T£ ,
the latter depends on particle parameters and Planck's constant, rather

The Cosmological "Constant" and Vacuum 165

than fluid parameters and Newton's constant as usual (see Wesson,
Liu and Seahra 2002). In a formal sense, (5.43) and (5.44) show how
a local particle can be viewed as a global wave.

The solution (5.43) is special, in that it describes a wave
which moves only along the z-axis and has E, p constant so that m is
also constant. However, it is not difficult to see how to generalize the
approach while keeping its main features. Thus a general correspon-
dence between the geometry and the particle properties would be
given by a relation of the form R a p = - 6s 2 p a Pp / h 2 , where s is a di-
mensionless coupling constant (Wesson 1999 p. 197). We have ex-
amined the dynamics which follow from such a relation. The usual
law is modified to read p a p"-p - m(dm / dx? ), which gives back the
4D geodesic if the mass is constant. The 4-momenta are conserved
via p p - p = 0, even if m = m (x a ). The last equation, if it could be
evaluated, would effectively realize Mach's principle by telling us
how to calculate the mass of a particle as a function of the coordi-
nates. What we have seen here is that this principle is indeed com-
patible with general relativity, provided we are willing to stretch our
understanding of metric and matter.

5.6 Conclusion

The cosmological "constant", when viewed from 5D, takes on
a drastically different nature from the true constant of 4D. The ca-
nonical coordinates introduced by Mashhoon treat x 4 = / in a way
analogous to x° = ct in the Robertson- Walker metric. When the 4D

166 Five-Dimensional Physics

subspace of the 5D manifold does not depend on /, we recover the
Weak Equivalence Principle as a geometric symmetry and the stan-
dard result A = 3 / L 2 . Here the cosmological constant is a true con-
stant, which measures the curvature of the embedded
4-space in the manner envisioned by Eddington (Section 5.2). How-
ever, if we carry out the simplest of coordinate transformations
wherein /—>(/- / ), in the context of the simplest cosmological mod-
els which are like those of de Sitter, then we find that A changes by a
factor I 2 (I - l )~ 2 as in (5.23). This corresponds for a null 5D path to
a decaying A, as in (5.33). The fifth force, which is generic to 5D
relativity, then takes the form (5.35). There is a velocity-dependent
acceleration which affects all objects in the universe, but which de-
creases in tandem with A as cosmic time increases. This has numer-
ous astrophysical consequences, of which a half-dozen are accessible
to observation (Section 5.3). The effect on A which follows from a
translation in / has more general implications (Section 5.4). Most no-
tably, there is a relation between the change in the cosmological "con-
stant" and the elapsed proper time (5.41) which is scale-invariant.
We can interpret this to mean that "empty" space with a finite curva-
ture can produce a particle with finite mass in accordance with (5.42).
This should not be too surprising: empty space with an unseen elec-
tromagnetic field can produce particles via pair creation, and there has
to be a classical analog of the way in which particles acquire mass via
the quantum Higgs mechanism. What is perhaps more surprising is
that the vacuum / particle relationship inferred from 5D is compatible

The Cosmological "Constant " and Vacuum 167

with the formal interpretation of wave solutions like (5.43), which can
be derived from the straight 4D theory (Section 5.5). It may be possi-
ble to realize Mach's Principle within the context of Einstein's gen-
eral theory of relativity, though this requires some mental flexibility.

The implications of what we have learned in this chapter are
widespread. In general, a change in coordinates which involves the
extra coordinate x 4 = / will not leave the metric in canonical form, but
will instead introduce a significant scalar field via g 44 = ± O 2 where
<D = <J>(x y ,/). Then the 5D metric has the form (5.1), with an effective
4D energy-momentum tensor given by (5.2). The latter shows that
the source of the gravitational field is a mixture of what have hitherto
been called matter and vacuum. The conventional split into ordinary
material and a vacuum field (measured by a constant A) is just the
result of suppressing the scalar field. If we wish to measure the en-
ergy density of the latter with the traditional symbol, then we should
write A (x r , I ). More cogently, we realize that in 5D the historical
division between "matter" and "vacuum" is obsolete.

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6. EMBEDDINGS IN TV > 5 DIMENSIONS

"Embed? Don't you mean in bed?" (Madonna)

6.1 Introduction

Theories like general relativity can be approached via differ-
ential equations or differential geometry. There is, of course, an over-
lap. But the distinction is traditional, and has to do with whether we
wish to use exact solutions of the field equations to study physical
problems, or the equations and their associated metrics to study the
mathematical properties of manifolds (see the books by Kramer et al.
1980 and Wald 1984). In the present volume we have been mainly
concerned with the former approach. However, we now turn to the
latter, because the subject of embeddings is of considerable impor-
tance if we are to properly understand how 4D general relativity fits
into N>5T> field theory. We gave a primer on Campbell's embed-
ding theorem in Section 1.5, which is sufficient to underpin much of
the physics of extended gravity. We now wish to give a deeper ac-
count of this and related topics, partly to elucidate the connection be-
tween the induced-matter and membrane versions of 5D gravity, and
partly to see how these can be extended to even higher dimensions.
For those readers who are more interested in physics than mathemat-
ics, we note that we will return to practical issues in Chapter 7.

The plan of this chapter is as follows. In Section 6.2 we will
quote some relevant results, eschewing proofs which would slow the
discussion. Extensive bibliographies on embeddings are available

170

i N > 5 Dimensions 171

(Pavsic and Tapia 2000; Seahra and Wesson 2003). Several of the
results we will mention were obtained a long time ago, but there is
still some disagreement about the strength of embedding theorems
and their application to physics. In Section 6.3 we will analyse the
general embedding problem, establishing the algebra anew so as not
to bias ourselves by previous considerations. These results will be
used in Section 6.4 to re-prove the Campbell-Magaard theorem, (our
conclusions will agree with the summary in Section 1.5). Then in
Sections 6.5 and 6.6 we will apply our general work to the induced-
matter and membrane theories, before closing in Section 6.7 with
some comments on the implications of embeddings.

6.2 Embeddings and Physics

The abstract theory of embeddings dates back almost to the
original work of Riemann in 1868. The application of embeddings to
physics was implicit in the 5D extensions of general relativity by
Kaluza in 1921 and Klein in 1926, and was explicit in Campbell's
work which was published also in 1926 (see Chapter 1). The subse-
quent development of the subject was sporadic, but it is useful to note
some of the more significant results.

Thus Tangherlini and others proved in the 1960s that the 4D
Schwarzschild solution can only be embedded in a flat manifold of
dimensionality N>6. But it was not til the 1990s that it was realized
that Birkoff s theorem in its conventional form breaks down for N> 4.
This followed from the discovery of exact solutions of the empty 5D

1 72 Five-Dimensional Physics

field equations which are 3D spherically-symmetric and static, but
have non-Schwarzschild properties. These "solitons" were later ex-
tended to the non-static case. The more general problem of how to
embed any solution of Einstein's theory in a higher-dimensional flat
space showed that it requires N > 10. This result builds on work by
Schlafli, Janet, Cartan and Burstin. The last showed the importance
of the Gauss-Codazzi equations as integrability conditions for the
embeddings, a subject we will return to below. Campbell, as we have
noted, asked about the embedding of 4D spaces of the type used for
general relativity in 5D spaces which are Ricci-flat. This is important,
because setting the Ricci tensor to zero provides the most basic set of
field equations relevant to physics. The theorem that Campbell con-
jectured was essentially proved much later by Magaard. It is their
work which underlies the currently popular approach wherein Ein-
stein's field equations in 4D are viewed as a subset of the Ricci-flat
equations in 5D. While it is not particularly strong, the Campbell-
Magaard theorem is widely regarded as a kind of algebraic protection
for higher-dimensional theories of relativity which reduce to that of
Einstein in 4D. It is, however, only a local theorem.

The more difficult problem of global embeddings was con-
sidered by Nash, and extended to metrics of indefinite signature by
Clarke and Greene. Theories of physics in which spacetime is glob-
ally embedded in a flat, higher-dimensional manifold are yet to re-
ceive serious attention. But there is no reason why such should not be
developed, especially since the theorems have been considered for the

Embeddings in N > 5 Dimensions 1 73

cases where the 4D space is compact and non-compact, a difference
which some researchers have noted is open to observational test using
high-redshift sources like QSOs.

The possibility that higher dimensions might not be compacti-
fied was only taken seriously by the physics community in the 1990s.
Induced-matter theory dates from 1992, and essentially uses the non-
compact extra dimension to give a geometrical account of the origin
of matter, as we have discussed at length previously (see Wesson
1992a,b; Wesson and Ponce de Leon 1992). Membrane theory dates
from slightly later, and assumes that the non-compact extra dimension
is split by a singular hypersurface with Z 2 symmetry, so accounting
for why particle interactions in the brane are stronger than gravity
which propagates outside it, as we have summarized elsewhere (see
Arkani-Hamed, Dimopoulos and Dvali 1998, 1999; Antionadis et al.
1998; Randall and Sundrum 1998, 1999a,b). While these non-
compact theories are differently motivated, they are known to possess
essentially the same field equations and equations of motion (Ponce
de Leon 2001, 2004). Below, we will compare the induced-matter
and braneworld models from the perspective of embeddings in 5D.

For dimensionalities higher than 5, some results are known,
but only for special values of N. For example, Horava and Witten
(1996) showed that the compactification paradigm was not a prereq-
uisite of string theory, which latter is attractive in that it automatically
avoids the singularities associated with point particles. They discov-
ered an 1 ID theory, which has the topological structure ofR ^^S 1 1 Z 2 .

1 74 Five-Dimensional Physics

This is related to the 10D E & x E% heterotic string, via dualities. In
this theory, the endpoints of open strings reside on a (3 + 1) brane, so
standard-model interactions are confined to it, while gravity propa-
gates outside. This is technically a braneworld model, thought it pre-
dates the more popular theory with that name outlined above. As an-
other example of N> 5D physics, we can mention N= 26 (Bars and
Kounnas 1997; Bars, Deliduman and Minic 1999). Models of this
type can describe the interaction of a particle and a string, and employ
a two-time metric (see Section 3.4 for the 5D two-time metric). The
double nature of the timelike dimension can admit supersymmetry
and various dualities, and shows that the concept of dimensionality
can be taken as far as we wish.

However, even if we let the dimensionality run, the physics
we obtain is to a certain degree restrained. This is because we have to
take notice of the embedding, to which we now turn our closer
attention.

6.3 The Algebra of Embeddings

In this section, we will study the properties of AD Riemann
spaces anew, without preconceptions concerning the signature or
whether there is a singular surface (as for membrane theory) or not (as
for induced-matter theory). For our purposes, the main geometrical
object of such a space is the Ricci tensor, which has (N I 2)(N +1)
independent components. We will be largely concerned with spaces
whose dimensions differ by 1, and if there is a danger of confusion we

Embeddings in N > 5 Dimensions 175

will use a hat to denote the higher space, as for example R AB versus
R a p. Since our coordinates are numbered from zero and run to n, the
total dimensionality of the space is N = 1 + n. Then uppercase Latin
indices run 0...«, while lowercase Greek indices run 0...(n - 1).
Square brackets on indices will indicate antisymmetrization. The co-
variant derivative in the higher space will be denoted by V^, while
that in the lower space will be denoted by a semicolon as usual. The
Lie derivative will be indicated by £ with an appropriate subscript to
identify the dimensionality. We will mean by M any general mani-
fold, in which however we will often be interested in a hypersurface
E/ defined by the "extra" coordinate /. In the case where S/ has some
special properties we will denote it by S , which then in the 4D case is
shorthand for spacetime. We are aware that this notation, while stan-
dard, is cumbersome. As some small relief, we will adopt geometri-
cal units, so that the speed of light and Newton's gravitational con-
stant are unity.

On our (n + l)-dimensional manifold (M, g AB ) we place a co-
ordinate system x={x A }. In our working, we will allow for two pos-
sibilities: either there is one timelike and n spacelike directions tan-
gent to M, or there are two timelike and {n - 1) spacelike directions
tangent to M. The scalar function

/ = /(*) (6.1)

defines our foliation of the higher-dimensional manifold. If there is
only one timelike direction tangent to M, we assume that the vector

176 Five-Dimensional Physics

field n A normal to £/ is spacelike. If there are two timelike directions,
we take the unit normal to be timelike. In either case, the space tan-
gent to a given £/ hypersurface contains one timelike and in - 1)
spacelike directions. That is, each £/ hypersurface corresponds to an
^-dimensional Lorentzian spacetime. The normal vector to the E/ slic-
ing is given by

n A =£®d A l , n A n A =s . (6.2)

Here s = ± 1 . The scalar <I> which normalizes n A is known as the
lapse function. We define the projection tensor as

h AB=g A B- £n A n B ■ (6-3)

This tensor is symmetric (Hab = h B A) and orthogonal to ha-

We place an ^-dimensional coordinate system on each of the
£/ hypersurfaces y= {y a }. The n basis vectors

e t = ' n A e i = ^ (6-4)

are by definition tangent to the Z/ hypersurface and orthogonal to n A .
It is easy to see that e A behaves as a vector under coordinate trans-
formations on M[^:x— »x(x)] and a one-form under coordinate
transformations on X, \y/ : y -> J> ()>)] . We can use these basis vec-
tors to project higher-dimensional objects into E/ hypersurfaces. For
example, for an arbitrary one-form on Mwe have

Embeddings inN>5 Dimensions 177

T a = e A J A . (6.5)

Here T a is said to be the projection of T A onto £/. Clearly T a behaves
as a scalar under <f> and a one-form under y/. The induced metric on
the S/ hypersurfaces is given by

K P = e t4SAB = e t4 k AB ■ ( 6 - 6 )

Just like g A B, the induced metric has an inverse:

h ay h rp = S a p . (6.7)

The induced metric and its inverse can be used to raise and lower the
indices of tensors tangent to 2/, and change the position of the space-
time index of the e A a basis vectors. This implies

e a A ej=8° . (6.8)

Also note that since h A B is entirely orthogonal to n A , we can write

At this juncture, it is convenient to introduce our definition of the ex-
trinsic curvature K a p of the £/ hypersurfaces:

K a s = <4 V a"b =-2<4l n h AB . (6.10)

Note that the extrinsic curvature is symmetric (K a p=Kfi a ). It may be
thought of as the derivative of the induced metric in the normal direc-
tion. This n-tensor will appear often in what follows.

178 Five-Dimensional Physics

Finally, we note that {y, I } defines an alternative coordinate
system to x on M. The appropriate diffeomorphism is

dx A =e A a dy a +l A dl . (6.11)

'-(*)

(6.12)

is the vector which is tangent to lines of constant y a . We can always
decompose 5D vectors into the sum of a part tangent to £/ and a part
normal to S/. For l A we write

l A =N a e A +$>n A . (6.13)

This is consistent with l A d A l = 1, which is required by the definition
of l A , and the definition of nf 1 . The ^-vector N " is the shift vector,
which describes how the y a coordinate system changes as we move
from a given 2/ hypersurface to another. Using our formulae for dx 4
and l A , we can write the 5D line element as
dS 2 = g AB dx A dx B

= h a/3 (dy a +N a dl)(dy^+N fi dl) + £0 2 dl 2 . (6.14)

This reduces to ds 2 = h a p dy a dy p if dl = 0, a case of considerable
physical interest.

Let us now focus on how ^-dimensional field equations on
each of the 2/ hypersurfaces can be derived, given that the

Embeddings in N > 5 Dimensions 1 79

(n + l)-dimensional field equations are

R AB =Ag AB , l = 2±- . (6.15)

1-/7

Here A is the "bulk" cosmological constant, which may be set to zero
if desired. In what follows we will extend the 4-dimensional usage
and call manifolds satisfying equation (6.15) Einstein spaces.

Our starting point is the Gauss-Codazzi equations. On each
of the £/ hypersurfaces these read

Rabcd&K^ = Kp r s + 2eK a[S K r]fi

^MABcn M e A a e B p e c y =2K a{p . n . (6.16)

These need to be combined with the expression for the higher-
dimensional Ricci tensor:

R AB = (h^ejfe? + sn M n N ) R AMBN . (6. 1 7)

The components of this tensor satisfy equations which may be
grouped strategically by considering the following contractions of
(6.15):

R AB e A a4= Xh aB
R AB e A a n B =
R AB n A n B =sX . (6.18)

180 Five-Dimensional Physics

The first member of these gives (n I 2) (n + 1) equations, the second
gives n and the last is a scalar relation. The total number of equations
is (1/2) («+ 1)(« + 2).

Putting (6.17) into (6.18) and making use of (6.16) yields the
following formulae:

R a e=M ap -z[E aP +KJ'{K l3fl -Kh Pf )\
= (K a/s -h aP K)

sA = E MV h MV . (6.19)

In writing down these results, we have made the following defini-
tions:

K = h aP K afj (6.20)

E aP ^RmNsn M e A a n N e B p , E ap =E pa . (6.21)

The Einstein tensor G ap = R ap - g a/3 R / 2 on a given S; hypersurface
is given by

G ap = -e (E ap + K^P* 1 - \h ap K» v P MV ) + A [l - \{n + s )] h aP ,

(6.22)
where we have defined the (conserved) tensor

P a p=K afi -h ap K

P aP =0 . (6.23)

Embeddings inN>5 Dimensions 1 8 1

This tensor is essentially the one which appears in other studies. For
example, it has the same algebraic properties as the momentum con-
jugate to the induced metric in the ADM formulation of general rela-
tivity (Wald 1984; though note that here the direction orthogonal to S/
is not necessarily timelike, so P a p is not formally a canonical momen-
tum in the Hamiltonian sense). Alternatively, it is the tensor which
appears in the vector sector of the induced-matter theory (Wesson and
Ponce de Leon 1992), and which contains the non-linear terms for the
matter in membrane theory (Ponce de Leon 2001). To complete our
analysis we recall that the condition that the 4D Einstein tensor have
zero divergence imposes a condition on E a p of (6.21). Making use of
the second member of (6. 1 9), this is

s(K MV K MV '- a -K^K^J) . (6.24)

This may be regarded as a condition, satisfied by the geometric quan-
tities on the £/ hypersurfaces described by (6.19), when we ask that
the physical quantities associated with matter be conserved.

In fact, the preceding analysis shows that (6.19) are in essence
the geometric formulation of the field equations for ND relativity.

We will make use of the field equations below, where we will
use them to infer general properties of physics, on the assumption that
it is the £/ hypersurface which we experience. For now, we remark
that it is possible to solve the first member of (6.19) for E a p and sub-
stitute the result into the last member of those relations. The result is

1 82 Five-Dimensional Physics

(n-l)A = R + £(K MV K MV -K 2 ) . (6.25)

This scalar relation, together with the n relations of the second mem-
ber of (6.19), provide (l+«) generalizations of the well-known con-
straints on the Hamiltonian approach to field theories like general
relativity. In the (1 + 3) theory, they are the constraints attendant on
the initial-value problem and numerical work on the Einstein equa-
tions. In the (1+4) theory, they are similar to the relations found for
the Randall-Sundrum braneworld scenario (Shiromizu, Maeda and
Sasaki 2000). However, we will in what follows not be so much con-
cerned with applying our algebra to constraints, but more with count-
ing the number of independent relations concerned so as to establish
results on embeddings.

6.4 The Campbell-Magaard Theorem

With the algebra we have established, it is possible to re-
prove Campbell's theorem, independent of previous considerations
(Campbell 1926; Magaard 1963; Lidsey et al. 1997; Wesson 1999).
Our approach will be heuristic, serving to point towards the physical
applications we will take up in the section following.

In (6.19) we have a set of AD field equations which are the
geometrical-language analogs of the physically-motivated ones
(6.15). The equations (6.19) are defined on each of the £/ hypersur-
faces, which are labeled by a coordinate (/ ) that we expect on physi-

Embeddings inN>5 Dimensions 1 83

cal grounds to prove special compared to the others (y). The essential
geometrical objects, which can be thought of as spin-2 fields, are

M*0. *ap(y>i)> E a P {y> 1 ) ■ ( 6 - 26 )

Each of these tensors is symmetric, so there are 3(w / 2) (w + 1) inde-
pendent dynamical quantities governed by (6.19). For book-keeping
purposes, we can regard these components as those of a dynamical
super- vector ¥" = x ¥ a (y, I ). Now the field equations (6.19) contain
no derivatives of the tensors (6.26) with respect to /. This means that
the components v F a (y, / ) must satisfy (6.19) for each and every value
of /. In alternative language, the field equations on 2/ are "conserved"
as we move from hypersurface to hypersurface. That is, the field
equations (6.19) for (n + 1) D are in the Hamiltonian sense constraint
equations. While this is important from the formal viewpoint, it
means that the original, physical equations (6. 1 5) tell us nothing about
how ¥" varies with /. If so desired, equations governing the
/-evolution of *¥" could be derived in a number of equivalent ways.
These include isolating /-derivatives in the expansion of the Bianchi
identities V A G AB = 0; direct construction of the Lie derivatives of h AB
and Kab = ^ c Vc«b with respect to l A ; and formally re-expressing the
gravitational Lagrangian as a Hamiltonian (with / playing the role of
time) to obtain the equations of motion. Because the derivation of
diY" is tedious and not really germane to our discussion, we will omit

1 84 Five-Dimensional Physics

it from our considerations and turn to the more important problem of
embedding.

Essentially, our goal is to find a solution of the higher-
dimensional field equations (6.15) or (6.19) such that one hypersur-
face S in the 2/ foliation has "desirable" geometrical properties. For
example, we may want to completely specify the induced metric on,
and hence the intrinsic geometry of, S . Without loss of generality,
we can assume that the hypersurface of interest is at / = 0. Then to
successfully embed L in M, we need to do two things:

(a) Solve the constraint equations (6.19) on So for *F"(y, 0)
such that H has the desired properties (this involves physics).

(b) Obtain the solution for ^(y, I ) in the bulk (i.e. for / * 0)
using the evolution equations df¥ a (this is mainly mathematics).

Both of these things have to be achieved to prove the Camp-
bell-Magaard theorem, along with some related issues. Thus, we
have to show that (a) is possible for arbitrary choices of h a p on E , and
we have to show that the bulk solution for x F a obtained in (b) pre-
serves the equations of constraint on E; * E . The latter requirement is
necessary because if the constraints are violated, the higher-
dimensional field equations will not hold away from E . This issue
has been considered by several authors, who have derived evolution
equations for *F a and demonstrated that the constraints are conserved
in quite general (n + l)-dimensional manifolds (Anderson and Lidsey
2001; Dahia and Romero 2001a, b). Rather than dwell on this well-
understood point, we will concentrate on the ^-dimensional field

Embeddings inN>5 Dimensions 1 85

equations for S ; and assume that, given Y" (y,0), the rest of the

(n + 1)D geometry can be generated using evolution equations, with
the resulting higher-dimensional metric satisfying the appropriate
field equations.

With the preceding issues understood, and the weight of the
algebra of the previous section established, the Campbell-Magaard
theorem becomes close to obvious. The argument simply involves
counting.

From above, we recall that our super-vector W has a number
of independent dynamical components given by rid = (3n / 2) (n + 1)-
This can be compared to the number of constraint equations on £ ,
which as we have seen is n c = (1/2) (n + 1) (n + 2). For n > 2, it is
obvious that nj is greater than n c , which means that our system of
equations is under-determined. The number of free parameters is
rif = n d - n c = (n 2 - 1). Therefore, we can freely specify the func-
tional dependence of (n 2 - 1) components of *F a (y, 0). In other
words, since rif is greater than the number of independent components
of h a p for n > 2, we can choose the line element on So to correspond
to any w-dimensional Lorentzian manifold and still satisfy the con-
straint equations. This completes the proof of the theorem: Any
n-dimensional manifold can be locally embedded in an (n + 1)-
dimensional Einstein space.

This theorem has been used in the literature in several forms,
and there has been some discussion of the physical latitude afforded
by the just-noted {n 2 - 1) components of algebraic freedom (see

1 86 Five-Dimensional Physics

Section 1.5). This number can arguably be cut down, after the con-
straints are imposed and the induced metric is selected, by a further
(n I 2) (n + 1), leaving (n + 1) {n I 2 - 1) components. However, this
still leaves a significant degree of arbitrariness in the embedding
problem. It implies, for example, that the same solution on E can
correspond to different structures for M. Due to this and related con-
cerns, the Campbell-Magaard theorem has sometimes been criticized
as weak, a view which seems to us to be unjustified. If it were not for
the considerable algebraic apparatus we set up in Section 6.3, it would
not be clear that such a theorem exists. Indeed, it is due to that appa-
ratus that a simple counting of degrees of freedom suffices to estab-
lish the theorem. As counter-comments to the charge that the theorem
is weak, we can point out that it is possible to imagine theories of
gravity in which it does not hold; and that it has significant implica-
tions for so-called lower-dimensional theories of general relativity,
where workers ignorant of the theorem have blithely invented field
equations which are mathematically tractable but do not respect the
constraints handed down by differential geometry. Our view is prag-
matic: Einstein's theory is a highly successful theory of gravity based
on Riemannian geometry, and since we know that the real world has
at least 4 dimensions, we should use the Campbell-Magaard theorem
as a "ladder" to higher dimensions.

Embeddings inN>5 Dimensions 1 87

6.5 Induced-Matter Theory

In this section, and the following one on membrane theory,
we will assume that our (3+1) spacetime is a hypersurface in a 5D
manifold. We will therefore be able to use the relations of Sections
6.3 and 6.4 with N= in + 1) = 5. We will also employ some results
from the literature on dynamics (Ponce de Leon 2001, 2004; Seahra
2002; Seahra and Wesson 2003). Our aim in this and the following
section is to understand the nature of 4D matter in a 5D space which
is either smooth or divided.

Induced-matter theory is frequently called space-time-matter
(STM) theory, because the extra terms in the 5D Ricci tensor act like
the matter terms which balance the 4D Einstein tensor in general rela-
tivity (Wesson 1999). The field equations in our current approach are
(6.19) with X = 0. Despite the fact that the Einstein tensor in 5D is
zero, it is finite in 4D and is given by (6.22):

G aP = -e{E aP +K\P* P -±h afl K MV P MV ) . (6.27)

Matter enters into STM theory when we consider an observer who is
capable of performing experiments that measure the 4-metric h a p or
Einstein tensor G a p in some neighbourhood of their position, yet is
ignorant of the dimension transverse to the spacetime. For general
situations, such an observer will discover that the universe is curved,
and that the local Einstein tensor is given by (6.27). Now, if this per-
son believes in the Einstein equations G a p = %nT a p, he will be forced
to conclude that the spacetime around him is filled with some type of

188 Five-Dimensional Physics

matter field. This is somewhat of a departure from the usual point of
view, wherein the stress-energy of matter acts as the source for the
curvature of the universe. In the STM picture, the shape of the 2/ hy-
persurfaces plus the 5 -dimensional Ricci-flat geometry fixes the mat-
ter distribution. It is for this reason that STM theory is sometimes
called induced-matter theory: the matter content of the universe is in-
duced from higher-dimensional geometry.

When applied to STM theory, the Campbell-Magaard theo-
rem says that it is possible to specify the form of h a p on one of the
embedded spacetimes, denoted by Do. In other words, we can take
any known (3+l)-dimensional solution h a p of the Einstein equations
for matter with stress-energy tensor T a p and embed it on a hypersur-
face in the STM scenario. The stress-energy tensor of the induced
matter on that hypersurface E will necessarily match that of the
(3+l)-dimensional solutions. However, there is no guarantee that the
induced matter on any of the other spacetimes will have the same
properties.

We now wish to expand the discussion to include the issue of
observer trajectories in STM theory. To do this, we will need the co-
variant decomposition of the equation of motion into parts tangential
and orthogonal to spacetime. These are given by Seahra and Wesson
(2003, p.1338). We are mainly interested in the motion perpendicular
to S/, when the acceleration is:

'i = ^(K ap u a u p +d)-i[2u l3 (\n^\ i3 +in A V A ^\ . (6.28)

nN>5 Dimensions 189

Here / is the extra coordinate, ¥ is the lapse introduced above, and
$ = n A f A is the magnitude of any non-gravitational forces (per unit
mass)/ 4 which operate in the higher space. The other symbols are as
defined in Section 6.3. (An overdot denotes differentiation with re-
spect to proper time or some other affine parameter.) Equation (6.28)
is the covariant version of other relations for 5D dynamics, for exam-
ple (3.18). It clearly shows that the acceleration perpendicular to £/
(given by / ) is coupled to the motion in it (given by the 4-velocity
u a ). We proceed to consider 3 cases, specified by the extrinsic curva-
ture K^ and the non-gravitational force $ :

(a) K a p * and $ = . A sub-class of this case corresponds
to freely-falling observers. We cannot have / = constant as a solution
of the /-equation of motion (6.28) in this case, so observers cannot
live on a single hypersurface. Therefore, if we construct a Ricci-flat
5D manifold in which a particular solution of general relativity is em-
bedded on Do, and we put an observer on that hypersurface, then he
will inevitability move in the / direction. Therefore, the properties of
the induced matter that the observer measures may match the predic-
tions of general relativity for a brief period of time, but this will not
be true in the long run. Therefore, STM theory predicts observable
departures from general relativity.

(b) K a p = and $ = . Again, this case includes freely-
falling observers. Here, we can solve equation (6.28) with dl / dl =
(where X is an affine parameter). That is, if a particular hypersurface

1 90 Five-Dimensional Physics

So has vanishing extrinsic curvature, then we can have observers with
trajectories contained entirely within that spacetime, provided $ = .
Such hypersurfaces are called geodesically complete because every
geodesic on S is also a geodesic of M. If we put K a p = into (6.27),
we get the Einstein tensor on E/ as G a p = 87tT a p = - sE a p. This implies
by (6.24) that the trace of the induced energy -momentum tensor must
be zero. Assuming that T a p can be expressed as that of a perfect fluid,
this implies a radiation-like equation of state. Hence, it is impossible
to embed an arbitrary spacetime in a 5D vacuum such that it is
geodesically complete. This is not surprising, since we have already
seen that we cannot use the Campbell-Magaard theorem to choose
both h^ and K a p on S - we have the freedom to specify one or the
other, but not both. If we do demand that test observers are gravita-
tionally confined to So, we place strong restrictions on the geometry
and are obliged to accept radiation-like matter.

(c) K a p * and $ = -K a/3 u a u p . In this case, we can solve
(6.28) with dl / dX = and hence have observers confined to the E
spacetime. However, $ = -K a/3 u a u p is merely the higher-
dimensional generalization of the centripetal acceleration familiar
from elementary mechanics. Since we do not demand K a p = in this
case, we can apply the Campbell-Magaard theorem and have any type
of induced matter on So. However, the price to be paid for this is the
inclusion of a non-gravitational "centripetal" confining force, whose
origin is obscure.

Embeddings in N > 5 Dimensions 191

In summary, we have shown that the Campbell-Magaard
theorem guarantees that we can embed any solution of general relativ-
ity on the spacetime hypersurface S of the 5D manifold used by STM
or induced-matter theory. However, for pure gravity, particles only
remain on S if K a p = 0, which means that the induced matter has
T" = 0, which implies a radiation-like equation of state. They could
be constrained to £ if there were non-gravitational forces acting, but
in general particles which are not photons will wander away from any
given slice of spacetime. This confirms what we found in Chapter 3,
where (3.18) has no solution for / = constant, in the absence of a fifth
force which would violate the Weak Equivalence Principle. On this
basis, the departure of particles from 2 discussed here is equivalent
to the change over cosmological times in their masses discussed
before.

6.6 Membrane Theory

This exists in several forms, all treatable with our preceding
algebra. The simplest form imagines one, thin brane. This divides a
bulk 5D manifold into two parts separated by a singular hypersurface,
which we call spacetime. Gravity propagates outside the brane,
but other physics is concentrated on it, largely by virtue of the as-
sumption that there is a significant, negative cosmological constant
(i.e., the bulk is AdSs). This model is simple, but perhaps limited as
regards what can be expected for physics, since the latter is automati-
cally reproduced in almost standard form on the membrane. By con-

1 92 Five-Dimensional Physics

trast, the addition of a second, thin brane leads to effects on the branes
which are dependent on the intervening AdS 5 space. These effects
have to do with the characteristic energies of gravitational versus
other interactions, and can lead to a better understanding of the
masses of elementary particles. Other results may be obtained by
considering more branes, ones which collide, and thick branes. We
will concentrate on the original form of the theory.

The hypersurface E is located at / = 0, about which there is
symmetry (Z 2 ). Since the membrane is thin, the normal derivative of
the metric (the extrinsic curvature) is discontinuous across it. This is
like the thin-shell problem in general relativity, and we can take over
the same apparatus, including the standard Israel junction conditions.
These imply that the induced metric on the £/ hypersurfaces must be
continuous, so that the jump there is zero:

[V] = . (6.29)

This uses the common notation that X± = lirm / — > 0* jX and [X] =
X + -X ~. In addition, the Einstein tensor of the bulk is given by

T { S=S{l)S ap e a y B . (6.30)

Here the 4-tensor S a p is defined via

Embeddings inN>5 Dimensions 193

[K ap y-Kls{S ap -\Sh ap ) , (6.31)

where k\ is a 5D coupling constant and S = h ^S^. The interpreta-
tion is that S a p is the stress-energy tensor of the standard fields on the
brane. To proceed further, we need to invoke the Z 2 symmetry. This
essentially states that the geometry on one side of the brane is the mir-
ror image of the geometry on the other side. In practical terms, it
implies

K P =-K P , [K a ,] = 2K: fi . (6.32)

Then we obtain

S ap = -3sK- 2 K + ap . (6.33)

This implies that the stress-energy tensor of conventional matter on
the brane is entirely determined by the extrinsic curvature of So
evaluated in the / — > limit.

This is an interesting result, because it shows that even if
spacetime is a singular hypersurface, what we call ordinary matter
depends on how that hypersurface is embedded in a larger world.
However, the result could have been inferred from the STM approach
(where there is no membrane and matter is a result of the extrinsic
metric) plus the junction conditions (which imply that even if there is
a membrane the metric is continuous). This correspondence has been
commented on in the literature, and has the happy implication that the
extensive inventory of exact solutions for induced-matter theory
(Wesson 1999) can be taken over to membrane theory. Indeed, we

1 94 Five-Dimensional Physics

encountered an example of this in Section 2.4, where we noted that a
5D bouncing cosmology could be interpreted as a (4+l)D membrane
model. A detailed account of this kind of problem is given elsewhere
(Seahra and Wesson 2003 pp.1334 - 1337), along with comments on
multiple and thick branes. For a single, thin brane it should be noted
that the derivation of models with Z% symmetry is helped by the fact
that the constraint equations (6.19) are invariant under K a p — »• - K a/S .
This can be used to construct an algorithm for the generation of
braneworld models.

The field equations on the brane are given by (6.22) with K a p
evaluated on either side of £ . Usually, equation (6.33) is used to
eliminate K* p , which yields the following expression for the Einstein
4-tensor on £ :

'3S MV S„ -
SS„„ -3SS\ + ^~

-sE a/} -^X{2 + s)h ap . (6.34)

Since this expression is based on the equations of constraint (6.19), it
is entirely equivalent to the STM expression (6.27) when 1 = 0.
However, it is obvious that the two results are written in terms of dif-
ferent quantities. To further complicate things, the braneworld field
equations are often written with the stress-energy as the sum of a part
proportional to the cosmological constant for the hypersurface and
another part,

Embeddings inN>5 Dimensions 1 95

S aP =r ap -Ih ap , (6.35)

which however is non-unique. On the other hand, the STM field
equations are often written in a non-covariant form, where partial
derivatives of the induced metric with respect to / appear explicitly
instead of K a p and E a p. We believe that this disconnect in language is
responsible for the fact that some workers have yet to realize the sub-
stantial overlap between induced-matter theory and membrane theory.
As we did in the previous section, let us now turn our atten-
tion to observer trajectories. We will use the same equation for the
motion in the direction perpendicular to the hypersurface, namely
(6.28), which is common to all 5D embeddings. However, to simplify
things we will set the lapse via O = 1 (this is a 5D gauge choice, so
our 4D results will be independent of it). Then in the / direction, the
acceleration reads

/ =e{K ap u a u p +$) , (6.36)

where as before $ denotes the magnitude of non-gravitational forces.
By using (6.31), the last relation gives

K- ap u a u p =+\sK]\S ap u a u p -\[k-s1 2 )s\ . (6.37)

We can view this as the zeroth-order term in a Taylor-series expan-
sion of Kap u a u p in powers of /. In this spirit, the acceleration can be
rewritten as

1 96 Five-Dimensional Physics

1 =-^s & i(l)K 2 5 [S ap u a u fi -^(tc-s! 2 )s'\ + e!S + 0(l) , (6.38)

where sgn(/) =

(6.39)

Here we are using u A u A = k (we assume that u A is timelike). From
(6.38), it is obvious that freely-falling observers (# = 0) can be con-
fined to a small region around the brane if

Sapifu 1 ' -\(tc-ei 2 )S>Q . (6.40)

Of course, if the quantity on the left is zero or the coefficient of the
0{l ) term in (6.38) is comparatively large, we need to look at the sign
of the latter term to decide if the particle is really confined. To get at
the physical content of (6.40), let us make the low-velocity approxi-
mation l 2 <K 1 . With this assumption, (6.40) can be rewritten as

J*{

TS'-iTr\T^]gJuV>0 . (6.41)

This is an integrated version of the 5D strong energy condition as ap-
plied to the brane's stress-energy tensor, which includes a vacuum
energy contribution from the brane's tension. Its appearance in this
context is not particularly surprising, since the Raychaudhuri equation
asserts that matter which obeys the strong energy condition will gravi-

Embeddings inN>5 Dimensions 197

tationally attract test particles. What we have shown is that slowly-
moving test observers can be gravitationally bound to a small region
around E if the total matter-energy distribution on the brane obeys
the 5D strong energy condition.

Finally, we would like to show that the equation of motion
(6.38) has a sensible Newtonian limit. Let us demand that all compo-
nents of the particle's velocity in the spacelike directions be negligi-
ble. Let us also neglect the brane 's tension and assume that the den-
sity p of the confined matter is much larger than any of its principle
pressures. Under these circumstances we have

S ap u a u p ^p, h af> S ap ^Kp . (6.42)

The 5D coupling constant /r 5 2 is taken to be

A=\V& , (6.43)

where F 3 is the dimensionless volume of the unit 3 -sphere and G 5 is
the 5D "Newton" constant. With these approximations, we get the
acceleration for freely-falling observers:

l*-±sgn(l)V 3 G 5 p + 0(l) . (6.44)

This is precisely the result we would obtain from a Newtonian calcu-
lation of the gravitational field close to a 3D surface layer in a 4D
space, if we used the Gauss law in the form

198 Five-Dimensional Physics

-j dr g-dA = V 3 G 5 \ y pdV . (6.45)

Here the integration 4-volume is a small "pill-box" traversing the
brane. Thus we learn that the full general-relativistic equation of mo-
tion in the vicinity of the brane (6.38) reduces to the 4D generaliza-
tion of a known result from 3D Newtonian gravity in the appropriate
limit.

In summary, we have seen that the Campbell-Magaard theo-
rem says that we can embed any solution of 4D general relativity in a
5D space with a membrane. However, the matter associated with the
brane is not then freely specifiable. This should not be considered a
serious concern, though. Particles near the brane will be confined to a
small region near it if the 5D strong energy condition is satisfied, and
their motions are Newtonian in the appropriate limit.

6.7 Conclusion

To Newton, it would probably have appeared strange to sug-
gest that ordinary 3D space should be embedded in a 4D spacetime
continuum. But Einstein showed that a fourth dimension actually
simplifies physics while also extending its scope. The current
situation is intriguing: we do not know if all of the consequences of a
fifth dimension will prove to be desirable, even though 5D relativity
exists in two apparently acceptable versions. Induced matter (or
space-time-matter) theory explains 4D matter as a geometrical conse-
quence of the fifth dimension, like when we view a movie projected

Embeddings inN>5 Dimensions 1 99

onto a 2D screen from an unperceived depth. Membrane theory
views 4D spacetime and its contents as a special surface in the fifth
dimension, like when we walk across the 2D surface of the Earth
without knowledge of the underlying geology.

In this chapter, we have looked at the possible connections
between embeddings and physics (Section 6.2); developed the algebra
which necessarily attaches to embeddings if we are to relate higher
dimensions to what we already know (Section 6.3); and argued that
the Campbell-Magaard theorem, despite its weaknesses, represents
the ladder which allows us to move up or down between dimensions
(Section 6.4). The induced-matter and membrane theories are frater-
nal twins, in the sense that they share the same algebra but have dif-
ferent physical motivations (Sections 6.5, 6.6). Both depend on the
fifth dimension, and ascribe real meaning to it.

If the fifth dimension becomes a standard part of physics,
there is no reason why we should not employ embedding theory to
proceed to even higher levels.

References

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Antoniadis, J., Arkani-Hamed, N., Dimopoulos, S., Dvali, G.R. 1998,

Phys. Lett. B 436, 257.
Arkani-Hamed, N., Dimopoulos, S., Dvali, G.R. 1998, Phys. Lett. B

429, 263.

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59, 086004.
Bars, I., Kounnas, C. 1997, Phys. Rev. D 56, 3664.
Bars, I., Deliduman, C, Minic, D. 1999, Phys. Rev. D 59, 125004.
Campbell, J. E. 1926, A Course of Differential Geometry (Clarendon,

Oxford).
Dahia, F., Romero, C. 2001a, gr-qc / 0109076.
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1980, Exact Solutions of Einstein's Field Equations (Cambridge

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Magaard, L. 1963, Ph.D. Thesis (Kiel).
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Wesson, P.S. 1999, Space-Time-Matter (World Scientific, Singapore).

7. PERSPECTIVES IN PHYSICS

"Given for one instance an intelligence which could comprehend all
the forces by which nature is animated and the respective positions of
the beings which compose it, if moreover this intelligence were vast
enough to submit these data to analysis, it would embrace in the same
formula both the movements of the largest bodies in the universe and
those of the lightest atom; to it nothing would be uncertain, and the
future as the past would be present to its eyes" (Pierre Laplace)

This quotation, while lacking the pithiness of a contemporary
sound-bite, still condenses much of what preoccupies physicists. And
whether we agree with Laplace (1812) or not, the issues he raises
have resonance with what we have discussed in preceding chapters.
Physics, while archival by nature, is not a done deal like the Dead Sea
scrolls. It is an evolving subject, and certain questions inevitably re-
cur to occupy the serious student. These include the connection be-
tween the macroscopic and microscopic domains, the puzzle of de-
terminacy, and the subjective issue of how much we know (or do not
know) at any given stage. It is reasonable to ask in broad terms about
the future of physics, given that we have already given a fairly fine-
grained account of its present status.

We have looked, by turn, at several issues. In higher than 4
dimensions, field equations should be constructed around the Ricci
tensor (Chapter 1). For 5D, this means that the 4D big bang is not
merely a singularity but has physics attached to it, even when the

Perspectives in Physics 203

higher-dimensional space is flat (Chapter 2). The paths of particles
are in general affected by a fifth force in 5D relativity, which however
vanishes for the canonical metric, when we recover the Weak Equiva-
lence Principle as a geometric symmetry (Chapter 3). The determi-
nistic laws of 5D when regarded inexactly in 4D result in a Heisen-
berg-type relation, and imply a mass quantum, with an associated
(broken) symmetry between the spin angular momenta and squared
masses of gravitationally-dominated systems which has some empiri-
cal support from astrophysics (Chapter 4). There is also astrophysical
support for a decaying cosmological "constant" of the type allowed
by 5D theory, where if we view this parameter as a measure of the 4D
energy density of the vacuum we infer that the latter is unstable,
though in a manner consistent with Mach's Principle (Chapter 5).
The connection between 4D general relativity with matter and 5D
theory in apparent vacuum is Campbell's theorem, which is a kind of
algebraic ladder that spans manifolds of different dimensionality, and
thereby yields the physics of induced-matter and membrane theory
(Chapter 6). These two versions of 5D gravity are essentially the
same, though motivated respectively by cosmology and particle phys-
ics. Both are in agreement with observations. The fact that the two
approaches have been viewed differently is largely to do with issues
of philosophy and language rather than mathematics.

It was Wittgenstein's opinion that much of modern philoso-
phy is unproductive, because it is concerned with language rather than
the meaning which lies behind words. Russell and others pointed out

204 Five-Dimensional Physics

that mathematics is a kind of language, and indeed the natural one for
science (see Chapter 1). It has certain advantages. For example, it is
sanitary, enabling us to put forth hypotheses about the world without
emotion. It is also remarkably exact; and in this regard physicists are
fortunate in that they have a common set of algebraic rules - as if
science were a grandiose game of chess, played according to inviolate
laws on which everyone agrees. (A theoretical physicist who does a
faulty calculation is like a concert pianist who plays a bum note: it is
discordant, easily detected and usually followed by censure.) But
unlike the rigid and therefore somewhat sterile game of chess, physics
as based on mathematics has a growing edge, where material can be
added to widen its scope.

Imagination, however, is at least as important for advancing
physics as a sound knowledge of mathematical technique. Einstein is,
of course, frequently quoted in this regard. Unfortunately, modern
physics frequently gets bogged down in technicalities: there are too
many studies aimed at mere algebra or the addition of a decimal point.
Perhaps paradoxically, there are also too many attempts at recreating
the universe from time-zero, ignoring established theory and newly-
acquired data. It is the middle ground which is the most fertile for
advancing physics. This middle ground is also the one which leads
the way to new pastures. Current interest in the fifth dimension is a
good example of a track which starts in established farmland (Ein-
stein's general relativity) and is already in fresh and fruitful territory.

Perspectives in Physics 205

It was Minkowski who showed how to weld time and space
together; but as we have remarked, it was Einstein who realized how
inextricably they are intertwined through the Principle of Covariance.
The algebra of general relativity can be carried out in any system of
coordinates, and the standard solutions owe their applicability to the
fact that we choose to present them in terms of time and space coor-
dinates that are close to our primitive notions of those quantities.
This applies to the Schwarzschild solution for the gravitational field
outside the Sun, where the radial coordinate is chosen to agree with
the body of knowledge accumulated by that most venerable of sci-
ences, astronomy. It also applies to the Friedmann-Robertson- Walker
cosmologies, where the time coordinate is chosen to agree with the
one familiar from special relativity, and the body of knowledge which
supports this from particle physics. However, relativity really is rela-
tive: we can if we wish express either of these standard solutions (and
others) in different coordinates.

Covariance is both the greatest asset and the greatest draw-
back of field theory in more than the 4 dimensions of spacetime. It is
an asset, insofar as we can change coordinates at will, to assist in
finding a solution of the field equations. It is a drawback, insofar as a
solution may not correspond to something we recognize from the an-
nals of physics, expressed as it is in terms of our primitive notions of
time and space. It can be argued that we are at a rather exceptional
stage in the development of the subject. It is not enough to have the

206 Five-Dimensional Physics

mathematical tools necessary to solve ND field equations. Also re-
quired is the skill to show that they are relevant to the real world.

The question of relevance is central to modern physics, which
is preoccupied with finding a unification of the interactions of particle
physics and gravity. Let us leave aside, for now, the issue of whether
extra dimensions provide the best approach to unification. Let us take
it as understood that adding one or more extra dimension does no vio-
lence to theories like general relativity (whose field equations do not
restrict the dimensionality of the manifold), and that the consequent
increased richness of the algebra may be directed at explaining new
physical phenomena. Then, a basic issue is whether we accept all of
the solutions of extended theory as relevant to the real world, or only
a subset of them.

This is not a trivial concern. From the practical side, most
physicists who earn their living by solving field equations know that
solutions frequently turn up which are discarded as being "unphysi-
cal". The criteria for this are diverse: a solution may contain an un-
palatable geometrical attribute (e.g. a singularity in the manifold), or
an unacceptable dynamical result (e.g. a local velocity which exceeds
lightspeed), or a strange property of matter (e.g. a negative energy
density). The act of discarding such solutions is commonplace, but
most physicists would prefer that they were not obliged to exercise
this kind of scientific censorship. They would rather deal with a the-
ory whose results are always acceptable, by virtue of it being set up to
be complete . Milne commented on the need for a theory to be alge-

Perspectives in Physics 207

braically complete during the early development of Einstein's theory,
and other scientists like Dirac were aware of the problem (see Kragh
1990 and Chapter 1). Unfortunately, general relativity is not com-
plete in the noted sense. The most obvious instance of this concerns
the energy-momentum tensor. Some information on this usually has
to be imported from the non-gravitational domain in order to obtain a
solution. It might, for example, be the equation of state, as deter-
mined by atomic physics for a classical fluid or quantum theory for
relativistic particles. Most dimensionally-extended versions of gen-
eral relativity share this fault. In some cases, the fault is quite blatant.
Again using the example of the energy-momentum tensor, the com-
ponents of this for the extra dimensions are frequently just guessed.
The resolution of this problem is, of course, obvious: When we set up
the theory, the number of field equations and the number of un-
knowns should exactly balance.

This criterion is satisfied by induced-matter theory. As we
have seen, this involves setting the 5D Ricci tensor to zero, solving
for the potentials, and interpreting the solution as one where the fields
and the matter are both aspects of the geometry. Hence the alterna-
tive name, space-time-matter (STM) theory. This is complete in the
sense noted above, by construction. In 5D there are 15 independent
components of the Ricci tensor and the same number of components
of the metric tensor. There is no extraneous energy-momentum ten-
sor as such. Rather, the properties of matter are contained implicitly
in the first 10 components of the field equations, which by Camp-

208 Five-Dimensional Physics

bell's theorem can be written as the Einstein equations with sources.
The other 5 components of the field equations can be written as 4
conservation laws plus a scalar wave equation. The 15 potentials,
following the traditional view, are related to gravitation (10), electro-
magnetism (4) and a scalar field (1). This theory unifies the physics
normally associated with Einstein and Maxwell, and it is logical to
infer that its scalar field is related to the Higgs field of quantum me-
chanics and determines the masses of particles. It is a unified theory
of matter in 5D.

There is nothing sacrosanct, however, about 5D. Indeed, if
we are to incorporate the weak and strong interactions of particles -
along with their internal symmetry groups - we should consider TV > 5
dimensions.

In such a theory, we should again proceed by setting the Ricci
tensor to zero in order to obtain field equations. (In the absence of an
energy-momentum tensor, there is no ambiguity about whether the
Ricci tensor or the Einstein tensor should be set to zero, since the one
thing implies the other.) Then we can again use Campbell's theorem
to go between dimensions, and thereby identify the lower-
dimensional sources that are induced by the higher-dimensional vac-
uum. While this is easy to state, however, it may not be so easy to
accomplish. We remarked above that covariance is both the strength
and the weakness of ND field theory. We also remarked that space
and time are primitive sense concepts, which for that reason we use as
the independent variables in our theories. (The dependent quantities

Perspectives in Physics 209

are usually more abstract in nature, such as potentials.) But to solve
the field equations in theories of this type, it is necessary to make a
starting assumption about the form of the metric (or distance measure).
That is, it is necessary to make a choice of coordinates, or gauge. In
4D, we are helped in this by our knowledge of how time and space
"behave" in certain circumstances. In 5D, we are helped by our
knowledge of mechanics, which leads to the canonical metric and the
inference that the fifth dimension is related to mass. (See Chapters 1
and 3. For membrane theory as opposed to STM, we are helped by
our knowledge of the hierarchy problem to the warp metric, with the
inference that the extra coordinate measures distance from the brane.)
In a theory with more than 5 dimensions, however, we are in un-
charted territory. How can we make a sensible choice of gauge when
we do not even have a clear idea of the nature of the higher
coordinates?

This is a question which haunts workers in ND field theory. It
is largely our ignorance of the nature of the higher coordinates which
is responsible for the plethora of papers in the subject, most of which
are long on algebra but short on physics. However, it should be re-
called that the Covariance Principle is essentially a statement about
the arbitrariness of coordinates. As such, it applies even to our stan-
dard labels of space and time. At the risk of making a vice into a vir-
tue, it is instructive to reconsider these referents.

Time is probably more discussed than any other concept in
physics. This is because everybody has a sense of its passing in the

210 Five-Dimensional Physics

macroscopic world, and because it has to be handled carefully to
make sense of the equations which describe the microscopic world.
The nature of time has been treated extensively in the books and re-
view articles by Gold (1967), Davies (1974, 2005), Whitrow (1980),
McCrea (1986), Hawking (1988), Landsberg (1989), Zeh (1992) and
Wesson (1999, 2001). These sources take as their starting point the
view of Newton, who in Principia (Scholium I) stated that "Absolute,
true and mathematical time, of itself, and from its own nature, flows
equably without relation to anything external, and by another name is
called duration." This sentence is often quoted in the literature, and is
widely regarded as being in opposition to the nature of time as em-
bodied later in relativity. However, prior to that sentence, Newton
also wrote about time and space that ". . .the common people conceive
these quantities under no other notions but from the relation they bear
to sensible objects." Thus Newton was aware that the "common"
people in the 1700s held a view of time and other physical concepts
which was essentially the same as the one used by Einstein, Min-
kowski, Poincare and others in the 1900s as the basis for relativity.

Nowadays, a common view is that the macroscopic arrow of
time is set by the evolution of the universe (Gold 1967; Davies 1974;
Whitrow 1980). This sounds plausible, given that the universe started
in a big bang and has a thermodynamic direction thereby stamped on
it. However, closer inspection shows that this view is flawed, because
in the comoving frame of standard cosmology we can set the relative
velocities of the galaxies to zero (see Chapters 1 and 3). This implies

Perspectives in Physics 2 1 1

that the cosmological arrow of time is gauge-dependent. It is also
difficult to see how events at the distances of QSOs - even if they are
temporally directed - can have an influence locally which explains
our human sense of the passage of time. For this, we must look to
biological processes. In this regard, there appears to be a dilemma:
biological processes, such as the building of the human genome, tend
to create information over time; whereas the associated thermody-
namical processes still produce entropy (Davies 2005). Insofar as
information is negative entropy, there is the suspicion of a paradox.
The same suspicion, it should be noted, hangs over other physical
processes which tend to create order out of chaos, such as the opera-
tion of gravity to produce a structured Earth from an amorphous
protostellar cloud.

Due to ambiguities of this type, other views on the nature of
time have become popular in recent years, wherein the temporal sense
is not simply linked to the increase of entropy. One of these is the
many-worlds interpretation of quantum mechanics due originally to
Everett (1957). In this, all outcomes of quantum-mechanical proc-
esses are possible, but we are only aware of those which we call real-
ity. This view, according to several workers including De Witt
(1970), is both mathematically and physically consistent. This ap-
proach does not directly account for the human perception of the pas-
sage of time, but Penrose (1989) has suggested that quantum effects
might be amplified by the brain to the level at which they become
noticeable.

212 Five-Dimensional Physics

Another view is that time is an illusion, in the sense that it is
an ordering mechanism imposed by the human brain on a world in
which events actually happen simultaneously. This idea may sound
precocious, and leads to the question: Is time instantaneous? How-
ever, it can be formulated in a meaningful way (Wesson 2001), and
has been taken up independently by a number of deep thinkers. Thus
from Einstein as reported by Hoffman (1972): "For us ... the distinc-
tion between past, present and future is only an illusion, albeit a stub-
born one." While a parallel opinion is from Ballard (1984): "Think of
the universe as a simultaneous structure. Everything that's ever hap-
pened, all the events that will ever happen, are taking place together."
And from Hoyle (1963, 1966): "All moments of time exist together,"
and "If you were aware of your whole life at once it would be like
playing a sonata simply by pushing down all the notes on the key-
board." The latter author points out that this view of time need not be
mystical. He considers a 4D world with coordinates (t, xyz) and a
surface defined by <f> (t, xyz) = C, where "We could be said to live our
lives through changes of C." This approach to time clearly has an
overlap with the many-worlds interpretation of Everitt noted above.
It is basically saying that reality is simultaneous, and that time is our
way of separating events in it.

The concept of simultaneity is gauge invariant, in the sense
that a null interval remains so no matter how we change the coordi-
nates, including the time. In 4D special relativity, ds - specifies
zero separation, which we interpret to mean that particles in ordinary

Perspectives in Physics 213

3D space exchange photons along straight paths. (Here ds as the ele-
ment of "proper" time takes into account the velocity in ordinary 3D
space, as embodied in the Lorentz transformations, without which
particle accelerators would be mere junk.) In 4D general relativity,
we use the same prescription to argue that in the presence of gravita-
tional fields, particles are connected by light rays which follow
curved paths. In 5D relativity, as developed in earlier chapters of this
book, we used the null path dS = to describe the paths of all parti-
cles, whether massive or not. The crucial point is that the interval is
null, as realized by Einstein in 1905 when he used this as a definition
of simultaneity. The rest of the physics flows from this definition.

The notion of a null path is mathematically precise, and is
central to modern cosmology. We use it regularly to calculate the
size of the horizon, which is the imaginary boundary that separates
those particles which are in causal contact from those which are not
(Rindler 1977; Wesson 1999). However, the physics which we build
around the null path is to a certain extent subjective. We find it diffi-
cult to think in a 4D manner, and prefer to split spacetime into its
(3+1) component parts. This is why we do not normally say that
ds = means that particles have zero separation in 4D, but instead say
that a photon propagates in time across a portion of ordinary 3D space.
It is even harder to think in a 5D manner, so we again find it conven-
ient to decompose the manifold into its component parts, and imagine
some influence which propagates through them. However, while the
notion of a null path is independent of how we choose coordinates,

214 Five-Dimensional Physics

the splitting of the manifold is not. To this extent, the physics con-
tains a subjective element.

Eddington was the first person of stature to suggest that phys-
ics might, at least in part, be subjective. He wrote extensively on this
at a time when it was an unpopular view. His book The Philosophy of
Physical Science, which came out in 1939, was criticized by both
physicists and philosophers. Yet a person who reads Eddington with
an open mind cannot but find his arguments compelling; and recent
developments in cosmology make his views more palatable now than
they were before. (There has been a comparable increase in the ac-
ceptability of counter-intuitive consequences of quantum mechanics,
as considered for example by Bell 2004.) It would be inappropriate to
go here into the details of Eddington' s philosophy, especially as re-
cent reviews are available (Batten 1994; Wesson 2000; Price 2004).
But he basically held the view that while an external world exists, our
perception of it is strongly influenced by the physiological and psy-
chological attributes that make us human. He used the analogy of a
fisherman, who notices that all the fish he catches are larger than a
certain size, and assumes that this is a fact of nature, whereas it is due
to the size of the mesh in the net he is using. As a more physical ex-
ample, in the situation we considered above - where events are simul-
taneous in 4D or else connected by photons which travel in time
through 3D - Eddington would have taken the view that we had in
some sense invented the idea of a photon, in order to give an explana-
tion of the situation in a way which fits our perceptions. He was led

Perspectives in Physics 215

to the conclusion that biology is actually the most "valid" form of sci-
ence, in that there is less obstruction between the data and our inter-
pretation of the data. Contrarily, Eddington regarded much of phys-
ics as invented rather than discovered.

The views of Eddington overlap somewhat with those of Ein-
stein, Ballard and Hoyle on time, which we outlined above. These
and others have clearly regarded with some doubt the way in which
physics has traditionally been presented. In particular, there has
apparently always been some scepticism about the status of coordi-
nates. It is therefore not surprising that modern physics - which is
focussed on unification through extra dimensions - finds itself beset
with questions of interpretation. For our present concerns, there is
one question which is paramount. In loose language, it is just this:
Are extra dimensions "real"?

We believe that the answer to this is : Yes, provided they are
useful.

Lest this be considered specious, it should be recalled that the
history of physics is replete with examples of ideas which were
adopted because they were useful, or discarded because they were not.
It is better to treat the dynamics of a particle with a 4-vector than to
deal separately with its energy and momentum. And the aether had
become unworkably complicated before the Michelson-Morley ex-
periment terminated its tenure. In the case of extra dimensions, the
requirement is simply that they earn their keep.

216 Five-Dimensional Physics

We have in this volume concentrated on the case of 5 dimen-
sions, which has been known since the 1920s to provide a means of
unifying gravity with electromagnetism. This, apparently, was not
enough to earn the fifth coordinate the legitimacy of being "real".
However, in the last decade there has been an explosion of new inter-
est in 5D relativity, due largely to how the extra dimension may be
used to consider problems related to mass. STM theory views the
extra dimension as being all around us in the form of matter, and in
the special canonical gauge the extra coordinate is essentially the rest
mass of a particle. Membrane theory views the extra dimension as
perpendicular to a hypersurface which is the focus of particle interac-
tions, gravity spreading away from this in a manner which helps ex-
plain why typical masses are smaller than the Planck one which
would otherwise rule. The above-posed question about the "reality"
(or otherwise) of the fifth dimension is seen to be connected to how
useful it proves to be in describing mass.

In this context, there are some specific questions that need to
be addressed:

(a) Is the inertial rest mass of a local particle governed by the
scalar potential of a global field? This would be a concrete realization
of Mach's Principle. But while we have given special results which
show the plausibility of this view, a general demonstration is needed.

(b) If the foregoing conjecture is true, how do we account for
the disparate masses of elementary particles which have otherwise

Perspectives in Physics 217

identical physical properties? For example, how do we calculate the
muon / electron mass ratio?

(c) Given that 5D relativity unifies gravity and electromagnet-
ism, we can use the 5 degrees of coordinate freedom to "turn off the
electromagnetic interaction; so where does the physics of the latter
"go"? Insofar as we cannot suppress the 10 gravitational potentials in
the same manner, it is reasonable to infer that the scalar field takes up
the slack. But if so, a simple system like the hydrogen atom ought to
have two equivalent descriptions: one where the structure is due to
electromagnetism (like the original Bohr model), and one where the
electron orbits are due to the form of the scalar field. We lack the
second picture.

(d) The laws of mechanics are different in 5D versus 4D, and
if the fifth dimension is related to rest mass, we can ask: Is the extra
dimension likely to be revealed by more exact tests of the Weak
Equivalence Principle? This is a geometric symmetry of 5D relativity
for the induced-matter scenario. But as in particle physics, we expect
the symmetry to be broken, in this case at a level dependent on the
ratio of the test mass to the source mass. This can be examined by
upcoming experiments, such as the satellite test of the equivalence
principle, which will use test masses in orbit around the Earth.

(e) By simple arithmetic, there are 5 laws of conservation in a
5D world, but if we take a 4D view then discrepancies are inevitable;
so can we detect these? On a microscopic scale, these discrepancies
mimic the phenomena usually attributed to Heisenberg's uncertainty

218 Five-Dimensional Physics

principle. But on a macroscopic scale, they should be detectable by
studying the dynamics of objects such as galaxies. Cosmology is in
an era of precision measurements, and we should be able to assess the
dimensionality of the universe using astrophysical data.

(f) The universe may well be flat in 5D (even though it is
curved in 4D), so it is natural to ask: can we show that its total energy
is zero? This question is easy to formulate, but is difficult to answer
in a practical sense, given the non-locality of gravitational binding
energy. However, a corollary is that the paths of particles in a 5D
universe should be null geodesies, which implies that all of its parts
are in contact. This leads us to expect that SETI may be a done deal
(though we do not know the mechanism); and that all objects in the
universe should have identical properties (irrespective of the nature of
the standard horizon). Preliminary studies of the spectroscopic prop-
erties of QSOs support this view. In a 5D universe where intervals
are null, all mass-related properties of objects should be identically
the same, everywhere and for all times.

The preceding half-dozen questions provide practical ways to
test for the existence of a fifth dimension. It should be recalled, in
this connection, that we do not need to "see" an extra dimension in
order to admit that it exists. We do not see time, but few of us would
deny that it exists. In closing, we point out that the existence or not of
a fifth dimension is pivotal to physics. Because if there is a fifth,
there should be more . . .

Perspectives in Physics 219

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(Springer, Berlin).

ADM formalism

25, 181

Big bang 34, 38, 45, 46, 52, 202
Big bounce 47, 52

Birkhoff theorem 54, 156

Black hole 54

Buddhism 34

Campbell theorem 7, 20, 24, 36,
182
Canonical gauge 18, 87, 103

Causality 101, 147, 212

Compactification 4

Cosmological constant 10, 20, 48,
108, 134
Covariance principle 66, 68, 205
Cylinder condition 4

Dark energy
Dark matter
Dimensions
Dirac equation

47,53
47,53
2,215
69,116

Eddington 134, 214

Einstein equations 6, 10, 86, 140
Einstein gauge 88, 103, 1 10, 120
Embeddings 23, 170

Energy-momentum tensor 1 6, 27
Entropy 52,211

Equivalence principle 67, 68, 8 1 ,
88, 191

Fifth force 70, 74, 83, 203

FRW models 19,60,136,205
Fundamental constants 87,108

Galaxies 150, 154

Gauss-Codazzi equations 1 79
General relativity 80

Geodesic principle 14, 41, 66, 68,
71,85

Hamilton- Jacobi formalism 76

Heisenberg relation
Higgs field
Horizons
Hubble parameter

Inflation models
Isometries

22, 106

13,41,92

52,91

43

140, 151
57, 58, 124

Kaluza-Klein theory 5, 73

Klein-Gordon equation 69, 1 13

Large Numbers Hypothes

135

Mach principle 67, 92, 1 62, 2 1 6
Mass quantum 107

Maxwell equations 4, 7, 17

Membrane theory 8, 22, 173, 174,
191
Microwave background 46, 1 52
Milne model 19,37,126

Occam

1

Phase transition 54

Pioneer spacecraft 157

Planck gauge 88, 103, 1 10, 120
Planck mass 5

222 Five-Dimensional Physics

QSOs 65, 151, 173,

211

218

Time

209

Quantum tunneling

34

Two-time models

77

Reincarnation

47

Vacuum (4D)
Vacuum (5D)

11,134,158
90, 134, 138

Schrodinger equation

69

Schwarzschild solution

156

Wave models

79, 129, 162

Shock wave

43

Wave-particle duality 124,164

Solitons 54,91,

156

172

STEP SI

,97

217

Z 2 symmetry 18,

22,28,53,192

String theory

9

Zero-point fields 9

102, 109, 136

Supergravity

9

Supersymmetry

9