GRAVITY WITH A TWIST - University of Toronto T-Space · ported reference vector [2]. This thesis is devoted to the study of a theory of gravity that employs both curvature and torsion

G R A V IT Y W IT H A T W IST

by

Neil John Cornish

A thesis submitted in conformity with the requirements

for the degree of Doctor of Philosophy

Graduate Department of Physics

University of Toronto

Copyright ©1996 by Neil J . Cornish

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Gravity with a Twist by Neil John Cornish Department of Physics, University Toronto 1996 November

A bstract

A theory of gravity which describes spacotime in terms of non-Riemannian

geometry is presented and studied. This Nonsymrnetric Gravity Theory (NGT) is a

close relative of Einstein's general relativity, differing only fry a fundamental torsion

or twist in the spacetime structure. The principal aim of the study is to investigate

the existence, or otherwise, of black holes in NGT. Along the* wav many issues have to

be addressed including the physical description of spacetime geometry and the inter

action of gravity with m atter. At least a partial revolution of these* issue's is offered.

In places results will be presented in a historical fashion, reflecting the* evolution of

ideas that occurred as the study progressed. The definitive* re*sult of the? stuely is to

show that there will be no Schwarzschild-type black holes in spae:etime*s with torsion.

i


D edication

This work is dedicated to my parents, Dr. E. .1. ic Mr. J. N. Cornish.

N.J.C — February 1996

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A cknow ledgm ents

My deepest thanks go to ray advisor. .John \V. MotFat. for teaching me how

to ask the right questions. It was a pleasure to be treated like a collaborator, not

a conscript. On a personal note. I want to thank John and his wife Patricia for

welcoming me into their family and making me feel at home', so far from home.

To my friend and mentor. Norm Frankel. who launched me on this journey,

what else can I say but “may the force be with you” .

I want to thank my friends from the Canadian Institute for Theoretical Astro

physics, Dick Bond, Nick Kaiser, .Janna Levin and Glenn Starkm an for always being

in my corner.

My time in Toronto wouldn't have been the same without my mates from

the Toronto Eagles (& Panthers) Australian Rides football team. Thanks for the

memories, and for the times I can’t remember.

I have enjoyed wrorking with my fellow graduate students, Mike Clayton, Tony

Demopoulos, Brian Hand, Jacques Legare and Dariusz Tatarski. I also bcncfittnd

greatly from talking writh Pierre Savaria on numerous occasions.

I am very grateful for the financial support of a Canadian Commonwealth

Scholarship.

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C ontents

A b s tra c t i

D ed ica tio n ii

A ck n o w led g m en ts iii

1 Introduction 1

1.1 Gravity with a T w is t ....................................................................................... 1

1.2 Black Holes and Singularities ....................................................................... 2

1.3 The Jordan-Brans-Dicke A n a lo g y ............................................................... 5

1.3.1 The equivalence principle................................................................... 6

1.3.2 Monopole excita tions.......................................................................... 7

1.3.3 Nouperturbative departures from G R ............................................ 7

1.4 Thesis O u tlin e ................................................................................................... 9

2 N G T Formalism 12

2.1 Non-Riernannian Geometry .......................................................................... 13

2.2 Vacuum Field E q u a tio n s ................................................................................ 20

2.2.1 N G T -7 C ................................................................................................. 21

2.2.2 N G T -9 5 ................................................................................................. 23

2.3 Coupling to M a t te r .......................................................................................... 25

2.4 Particle Trajectories and Causal S tructure.................................................... 28

2 5 Linearised N G T ................................................................................................. 30

iv


3 Vacuum Solutions

3.1 A (2+l)-dimensional P r im e r .................................

3.2 Wyman-Schwarzschild M e tr ic ..............................

3.2.1 The approximate s o lu t io n .......................

3.2.2 The exact s o lu t io n ....................................

4 Analysis o f the W yman-Schwarzschild Solutiou

4.1 The Wyman G eo m etry ..........................................

4.2 Curvature and R e d sh if ts .......................................

4.3 Causal S t r u c t u r e ....................................................

5 Electrovac Solutions

5.1 The Mann M e tr ic ....................................................

5.2 Analysis of the S o lu tio n s .......................................

6 The Short Range W yman Solution

6.1 Field E q u a tio n s .......................................................

6.2 The Far-Field M e tr ic ..............................................

6.3 The Mid-Field M etric ..............................................

6.4 The Near-Field M e t r ic ...........................................

7 Gravitational W aves

7.1 Spherical S y s tem s ....................................................

7.2 Axi-Symmetric Systems .......................................

7.2.1 The Axi-symmetric M e tr ic .......................

7.2.2 The Electric S e c to r ....................................

7.2.3 The Magnetic S e c to r .................................

8 Interior Solutions

8.1 Equilibrium E q u a tio n s ..........................................

8.1.1 Matching C o n d itio n s .................................

8.1.2 Binding E n e rg y . . . .

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9 Summary and Conclusions 109

10 Addendum 111

A Notation and Conventions 113

Bibliography 114

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C hapter 1

In troduction

1.1 G ravity w ith a T w ist

Einstein developed his theory of gravity [1] by striving for beauty am I simplicity

in the mathematical expression of a guiding physical principle - the equivalence of

inertial and gravitational mass. This lead Einstein to formulate general relativity as

a geometrical theory in which particles follow the geodesics of a curved spacetime

geometry, with the curvature of spacetime determined by the distribution of mass.

When formulating general relativity, Einstein demanded that spacetime be lo

cally invariant under homogeneous Lorentz transformations. This naturally led him

to describe the geometry in terms of a pseudo-Riemannian spacetime where the met

ric (jft,, and connection are symmetric under ft *— /./ in a coordinate b;isis. In

geometrical terms Riemmaniau geometry describes a manifold with curvature but no

torsion. Loosely speaking, curvature measures the divergence of a bundle of geodesics,

while torsion measures how a bundle of geodesics twists relative to a parallel trans

ported reference vector [2]. This thesis is devoted to the study of a theory of gravity

that employs both curvature and torsion in its description of the spacetime geome

try'. Historically two classes of gravity theory with torsion have been developed: the

Einstein-Cartan [3] theory where the metric remains symmetric, and non-symrnetric

gravity [4] where both the metric and connection are non-symmetric. In Einstein-

C artan theory the torsion plays no direct role in the gravitational dynamics, making it

1


1.2. Black Holes and Singularities 2

a relatively uninteresting generalisation of general relativity. In contrast, the torsion

introduced in non-symmetric gravity can have a profound effect on the dynamics. In

the following chapters we will see that nonsymmetric gravity - gravity with a twist -

can cause an interesting twist in fate for stars undergoing catastrophic gravitational

collapse.

1.2 B lack H oles and Singularities

“The star has to go on radiating and radiating and contracting and contracting until.

I suppose, it gets down to a few kin radius, when gravity becomes strong enough to

hold in the radiation, and the star can a t last find peace....Various accidents may

intervene to save the star, but I want more protection than that. I think there should

be a law of Nature to prevent a star from behaving in this absurd way!”

A. Eddington. 1935.

In this thesis we ask whether NGT might allow Eddington, or at least his ghost,

to at lust find peace.

Of all the predictions of modern physics, none has so universally captured the

popular imagination as black holes. For example, the starship Enterprise encountered

a black star in an episode of Star Trek, first aired in 1967, several months before

Wheeler coined the term “black hole” [5]. Although Schwarzschild [6] published his

famous solution in 1916, it was another 40 years before its physical significance was

fully understood. For many years the “singularity” in Schwarzschild’s solution at

r = 2M was a m ajor source of confusion. Schwarzschild remarked that no static fluid

body could have a radius less than 9/S(2A/), so he considered the surface r = 2M

physically irrelevant. This view was to remain the official dogma until 1939 when

Oppenheimer and Snyder [7] showed that black holes could form as the result of

gravitational collapse. This again raised questions about the Schwarzschild radius,

although Einstein still (wrongly) maintained tha t “the ‘Schwarzschild singularity’



does nor exist in physical reality for the reason that m atter cannot he concentrated

arbitrarily" [$]. The singularity in Schwarzschild's coordinate svsrem at r = 2 M

was commonly thought to he a physical singularity until the early sixties, although

Lemaitre realised as early as 1933 that r = 2.\f was a non-singular surface much

like the de Sitter horizon [9], Based on Lemaitre's work. Robertson [1(1] delivered a

lecture in Toronto in 1939 explaining why the surface' at r ~ 2.M was neither singular

nor inaccessible.

It was not until the publication of the Hawking-Penrose [ l l | singularity theo

rems in the mid-sixties that, the entire relativity community was dragged, with many

still kicking and screaming, to accept black holes as an inescapable prediction of gen

eral relativity. By this time, many alternative coordinate systems had been found in

which the Schwarzschild radius was clearly non-singular. It was also understood that,

physical singularities were characterised by having infinite curvature. Hawking and

Penrose showed that physical singularities, such ms the origin of the Schwarzschild

metric, were inevitable in general relativity coupled to ordinary matter. A central

feature o f the singularity theorems is the role played by trapped surfac(*s. Loosely

speaking, a trapped surface is the outer boundary of a region from which nothing,

not even light, can escape. During gravitational collapse, a trapped surface forms at

the center of an object and sweeps out to encompass all of the matter. Once m at

ter is inside this trapped surface a singularity must form as all particles are driven

inexorably inward, no m atter how hard the m atter pushes back.

For this reason, any theory th a t hopes to get rid of black holes and singularitii>s

must avoid the formation of trapped surfaces. But what kind of theory can hope to

do this? As emphasised by Zeldovich and Novikov [12], any theory that obeys the

equivalence principle cannot hope to get rid of black holes. They pointed out th a t

for objects 10® times as massive as the sun, the density of m atter when compressed

almost to its Schwarzschild radius is only a few times that of water. “Under these

conditions, which are in no way remarkable, certainly nothing fantastic can take

place. The only thing th a t is unusually large is the gravitational field, but according

to the principle of equivalence the gravitational field itself does not produce local

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changes in the laws that govern physical processes". As we will see. NGT violates the

equivalence principle and thereby provides a mechanism by which the local laws of

physics can change dramatically when gravitational fields become strong1 . This does

not prove tha t NGT gets rid of black holes, it simply assures us tha t the possibility

exists. Moreover, it is a simple m atter to show that the Hawking-Penrose singularity

theorems do not apply to NGT due to the additional torsion term s [14]. Thus, NGT

offers the exciting possibility that it is a non-singular theory, free from black holes.

But why get rid of black holes? In Eddington ?s words, a black hole would seem

to offer a s ta r a peaceful resting place, safely cut off from the rest of the universe. This

peace was shattered in 1974 when Hawking showed that black holes radiate thermally

via a quantum tunneling process [15]. Hawking immediately realised he had created

a paradox: Hawking radiation is purely thermal and contains no information about

the collapsed star. As the black hole evaporation proceeds one is eventually left with

a thermal bath of radiation and no black hole. In this way, a pure quantum state can

be converted into a mixed state leading to the black hole information loss paradox.

The reason for the paradox can be traced to the causal topology of a black hole

spacetime. By definition, a black hole divides a spacetime (M , gû) into two causally

disconnected regions such tha t M = B + J _ (T+), where B is the black hole region

and is the causal past of future null infinity. This causal disconnection is the

root cause of the information loss problem in black hole spacetimes. Looked at as a

scattering process, the outgoing and incoming density matrices describing the initial

and final states are related by

Pout = $ Pin • (IT )

If the evolution were unitary, $ would be factorizable as the product S S T When a

black hole is present quantum mechanics is no longer unitary as the factorizability of

$ is lost when we trace over the quantum fields contained inside B [16]. This is an

alternative way of describing the black hole information loss problem.

1This also happens in a theory where local Lorentz invariance is explicitly broken in strong gravitational fields [13]. There are no black holes in this theory as the event horizon becomes an impenetrable barrier.


1.3. The Jordan-Brans-Dicke Analogy 5

In the past five years, several hundred papers have been written in an attem pt

to resolve the information loss paradox. Most of the proposals require that we refor

mulate quantum mechanics, or give up the notion of locality. We feel that a simpler

solution is to banish black holes. In Chapter 4 we will see that NGT does away with

black holes and offers a resolution to the information loss problem - at least for the

spherically symmetric case we studied.

1.3 T h e Jordan-B rans-D icke A nalogy

In 1959 Jordan [17], and later Brans and Dicke [18], introduced a scalar-tensor theory

of gravity. This theory differs from general relativity by way of a scalar field that

couples to both m atter and the spacetime geometry. The action for Jordan-Brans-

Dicke gravity (JBD) is given in vacuum by

S = J d ' x . (1.2)

The field equations th a t come from varying this action are

= 9pv,a ~ 9pt^%r ~ dppâv ~ ® » (1-3)

V "V p0 = 0 , (1.4)

0 (Rpv - 7}9pi>Rj = ^ (v M0V„0 - + V,t0 V „ 0 . (1.5)

In these units c = 0oo = 1, where 0 ^ is the value of 0 far from any sources. Newton’s

constant is then given by

G = i ± | £ . (1.6)3 -F 2 uj

The vacuum field equations of general relativity are recovered when cj » 1 and

0 = const. + O ^ . (1.7)

While the field equations for Jordan-Brans-Dicke gravity look nothing like the



field equations for NGT, there are some interesting similarities in the phenomenology

of the two theories. The most interesting ways in which JD B and NGT differ from

general relativity are through:

• Violations of the equivalence principle.

• Monopole radiation (no analog of BirkhofFs theorem).

• Solutions that do not reduce to those of general relativity.

1.3.1 The equivalence principle

The equivalence principle is often divided into the strong and weak versions. The

strong (weak) equivalence principle asserts that within a sufficiently small region of a

given spacetime point the laws o f nature (laws of motion o f free falling test particles)

take the same form as they do in special relativity. P u t another way, the weak equiva

lence principle states th a t the trajectory of an uncharged test particle with negligible

self-gravity is independent of all properties of the particle. In their standard form

both JBD gravity and NGT satisfy the weak equivalence principle. If one is perverse

enough to introduce non-universal, non-minimal couplings of 4> or g[tw] to m atter, it

is possible to violate the weak equivalence principle in both theories. In practice it is

unwise to introduce any form of non-universal couplings as there exists very strong

experimental evidence in support of the weak equivalence principle [19j. On the other

hand, there is only indirect experimental evidence for the strong equivalence principle.

Both JBD and NGT explicitly violate the strong equivalence principle. This can be

understood as follows: by a coordinate transformation one can always locally reduce

<7(;iI/) to where is the metric for Minkowski space. However, neither (<j> — 4><x)

nor flf/ji/] can be made to vanish by a coordinate transformation. Consequently, the

laws of nature in JBD and NGT differ from special relativity in a freely falling frame

of reference when masses are present.


1.3. The Jordan-Brans-Dicke Analog}' 7

1.3.2 M onopole excitations

In general relativity the gravitational field is described by a second rank symmetric

tensor. In field theory language, the gravitational force is mediated by a spin-2 gravi

ton. Angular momentum conservation thus forbids general relativity from supporting

monopole or dipole radiation. In JBD gravity there is an additional spin-0 scalaron,

making it possible for spherically symmetric sources to emit monopole radiation [20],

In NGT the skew fields </[,,„] introduce a number of skew excitations or “.skewons".

The exact number of skewons depends on the version of NGT we are working with,

but it is always in the range 3 —*• 6. The skewons come in a range of spins and

parities, one of which behaves like a spin-0 pseudo-scalar particle. Like the scalaron

of JBD, the pseudo-scalaron of NGT allows monopole excitations to be produced by-

time dependent spherically symmetric bodies. We study the wave properties of NGT

in C hapter 7.

1.3.3 N onperturbative departures from G R

The th ird , and most interesting departure from GR occurs in the strong field limit.

Naively one might assume that all solutions to the JBD and NGT field equations go

over to those of GR in the limit u —► oo or <j tu\ —► 0. This is not the c.ise. If it

were, the main motivation for this thesis would be lost as black holes could always

be recovered by “turning-off” the skew fields. To prove tha t JBD and NGT do not

reduce to GR we will begin by assuming the opposite situation and show that it is

false. If JBD and NGT always smoothly recover GR we need only consider <-/» or

as small perturbations on a GR background solution. Consider now the linearised

equations of motion in the spacetime geometry of a Schwarzschild black hole:

VPV > = 0 ( 1-8)

(1.9)

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where S = is the pseudo-scalar skew field in orthonormal coordinates. Near the

event horizon (r = 2M ) we find

f 2 M \0 ~ S ~ In f 1 - —— J . (1.10)

This tells us th a t the linear approximation has broken down and our assumption of

a smooth limit is false. In Jordan-Brans-Dicke theory the event horizon of a black

hole is converted from a coordinate singularity into a curvature singularity. The exact

solution is given by [21]

ds2 = - ( l + £ ) > - > (dr2 + f W ) , <b = e • (1-11)

Here; £ is given by

(1.12)r + r0

and r 0, « and (5 are real constants subject to the constraint

a'2 + /32 ( l + | ) —a/? —1 = 0 . (1.13)

For large u we can write

a = l + 0 (1.14)u UJ

where (1.13) demands a = 6(2 — 6)/4 and we require a ^ O and 6 ^ 0 for a non-trivial

solution. When a = 1 and 0 = 0 the metric (1.11) reduces to the Schwarzschild

metric in isotropic coordinates and we can identify 2r0 = M as the mass of the black

hole. However, for any u ^ oo, no matter how large, the JBD spherically symmetric

vacuum solution differs dramatically from its GR counterpart. The surface r = r0 is

still a surface of infinite redshift, but it is also a surface of infinite curvature. Writing

f = r 0( l + e), where e 1, we find the horizon to be characterised by infinite

curvature,

R a ~ ---\~n ’ U*15)oTjje2


1.4. Thesis Outline 5

and infinite or zero surface area (depending on the sign of a)

-4 = 64-r0V-“ . (1.16)

Only solutions with a > 0 appear to be physically viable. They lead to what is known

as the truncated Schwarzschild solution of Janis, Newman aud Winicour [22]. The

non-singular, finite area, infinite redshift event horizon of GR lias been converted

into a singular point in JBD gravity. Since the singularity is not naked, there is no

violation of cosmic censorship for solutions of this type. In Chapter 4 we will see a

completely analogous result occurs in NGT. The GR event horizon is again converted

into a singular point with infinite redshift. In JBD gravity detailed numerical studies

have shown tha t the truncated Schwarzschild metric cannot form as the result of

gravitational collapse [23]. Instead, the scalar field of the collapsing star is radiated

away as the collapse proceeds, leaving behind a pure Schwarzschild black hole with

<t> = 0oo everywhere. A similar numerical study has yet to be done in NGT, but we

have reason to believe the conclusion will be different. Most importantly, the skew

fields are generally taken to be massive in NGT so the monopole excitations are unable

to carry off any energy. This seems to indicate tha t the collapsing star will have a

hard time shedding its skew fields in order to settle down to a Schwarzschild black

hole. There are however other ways to bleed off the skew fields, such as through their

non-linear coupling to the symmetric sector. For this reason, a definitive statem ent

about the end s ta te of gravitational collapse in NGT awaits a (very difficult) numerical

treatm ent.

1.4 T h esis O utline

In Chapter 2 we begin by reviewing non-Riemannian geometry and show how torsion

and curvature arise. We go on to derive the field equations for the two versions of

NGT, and discuss how m atter is able to give rise to curvature and torsion. Using a

scalar field as our guide, we derive the equations of motion for a minimally coupled



test particle. We explain how NGT comes to have a multiple lightcone structure

described by three symmetric metrics formed from gû. We conclude the chapter with

a discussion of linearised NGT and the existence of a stable vacuum. The majority of

the introductory material is not new, excent for the vorticitv based energy momentum

tensor, and the equation of motion for minimally coupled test particles.

In Chapter 3 we derived the static, spherically symmetric vacuum solutions

to the NGT field equations in (2+1) and (3+1) dimensions. To our knowledge, the

(2+1) dimensional solutions are new, while the solution in (3+1) dimensions was first

discovered in the context of unified field theory.

In Chapter 4 we analyse the static spherically symmetric solution in detail. We

find that the Schwarzschild radius is no longer a black hole event horizon. Instead, the

curvature and redshift get very large just inside the Schwarzschild radius, eventually

becoming infinite a t the origin. Since the Wyman metric only has a trapped point,

and no trapped surface, it does not describe a black hole. Moreover, since there is no

event horizon, there is no reason why m atter should be crushed to a point. For this

reason, we argue that the everywhere vacuum W yman solution may never be reached

by gravitational collapse. This conjecture is also needed to ensure the theory is free

of closed timelike curves. All the results in this chapter are original.

In Chapter 5 we derive and analyse solutions to the field equations describing

a massive, pointlike charge concentration. This solution was first derived by Mann

in the context of algebraically extended gravity. Like we found for the uncharged

case, the electrovac solution is free of finite area event horizons. We also derive the

remarkable result that the to tal energy' contained in the electromagnetic field is finite.

In Chapter 6 we return to consider the spherically symmetric vacuum solution,

bu t this time in the new version of NGT where the skew fields have a mass. As one

might expect, the near field behaviour of the solution is largely unchanged from the

massless theory, while the far field region sees the skew field fall off exponentially fast.

This solutiou is being presented here for the first time.

In Chapter 7 we leave the confines of static metrics to consider the wave

properties of NGT. We study both spherical and axi-symmetric sources and show for



the first time that NGT gives rise to monopole radiation.

In Chapter S we consider how stars behave in NGT and show how suitable

couplings to the energy density and vorticitv of a fluid can give rise to the curvature

and torsion seen in the exterior vacuum solution described in Chapter 4. We find

that N G T stars are stablised relative to their GR counterparts, lending support to

our conjecture that total gravitational collapse can be avoided in NGT.

In Chapter 9 we conclude the thesis by reviewing our main results and dis

cussing some of the outstanding issues tha t remain unresolved.


C hapter 2

N G T Form alism

Einstein’s theory describes gravitation in geometrical terms by employing Riemannian

geometry. In Riemannian geometry, both the metric g ^ and the affine connection

are symmetric in a coordinate basis: = gVfL. = r*;i. This symmetry ensures

that spacetime is locally Lorentz invariant.

Nonsymmetric gravity (NGT) relaxes the assumption of local Lorentz invari

ance and allows both the metric and the affine connection to be nonsymmetric. Local

Lorentz invariance is replaced by local invariance under the larger group of gen

eral linear transformations. As a result, NGT is a geometrical theory' described

by uou-Riemannian geometry. The field equations are formulated in terms of cur

vatures which employ the full, non-Riemannian geometry. In mathematical terms,

non-Riemannian geometry is formulated on the ring of hyperbolic complex numbers,

rather than the field of reals used in Riemannian geometry'. In analogy' w ith quantum

mechanics, where complex numbers appear in intermediate stages, we must extract

real results to make contact with what is actually measured.

YVe will begin by briefly reviewing the fundamentals of non-Riemannian ge

ometry and Moffat’s hyperbolic complex field equations [4, 24]. We develop for the

first time, a complete picture of the geometry, causal structure and particle motion.

Different facets of the hyperbolic complex metric are found to describe each aspect

of the theory'.

12


2.1. Non-Riemannian Geometry- 13

2.1 N on-R iem an nian G eom etry

General relativity is based on the postulate that spacetime is locally invariant under

homogeneous Lorentz transformations. 5 0 (3 .1 ). This postulate is implemented In-

demanding that at each point in the manifold there exist a real vierbein <•“ whose

components transform under the action of the local Lorentz group so as to leave the

tangent space scalar eê^rjab invariant. The vierbeins are given at each point in the

manifold by

where the A’"'s correspond to a set of locally inertial coordinates at r = //. The

metric on the manifold is given by

under the action of the element Z of 5 0 (3 ,1 ). The matrix Z is real, unitary and

antisymmetric, and thus has six components corresponding to the six generators of

the Lorentz group.

Moffat [4] proposed that this picture should be generalized to allow spacetime

to be locally invariant under general linear transformations, G L {\,R ) D 5 0 (3 ,1 ) .

Moffat’s proposal can be implemented by exploiting the isomorphism between G’L(4, R)

and the unitary group 0 ( 3 , l ,f i ) on the ring of hyperbolic complex numbers Q [25].

Hyperbolic complex numbers can be written as 2 = x 4- uiy, where x and y are real

and u) is the pure imaginary- element of il which satisfies u) = — u . with u/2 = 1. The

elements Ug of the unitary group comprise a hyperbolic complex, unitary, nonsym

metric matrix corresponding to the sixteen generators of GL(4, R). Under a local

G L(4, R) transformation, the vierbeins transform as

(2.1)

9m> = e'fA'lab . ( 2 .2 )

and the vierbeins transform as

(2.3)

(2.4)


2.1. Non-Riemannian Geometry 1 4

Clearly, the vierbeins must also be hyperbolic complex valued:

e* = Rfi(e“) +o,'Im(e“ ) , (2.5)

= K + u f t . (2-6)

The combination e“e*r/ a 6 represents a scalar under the action of GL(4, R) and

is identified as the metric on the hyperbolic complex manifold Mq:

<Jtxu = e f f i u . (2.7)

The vierbein has an inverse, e£. defined by

« = K , = 61 . (2.8)

so tha t

guXg„\ = gXflg\u = . (2.9)

We may define a hyperbolic complex analog of gab via

Xab = Calfyhb ■ (2.10)

In order for \ab to have a real determinant we require tha t it be hermitian symmet

ric, Xai, = \ba- This in turn ensures that the vierbeins posses a kind of hermitian

symmetry, and that the determinant of e“ is real. The reality of det(e“ ) follows from

the relation

d e tx = (dete)(det/i)(det 77) . (2 .1 1 )

We then find that

det(<7,It/) = - ( d e te )2 = - ( d e te ) 2 . (2.12)

These conditions ensure tha t a non-degenerate metric is described by non-degenerate

vierbeins. The need for herm itian symmetric vierbeins was not noted in previous

studies of the hyperbolic complex structure.

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2.1. Non-Riemannian Geometry 1 5

Under a coordinate transformation the hyperbolic complex vierbeins transform

as

tJ;; = . (2.i:nd.rv

Since the x^'s are purely real, we also have

Ox** Oj'1h"' = - — h" f Ul = — — P C> 14)

11 dx'» " ' J,t ' 1

Thus, bo th h“ and / “ form a set of four real, covariant vector fields.

The metric contains both symmetric and antisymmetric parts, which may be

written as

= c o s t

SIH = (llV ‘ ~ ’><* ■ 12 l(! l

The symmetric parts of g,w are pure real and the anti-symmetric parts are pure

hyperbolic complex imaginary. Moreover, the fidl metric is hermitian since y,,,, = y„,,.

which follows from the unitarity of the U{3 . 1,12) transformations.

The metric defines basic geometrical concepts such as volumes and areas. The

4-volume form is given by

r/‘f2 = eftUK\ d r '1 A d x l/ A d x K A dxx , (2.17)

where the Levi-Civita tensor is defined by

W = eêt eKefVabcd = det(e)f/«//cA] = s /^ ij [fiutcX] , (2.18)

R ep ro d u ced w ith p erm iss io n o f th e cop yrigh t ow ner. Further reproduction prohibited w ithout p erm issio n .

2.1. Noa-Riemannian Geometryr 1 6

and [fiunA] is the totally antisymmetric alternating tensor:

[nun Aj

+ 1 if even perm, of {1.2.3.4}

-1 if odd perm, of {1.2.3.41 (2-19)

0 if any two are eq u a l.

If we choose a particular time-slicing t. the 3-volume form of a spatial hypersurface

is given by

= f tijk cLr' A c1j j A dx* . (2.20)

Naturally, the three volume is not coordinate invariant as it depends on the choice of

spatial hypersurface. The proper surface area of a two-surface orthogonal to (t. x) is

similarly defined by the 2-forrn

dlA u = etxij dx* A d r 7 , (2.21)

where ( i . j ) label coordinates that lie in the surface.

While the definition of volumes and areas is relatively straightforward in non-

Ricmannian geometry, the measurement of lengths is less clear cut. The most obvious

choice is to use the symmetric part of g ^ to define an infinitesimal line element

ds2 = g ^ d x * dxu . (2.22)

However, this choice is not unique as we could also have chosen the symmetric part

of gtu/ (which is not equal to the inverse of g{(iv)) to define lengths. Indeed, there are

an infinite number of such metrics to choose from 1.

Our confusion about the choice of a “physical” metric for measuring lengths

goes away once we realise tha t we arc asking the wrong question. W hat we really care

about is the paths followed by physical excitations such as photons and gravitational

lOne interesting candidate for the line element is da2 = piiudxtldxu where p^v = e“e£rfob- While p^t, is not real, it is symmetric. Moreover, it has the same real determinant as gû. in contrast to 9uif)


2.1. Non-Blemamnan Geometry' 17

waves. In XGT these excitations respond to the entire non-Riemannian geometry

and not just some real sub-portion we can represent in a line element. In particular,

test particles will not follow geodesics of or anv other truncation of the full

geometry-. The equations o f motion should be derived directly from the field equations.

Demanding that particles follow some real. Riemanniau sub-manifold is analogous

to demanding that a quantum wave function is real at all times. As in quantum

mechanics, one should first evolve the system and then project out the real answers

as a final step. We shall return to our discussion of test particle motion in §2.-1.

Invariance of gfiu under the infinitesimal coordinate transformation

x't‘ = x / i+ ^ ( x ) . (2.23)

yields Killings equation:

+ v / = 0 • (2 .2 -1)

The real part of the above equation is the symmetric Killing equation

+ D ^ u = 0 , (2.25)

where the covariant derivative DM employs Christoffel connections constructed from

Q(fiu)- The hyperbolic complex imaginary part of (2.24) imposes the? isometrics of the

symmetric part of the metric upon the anti-symmetric part:

— y[ticr}£,,v "b !J[rjv] n + g[(u/],a£, — ft • (2.26)

We may define a spin connection (Dff)o6 such tha t

V„e« = - ( Q arcel , (2.27)

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2.1. Non-Riemannian Geometry 18

where the metric compatible connection Tû is given by

V .e “ = e“><r- r ^ e “ . (2.28)

The spin connection is skew-Hermitian and the connection is Hermitian symmetric:

(O rU - - ( f i , ) * , = f . (2.29)

By differentiating (2.7) and using (2.27), (2.28) and (2.29), we find

^ o3iii> = 9tiu,a ~~ 9pi^PficT 9pp ai> = 0 • (2.30)

Using the relation 9n\gvX = 8” and (2.30) we find

= 9 Z + 9pur ^ + g " r ap = 0 . (2.31)

Forming the symmetric and antisymmetric parts of (2.31) yields

V,a('“'1 = o = a(? + a''”')rf„, + a(w) rg,„ + aw r^, + aMrf„i, (2.32)

v.a'""1 = 0 = a1 + a''"’ , + gM + jWr*,, a1"1 . (2.33)

Contracting (2.30) with g ^ and using the result

Tr ( i \ r l M <x) = (In |Det M |) A (2.34)

for a general m atrix M , yields the useful relation

r (U) = • (2.35)


2.1. Non-Riemajinian Geometry 19

Contracting (2.33) on v and cr and using (2.35) yields

^ A’r M = 7 = • (2-36)

Covariant derivatives with respect to the f connection are defined by

VpVx = v-x — vprp , = vX fl - npr'^ • (2.37)

v.®* = «i + e'r*., V „ r ' = »j, + d T * , ,

where

ua. = (’av<i t '-.v = . (2.38)

The hat and tilde notation has been introduced as a useful book-keeping device to

distinguish between the two types of hyperbolic complex vectors. In many instances,

explicitly real vectors such as vx = l /2 {u x + v x) are encountered. From (2.37) we see

that

V , y = i-J, + ■u'Tf/J/l) . (2.39)

The torsion tensor is defined by

rfH , (2.40)

and the torsion vector, is defined to be

r /‘ = r jv i • (2 .-U)

Adopting the notation V u = u“VQ we find

V|„V», - [u, v] = u ° v » r , (2.42)

where as usual

[u, v] = | (u V Q - t/a< ) . (2.43)

To simplify the presentation, we will suppress the hat and tilde notation from


2.2. Vacuum Field Equations 20

here on. The curvature tensor arrived at by parallel transporting a vector using the

T connection is defined by

2V[pV I/]Up = R xpflt/(T)vx , (2.44)

where

= ri.,„ - + r ^ r j ,. (2 .45)

VVe can form two contracted curvature tensors:

= R%p = r ^,A - r JA,„ - r ^ r “A + , (2.46)

and

Ppi, = R pPfiU = — FAt,i/t • (2*47)

From these we can construct two Hermitian curvature tensors, the generalised Ricci

tensor R ftu, and the torsion field strength Hpv:

Rpu = r ^ , A - - ( r ^ A),/i + rÂj.t/) - r ^ r “A + r*QA)r “„ , (2.48)

and

Hfu/ — r VtP . (2.49)

In addition to these curvature tensors, we can also consider the Hermitian tensor

F ^ as a another curvature tensor. With our non-Riemannian geometry in hand, we

can move on and derive the NGT field equations.

2.2 V acuum F ield Equations

It has been said tha t physics is a rt in a straightjacket. Never is this more apparent

than when trying to lay down the field equations for a theory. From the basic physical

objects gpv, r*„ and RxKflt, we can form any covariant Lagrangian we want. However,

the field equations that follow from the Lagrangian must pass a plethora of tests.

R ep ro d u ced with p erm issio n o f th e cop yrigh t ow n er. Further reproduction prohibited w ithout p erm issio n .


First, for weak fields and slow velocities they must recover Newtonian gravity. Second,

for weak fields and arbitrary velocity they must recover Einstein gravity. In addition,

the linearised equations should describe well behaved fields (ie. not tachyons or

ghosts) w ith spin less than or equal to 2 .

2.2.1 N G T -79

The most general, Hermitian symmetric action we can construct linearly from the

curvature tensors is [26]

» + bT^T,,) d lx . (2.50)

Varying w ith respect to g!“' yields

Rfu> + ciRfii/ + = o . (2.5i)

Varying with respect to T* leads to

The above equation reduces to the compatibility equation (2.30) if and only if

Vp + 2(a - b ) g ^ r „ = 0. (2.53)

This equation cannot be satisfied for any a or b. Thus, a consistent set of field

equations cannot be derived directly in terms of the Hermitian connection T. From

equation (2.53) we see that we need to further restrict T to have a vanishing torsion

vector r „ . This constraint can be enforced in the Lagrangian by employing a Lagrange

multiplier field W^. A suitable term is

Cr = y / ^ g g ^ W p T u . (2.54)



From (2.36) we see tha t terms such as

Cw = ( V ^ g g ^ 1) (2.55)

or equivalently,

Cw = v ^ r n w , . (2.56)

will also ensure F;i = 0 .

W ith T,, constrained to vanish, the action becomes

S = f y / = j ( R + f < rW W ]) > (2.57)

where the factor of 2/3 has been inserted for historical reasons. We note that the

action is invariant under the U{1) gauge transformation

1 Vft —*■ \\ fl 4- X'ft. (2.58)

The vacuum field equations th a t result from varying the above action with

respect to g, F and IF , are

Rp» + | h v „ , = 0 , (2.59)

9pu,o ~ 9pv^pfur ~ 9p p = 0 , (2.60)

and

( n/=517[H ) , , = 0 ’ (2.61)

respectively. The divergence equation, (2.61), which comes from the variation of IVft,

has been used to simplify the compatibility equation (2.60).

The Lagrange multiplier IFM can be eliminated by breaking (2.59) into sym

metric and antisymmetric pieces and forming the cyclic curl of the antisymmetric

piece:

Riftu) = 0 , (2.62)

R[iiu,p\ = 0, (2.63)


2.2. Vacuum Field Equations 2 3

where

= ^Wi.p + 4l + ^[p/‘],„- (2 .6 -1)

The collection of equations (2.60). (2.62) and (2.63) are the vacuum field equations

of NGT-79.

It is possible to add a cosmological constant term. A, and a skew-mass term ft

to the NGT action:

£ = (A -

= ( a </(„„) 4- (A + ^ / / 2 )%,<]) ■ (2.65)

The only field equation to change is (2.59), and it now reads

R f i u d" A 9iti> d* 7 ^5[kA] d~ () ' ^ O i i k U X v ) = — TjHi/t,p] ■ (2.66)

In the remainder of this thesis we shall drop the cosmological constant term, hut on

occasions we will consider the effect of having massive skew fields, r/[/a,].

2.2.2 N G T-95

For reasons that will be discussed in §2.5, Moffat recently proposed a modified version

of NGT [24] in which the skew degrees of freedom #[,,„] are massive and the Lagrange

multiplier IF), becomes a physical torsion. The derivation of the new field equations

is very difficult using the Lagrange multiplier methods we used for NGT-79. Instead,

Moffat uses the non-Hemiitian connection and the generalised curvature

t U W ) = » A .a - 5 ( w S u + » ? , , ) - w ~ K x + ■ (2-87)

in his derivation. The action is taken to be [24]:

s = J (r(W) - l-n2<r<JWtl] + dllx . (2.68)


2.2. Vacuum Field Equations 2 4

Variation of the action leads to the vacuum field equations2

R,u,{W) - ^ R ( W ) + V = 0 • (2-69)

and

Opv,a ~ Opv^pcr ~~ O p = 0 ? (2 ./0)

with Sliu given by

s , lv = + o[a,Aapaup» - \opu<){ap\o[ai'] + \ v p\vp^ . (2 .7 1 )

The metric compatible T connection is related to the IT connection via

+ K "n» • <~72)

where K*w is given in terms of the torsion vector, I F/t = ITÂ]. by

O p v k p o "b O p p ^ - a v = {jJppO an ~ 0 per Opt' ~ OpuO{ap\) - ( 3)

In NGT-95 the metric compatible connection no longer has a vanishing torsion vector

since

^ = ^[p*] ^ 6 . (2 .74)

The notation used here differs from Moffat's [24] as he defines to equal the com

bination n'''„ + 2 /(d - l)5*l'F„. In Moffat’s notation the T connection is not metric

compatible, but it does have r ;( = 0. Consequently, care should be taken when

comparing results quoted here with those in Moffat’s papers.

When cr = 0 equation (2.69) simplifies to equation (2.66) of NGT-79. For

reasons that will be explained in §2.5, the coupling constant a of NGT-95 is fixed to

equal —4/(r/(r/ — 1 )) in d spacetime dimensions.

•A detailed derivation can be found in Ref. [27].


2.3. Coupling to Matter 2 5

2.3 C oupling to M atter

Phenomenological matter couplings are introduced via the Lagrangian density

t,S,, . (2.75)

Here Tttl/ is the m atter energy-momentum tensor and S ' 1 is a particle current. From

here on we adopt natural units G = c = 1 . The field equations in the presence of

m atter read

) + Sflv — S~T/W , (2.70)

ancl

i f 11 ,<7 ~~ U pv ~ U p p ^ a u — 0 • (2.1 I )

The metric compatible connection is related to and Sfl by

, (2.78)

where A'*„ is now given by

Opi'hpo T fjppkot/ = ~ ”3 ”^ ) — UpirUpu ~ !lpi'!J[(7p\ j • (2 .(9 )

From the compatibility equatiou we may derive the condition

(v '= M [H ) i„ = 47r5" + l a(JUU,)W» ■ C--80)

Taking the divergence of the above equation we find

{ y / = g s n * = . (2.81)

I11 infinite range NGT-79 (<x = /x = 0), this tells us that the current density is

conserved. This allowed to be interpreted as the baryon an d /o r fermion number

of a body (using suitable coupling constants). Astrophysical bodies could then be


2.3. Coupling to M atter 26

assigned an NGT charge. I2, defined in analog].' with electric charge by

Z2 = J S ld:'E t = J y / ^ S W x . (2.82)

The entire phenomenology of NGT from 1979 until 1994 was based on the gravita

tional effects of the NGT charge. This thesis marks a major departure in the history'

of NGT as we largely dispense with the NGT charge. This is fortunate as NGT-95

does not allow a conserved NGT charge to exist. The absence of NGT charge in

NGT-95 will be demonstrated for the first time in Chapter 6 . In the context of NGT-

79 we may also question the wisdom of introducing NGT charge. The N G T charge

current S'1 is introduced through a coupling to the Lagrange multiplier IT,,. This is a

rather suspect approach as IT,, is not a physical field in NGT-79. IT,, does not appear

in the field equations. The situation is even worse in massive NGT-79 as massive

fields do not like to couple to conserved charges. Moreover, the original T connection

is no longer metric compatible when S * # 0. Rather than coupling physical currents

to unphysical fields, it makes more sense to insist on a globally metric compatible

connection. In this case, the three equations

hold globally, and serve to eliminate three skew degrees of freedom. For historical

reasons, solutions with NGT charge will be discussed in later chapters, bu t we no

longer consider these solutions to be physically sensible.

Turning our attention to the m atter energy-momentum censor, we can consider

many phenomcuologic.il sources. Some of the more interesting ones are scalar fields,

electromagnetic fields and perfect fluids. For a minimally coupled scalar $ we have

(2.83)

t » = W d v ® - \ ^ 9 {a0)da^ d ^ . (2.84)


2.3. Coupling to M atter 2 7

The electromagnetic energy-momentum tensor is given by [2S-30]:

T q J = - ~ { { g a J H ^ F m - K g ^ F l l u F n J - \ g a j ( H ^ H ilt. - F,,,,)-)]. (2.85)4ii 4

where the skew tensor = —HL,tl is defined in terms of the skew electromagnetic

field tensor Ftiu by the equation:

Hi* + = 'Ig.^g0''Fih . (2 .8 6 )

The coupling constant k provides a uon-mininnil coupling of electromagiu'tism and

gravity. The energy momentum tensor for a perfect fluid is given by [31]

Tu = {p + - v K ■ (2.87)

where it" = dx,L/d r is the fluid’s four-velocity and p and p are the usual internal

energy density and pressure, respectively. A perfect fluid can also couple directly to

the skew metric through couplings of the form

£ = \f~g{F)iuk + A V , , (2.88)

where FpUK is the skew field strength

F>tUK = g\[U>\,K T g\Kfl\,!/ d" /7 [|/k],;i , (2 .89)

and is the dual of the particle number current:

• (2 .90)

Varying the above action leads to the energy momentum tensor

T ad = A2 [d0nkF \ x - a + 1 6 7 rA2(./°./" - ^ •/"./„)) •

(2 .91)


2.4. Particle Trajectories and Causal Structure. 28

An energy momentum tensors of this type was first considered by Damour and Pinto-

Neto [32] in the context of a linearised version of NGT. Since NGT introduces a

spacetime torsion it is natural to consider couplings to intrinsic spin or fluid vorticitv.

The vorticity of a fluid is defined by

iâ/J = Map + a[aU0 j , (2.92)

where n„ is the fluid’s 4-acceleratiou, an = \u a, and uiaj = U[oj|- In order to

avoid nasty derivative couplings in the action, we only consider couplings to A

suitable Lagrangian is

£ = 4 • (2.93)

Varying this action leads to the energy momentum tensor

T a(} = 4npeaaKXujKX. (2.94)

In Chapter 8 we will use an energy momentum tensor of this type as the source term

for skew fields inside a star.

2 .4 P article T rajectories and C ausal Structure.

In general relativity, causal structure, geometry and particle trajectories are all deter

mined by one quantity - the spacetime metric. Because NGT employs non-Riemmanian

geometry the situation is not so simple. In the past there has been a tendency to

“throw away” the non-Riemannian geometry after using it to derive the field equa

tions and adopt a Rieinauuian geometry based on gû)- A closer examination suggests

th a t test particles respond to the full NGT geometry', and do not just follow geodesics

In the massless theory, three symmetric metrics play a role in defining the

causal structure [33]. These are

l T f i n u — 9 ( f i u ) i (2.95)

R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.

2.4. Particle Trajectories and Causal Structure. 29

,171*“' = g W . (2.96)

- „«'“•> . (2.97)

Primary among these is the metric as it is the metric tha t appears in the

d ’Alembertian or wave operator □. As in GR, the propagation of fluctuations along

caustics of □ defines the causal structure. In NGT this is not the end of the story

though as the other two metrics play an auxilliary role in determining the causal

structure. The compatibility equation (2.30) can only be inverted when ,m t/ / 0 and

:irntt 0. If either of these components vanishes, we find that the field equations

reduce to initial value conditions on a characteristic surface. Since characteristic

surfaces define the causal structure, we see that all three metrics combine to give

NGT a triple lightcone stucture1.

As a simple example of how g ^ ^ arises in hyperbolic wave equations, consider

the example of a minimally coupled massless scalar field with Lagrangian density

£ = s /^T jg^ O^Pd^P . (2.98)

The equation of motion for <1> reads

( s / ^ i g {“u)du<p) = 0 . (2.99)

Using the identity (2.35) in the above equation we find

d„ (g ^ d ^ p ) + Tp{pu) (g^ O ^ p) = 0 . (2.100)

Since P * — g ^ ^ d ^ P is a purely real vector (ie. has no hyperbolic complex imaginary

3I thank Jacques Legare and Mike Clayton for sharing their unpublished work on the causal structure.


2.5. Linearised N G T 30

part.). wo find from (2.39) that the equation of motion becomes

( ^ c U ) = ( g ^ V „$) = 0. (2.101)

Here vve have used the fact that the covariant derivative of a scalar reduces to an

ordinary partial derivative. Finally, using (2.32), we find

□<f* = = 0 . (2 . 1 0 2 )

We see that caustics of >01 define the light-cone structure for scalar modes. However,

in the geometrical optics limit, scalar perturbations do riot follow geodesics of ->mw

Rather, scalar perturbations follow paths defined by the symmetric part of the full

NGT connection

The scalar field example suggests th a t minimally coupled test particles will

follows paths defined by the full NGT metric gû. Paths are the “straightest possible"

trajectories, while geodesics are the “shortest possible" trajectories. A path is defined

by the parallel transport relation

a1' = = 0 . (2.103)

A spacetime will be path incomplete if the curvature invariants formed from /?^(/«a(T)

are singular at any point.

2.5 L inearised N G T

There are a number of different weak field limits we can consider in NGT. These

include the Newtonian limit of weak fields and small velocities, the linear theory of

weak fields and arbitrary velocity (less than the speed of light!), and the limit of

weak skew fields in a GR background. We will be most interested in the la tter two

linearisations as they have played a major role in the development of the theory in

the last few years.


2.5. Linearised N G T 2 1

For economy of space we can consider NGT-79 and NGT-95 simultaneously

as the former can be obtained from the latter by taking <7 = 0 at any stage of the

calculation. By expanding about a Minkowski space flat background we are able to

adopt a field theory perspective and discuss the gravitational fields tj{lul) and </[,„,] in

terms of gravitons and skewous.

We begin by expand g,tu about Minkowski spacetime:

Uiiis l)n/ "t- hfiu t ••• • (J. 10-1)

where is the Minkowski metric tensor: rj)U, = d iag (+ l. - I. - 1 . - I). We find from

(2.9) that

</» = , r ~ h>lu + ..., (2.105)

where h,u' = To first order of approximation, (2.80) gives

2 7r(e/-.)/2 (/_ l

° hltlXl = r({d - l ) /2)S,t + ~ 2 ’ (2-U)G)

where we have generalised to d spacetime dimensions. The factor 27r(,i 1 / 1 '((</ -

l) /2 ) is the surface area of a {d — 2) dimensional sphere. From (2.77) and (2.78), we

obtain the result to first order

I = ',/* + dfin t/ — h v ,a ) . (2.107)

and1 0 / 2

K i . = - 5 <7(«;hv - « > ; ) - -i r ( ( d - \ ) / 2 ) i^ S" ~ s"s ") ■ (2108)

The antisymmetric and symmetric field equations derived from (2.76) decouple

to lowest order. The symmetric equations arc the usual Eiustein field equations in

the linear approximation [34]. The skew equations are given by

r m *>M. ( - 4 >\ r r - , 4 7 r (< i_0 / 2 ^ f^ _ 2 4 7r( ' t - , ) / 2 ^(□ + vr)h[M - (a + d _ i J ^ + d _ 3 r((f/ _ 1)/2)^M -

(2.109)



If wo tako tho divergence of (2.109)r we get

d - 1 , TTr ( a 2 \tD H - + 2 H> " ( m + + r ((d - 1 ) / 2 )

d - 2 47r( d - 1^ 2

d -F 1

d - 3 r ((d - 1)/2)T[M'% (2 .1 1 0 )

wheread 2

T = - + — ) - (2111)

In vacuum we have

t UW^ + ^ 9 1 a /i2\V/L = 0 . (2.112)

In NGT-79, a — 0 and we find

□W ;1 = 0 , (2.113)

while in NGT-95, r = 0 (ie. a = —4 /d /(d — 1 )) and

IF,, = 0 . (2.114)

The above equations highlight the fundamental difference between the old and new

versions of NGT. The new version of NGT was constructed with the express goal

of keeping Wfl non-dynamical. In NGT-79, W u enters as a non-dynamical Lagrange

multiplier (it has no kinetic term in the Lagrangian), but simple manipulations of the

field equations suggest th a t IF,, obeys a wave equation. Moreover, when the preceding

analysis is repeated on a curved background, the curvature becomes a source for this

wave equation[35|. Because of this, it has been argued that the effective dynamics

of IF,, leads to unstable, runaway solutions. The argument proceeds as follows: the

wave equation for II),, (2.113), has the generic solution II), ~ R ~ l, where R is the

distance from the wave source. Inserting this in (2.109) then tells us th a t h [,,„] ~ R°

if /i = a = 0 [35]. The conclusion is tha t NGT-79 will generically have badly behaved

radiativee solutions, producing an infinite flux of skew radiation. Moffat devised the

new version of NGT as a way around these concerns. In NGT-95 the torsion vector


2.5. Linearised N G T 3 3

IVft remains non-dynamical by construction, and is instead determined by algebraic

equations, thus evading the argument leveled against NGT-79.

The preceding arguments using IT;1 are actually rather ridiculous in the con

text of NGT-79 as the Lagrange multiplier U 't does not even appear in the final

field equations (2.60)..(2.63). The concern about runaway solutions is more sensibly

directed a t equation (2.63). which linearises to read

(□ +/r)A[„„,pi = 0 . (2.115)

The above equation for hyiv\ contains third order time derivatives, and is thus likely to

have unstable solutions. Curiously enough, one of the main critics of NGT-79, Stanley

Deser, proposed a gravitational theory in (2+l)-diineusions that leads to third order

field equations [36]. There it could be shown that an extra cron formal symmetry

removed any badly behaved modes. In NGT-79 there is generally no additional

symmetry to protect the theory from badly behaved solutions. In the weak field limit

there is an additional gauge invariance under the transformation

' fy/n*] "b [/i,v\ i (2T 16)

but this symmetry is only approximate and does not survive at higher orders. Thus,

massless NGT is likely to exhibit radiative instability.

The linearised vacuum equations of NGT-79 can be expressed ;is

( □ + / i 2 ) / > A = 0, (2.117)

and

/tt^J = 0 . (2.118)

We see th a t the field strength FIW\ = hy,Utx] obeys a regular wave: equation. The

radiative properties of the skew sector are controlled by the rate of fall-off of F;i„\

and hy,„]. In particular, the rate of energy loss for an isolated body in NGT is given



by [37]

- r ^ j t ' ^ n . d n . (2.119)

where the integration is over a sphere of radius R in the wave zone, h, is an outward

pointing unit vector, and is the energy-momentum pseudo-tensor. The skew

contribution to is given by [37]:

' E = + . (2 .1 2 0 )

When the skew field is massless (/1 = 0) the wave equation (2.117) indicates tha t

F,u,a ~ R ~ l . This in turn indicates tha t the three skew modes not constrained by

(2.118) will generically behave as ~ R°. Putting all this in (2.120) tells us tha t

4kew °nly falls of as R ~ l. From (2.119) we see that this gives rise to an infinite

flux of radiation in the limit R —► oo. The above analysis indicates that NGT-79 will

generically have badly behaved wave solutions. In Chapter 7 we will see that massless

NGT has a t least some well behaved wave solutions in the case of spherical or axi-

symmetry. However, this does not prove that massless NGT is a healthy theory as

the analysis described in Chapter 7 excludes badly behaved solutions with ~ R°

from the outset. In general, massless NGT can be expected to have badly behaved

wave solutions. The theory is radiatively unstable.

W hen the skew field is massive, NGT-79 looks to be in bette r shape. The wave

equation (2.117) then tells us that F ^ x falls off exponentially with R. and again

falls off like R°. In this case the fast fall-off of the field strength is enough to ensure

a finite flux of skew radiation.

The behaviour of Moffat’s new NGT-95 is very different from massive NGT-

79. In particular, the mass term of NGT-95 is necessary for a different reason. In

(3+l)-dim ensions the linearised vacuum field equations simplify to read

(□ + //2) V ] = 0 , (2.121)


2.5. Linearised N G T

and

(2.122)

A nice feature of these equations is that they only contain second order time deriva

tives. It may appear that we can smoothly take the limit /t —» 0 to obtain a massless

version of NGT-95. This is not the case. When there is m atter present we find

The same is true on a curved background as the curvature gives rise to an effective

source term 2],,^ oc Rftl/Kx h ^ . Unless the divergence of 7j;i„| vanishes, the limit

/.i —> 0 will be singular. Thus, we see th a t both NGT-79 and NGT-95 require // ^ 0

to be viable theories, but for very different reasons. The main advantage of NGT-95

is th a t it is free from third order time derivatives.

Note added: Since the completion of this thesis, Clayton [38] has uncovered a prob

lem with the Hamiltonian formulation of NGT-95. Clayton decomposed the field

equations into (3+1) form by foliating spacetime into spacelike hypersurfaces. The

resulting Hamiltonian constraint formalism described six degrees of freedom rather

than three. This disagrees with the linearised treatment described above, and is due

to a linearisation instability. Moffat [39] has since shown that an additional constraint

has to be included in the NGT-95 action (2.68) for a sucessful Hamiltonian formula

tion. The constraint ensures th a t there are always three skew degrees of freedom. An

addendum has been added describing the new constraint. The main results described

in this thesis are unaffected by the change.

TT'/‘ — 32/T . (t/|, /2 'v * (2.123)


C hapter 3

V acuum Solutions

3.1 A (2-j-l)-d im ensional P rim er

Before moving on to consider NGT in (3+l)-dimensions, it is instructive to consider

how the theory behaves in (2+l)-dimensions. The lower dimensional world is par

ticularly interesting as it is free of gravitons in the linear approximation. The skew

fields can thus be studied on their own, without any interference from the symmetric

sector.

The lack of gravitons in (2+l)-dimensional general relativity follows from the

identity

Outside of sources T^,, = 0 and space is completely flat. Consequently, the most

general, circularly symmetric vacuum solution is (2 +l)-dimensional Minkowski space

However, a point source of mass M a t the origin leads to a conical singularity at

r = 0 , evidenced by the restricted range of the angular coordinate <f>:

RT, = . (3.1)

da2 = dt2 — dr2 — r 2d(f>2 . (3.2)

0 < 4> < 2 tt ( 1 - 2 G M ) . (3.3)

36


3.1. A (2+l)-dimensional Primer 37

The above metric can be used to describe spacetime near a cosmic string if we suppress

mention of the dimension parallel to the string. The metric (3.2) may be transformed

to one with a angular coordinate that extends over the conventional range [0 . 2 <r]:

ds2 = dt2 - - _ T/GA-/ r- - rdO '1 . (3.4)

From the geodesic equation. P -I- T'ftujx r'x J = 0 , we see that initially static bodies

(xc = 1 . ir* = 0) do not accelerate since r« = 0. Thus, general relativity dues not

reduce to Newtonian gravity in (2-fT)-dimensions.

Moving along to NGT, we shall seek solutions to the various versions of NGT

in the simplifying setting of static, circularly symmetric spacetimes, it:, tin; metric

for an infinitely long and straight cosmic string. The most general metric of this form

is given by

(j = j ( r ) dt ® dt — ct(r) dr 0 dr — r2d0 0 d.0 -F A(r) dt A dr . (3.5)

Starting with NGT-79, we find the divergence equation (2.G1) demands that

(3.6)/ y ^ L \ / r r T T I ’

Substituting this into a ( R u + S tt) + 7 [R-rr + Srr) = 0 yields

2 Pa , 7 - 1----— ----- ( 2 , VI \ ’ ' 1j rfr 4-1l )

so that

(« )

Substituting (3.6) and (3.8) into R ^, + = 0 leads us to

4 ra ' + /1212a 1 = 0 , (3.9)


3.1. A (2+l)-dimensionaI Primer 38

so that

a = ( l — 2GM -r j f i 2l2 ln(/ir)) . (3.10)

However, substituting the above expressions for a, 7 and A into the (tt) or ( r r) field

equation tells us that

lt2l2 = 0 . (3.11)

Thus, the only solution to massive NGT-79 is the GR solution with A = 0. This

makes sense as I comes from a Gaussian surface integral being related to a volume

integral over S l:

h iT tj{lT]'fide=Ist(lv=1 ■ (3-12)The above identification only makes sense for massless fields. The static circularly

symmetric solution confirms that massive NGT-79 should not couple to NGT charge.

The massless theory can couple to NGT charge, giving rise to the solution

(j = (1 + ^ 7 ] dt ® dt - ---- \ dr ® dr — r 2d0 ® dd — . —- ^ d t A dr . (3.13)V r1) 1 — 2GM y (i _ 2GM)r

The acceleration of an initially static test particle is then given by

I2 (I — 2GM)r = - - — --------------------------------------------- (314)

showing that NGT charge gives rise to a repulsive 1 / r 3 force in (2+1) dimensions.

The solution in NGT-95 takes the form

a - T ^ G M + 0 ^ ' (3-15>7 = 1 + 0(fx2l2) . (3.16)

Here is a modified Bessel function of the second kind. For small fir the skew field


3.2. Wyman-Schwarzschild Metric 39

takes the form

A =I

y/1 - 2 G M r1 + 2 7 , - 1 + 2 In \/3 fir

2x/l - 2GM3/rr~

- 1(1 - 2

For large fir the skew field has the Yukawa-like form

A1 - 2 G M V 2 y /I V Sy/x

(3.18)

(3.19)

where x = >/3fir/y/1 — 2G M . The symmetric metric functions have the asymptotic

form

frl~ir / 3 13 \ . .~ 1 - 2 G M C ' ( l 6 + i ; + - ) + W f K

3 13

1a =

1 - 2GM1 -

“/27T1 - 2G M

(3.20)

. (3.21)

The identification of / <is the NGT charge is rather misleading in NGT-95. There is

no conserved NGT charge in NGT-95 as the U{1) invariance of the Lagrangian lias

been broken by the K+H'A term. Moreover, the skew field is massive and thus unable

to support a long range force. We ouly make the identification of I with the NGT

charge of massless NGT as there is an approximate ( / ( I) invariance in tin; near-zone

where fir <sC 1 and \Vti « 0. This can be seen from equation (3.18) where the leading

term recovers the massless NGT solution. It should be stressed that NGT-95 will

generically not have the smooth massless limit seen here.

3.2 W ym an-Schw arzschild M etric

In this section we will derive the NGT analog of the static, spherically symmetric

Schwarzschild solution. In GR the Sclnvarzschild solution describes a black hole

of mass M with an event horizon a t r = 2A/. The Schwarzschild metric in areal



coordinates is given by

2M \ / 2 A/ ' " 1c/.sa*2 = ^ 1 ---- — j dt2 — ^ 1 ---- — ^ — r2d02 — r2 sin2 ddcp2 . (3.22)

The radial coordinate has the property that two-spheres of constant r have area 4?rr2

(hence the name “areal” coordinate). Since the spacetime is static, the m etric (3.22)

is invariant under time translations generated by the timelike Killing vector

< % = (1 - 2 A / / r , 0,0,0) . (3.23)

The norm of the timelike Killing vector

{l)e l% = 1 - — , (3.24)r

vanishes a t the Schwarzschild radius r = 2M . This tells us tha t there is an event

horizon a t r = 2M . The redshift between r = r0 and r = oo,

= T,— ~w/~ “ 1 ’ <3 25)y 1 — 2 A //r 0

diverges a t the horizon. The horizon is characterised by a finite surface gravity

, _ 1 0gtt 1K — ~ *2 dr y/ljttfjrr “ T m ' (X26)r=2M

which gives rise to Hawking radiation with tem perature T = hK/(2-k[}). T he horizon

describes a 2 -sphere of area .4s = l 6 r M 2. The laws of black hole thermodynamics

relate the surface area of the horizon to the Bekenstein-Hawking entropy of the black

hole: S = (Aro//i)(.4/4). The black hole harbours a central singularity at r = 0 where

the curvature diverges as M /r 3. Thus, the point r = 0 must be topologically excised

from the Schwarzschild manifold.

YVe will be interesting in studying how the NGT analog of the Schwarzschild

solution differs from the GR solution. In order to study the strong field regime near



the origin we will need an exact solution to the NGT field equations. As a prelude

to our discussion of the exact solution, we will present our original derivation of the

weak field solution. After we derived the weak field solution. Pierre Savaria directed

us to a collection of papers written by Wyman [42], Bonnor [43], and Vanstone [44]

th a t studied static spherically symmetric solutions in Einstein-Straus [4 5 ] unified

field theory. Their solutions reduced to our approximate solution for weak fields, and

could be shown to provide exact solutions to the field equations of massless NGT-

79. The generalisation of these solutions to massive (// ^ 0 ) NGT is considered in

Chapter 6 . The next Chapter is devoted to a detailed study of the so-called Wyman

generalisation of the Schwarzschild solution. We will see that the Wyman solution

does not describe a black hole as no part of the Wyman-Schwarzschild manifold fit's

inside an event horizon.

3.2.1 The approxim ate solution

The most general, spherically symmetric metric for NGT was found by Papapetrou [46]

to bef

:

7 (r) w(r) 0 0

—w(r) — a(r) 0 0

0 0 —3 (f) / ( r ) s in f l0 n t f - \ a / ^ ^ 2 ,

\

0 — /(r )s in f l — f3(r)sin~0 y

The determinant of the is given by

(3.27)

x/ ^ 7 7 = sin0 (a 7 — w'2) l/'2(02 + f 2) l/2. (3.28)

We may choose the radial coordinate so that 0 ( r ) = r2 without loss of generality.

However, r should not be confused with the areal coordinate used to describe the

Schwarzschild metric. The NGT analog of the areal coordinate is given instead by

f = ( 0 2 + f 2) l/1-

Equation (2.61) yields the following expression for w(r) in terms of the other



metric functions:

The solution with I2 = 0 was first found by Wyman [42] in the context of Einstein-

Straits Unified Field Theory. Similarly, the general solution with I2 ^ 0 was found

by Bonnor [43] and Vanstone [44] for Unified Field Theory. Papapetrou [46] found

what was to become the accepted expression for the static, spherically symmetric

metric of NGT during the period 1979-1994. Papapetrou considered the special case

f ( r ) = C r2, Lu'(r) = l2/ r 2. In most cases the constant C was taken to vanish since

the solution with C ^ 0 is not asymptotically Minkowskian.

The Wyman, Bonnor and Vanstone solutions remained largely forgotten until

a detailed study 1 of radiative solutions in NGT was undertaken [37, 47]. By using

the Boudi-Sachs [48, 49] formalism we were naturally lead to discover the asymptotic

form of the Wyman-Schwarzschild metric. The Bondi-Sachs formalism finds solutions

to the field equations in term s of a power series in 1 f r , with the coefficients taken to

be functions of t, 0 and 0 . In the simplified context of static, spherically symmetric

spacetimes the method reduces to a straight power series solution in 1 / r . In terms of

such an expansion, the metric functions can be written as:

(3.30)

r r 2(3.31)

w (3.32)

(3.33)

/ = f lT + f i + — + . . . .r(3.34)

‘See Chapter 7 for an account of this work


3.2. Wyman-Schwarzschild Metric 4 3

The second expression for w comes from employing (3.29). which relates w to the

other m etric functions. The subscripts on the coefficients refer to the order in 1 / r of

the term in orthonorinal coordinates.

Since we are interested in the Wyman solution we shall take /- = 0 so that

w = 0. The vacuum field equations then read:

0 = (log 7 )" ~ \ (log 7 )' (log ( ^ ^ + .4' (log 7 ) ', (3.35)

0 = .4 '( lo g a ) ' - 2,1" - ((.-I' ) ' 2 + (£ ') '-) - (log 7 )" + ^ (log 7 )' (log , (3.36)

where

4 =2

The usual G R expressions are recovered when .4 = 2 log r and B = tt/2. The constant.

.4 = ^ log(r'' + f '2) . B = arct.au( r / / ) . (3.39)

q comes from the Lagrange multiplier field W# — Zq cos 0/2. W ith w = 0 , the field

equation (2.61) is automatically satisfied.

Inserting the expansions (3.30)..(3.34) into the NGT-79 field equations leads

to the following expressions at leading order:

0 = (472 — 7 j — Oi7i) + . . . , (3.40)

0 = _4(q, + 7 l) - <12 + + llQ -'» + ^ + 8n*> + . . . , (M l)r

, . \ , (2 a i7 i + 2 7 2 + f \ + 2 cvf)0 = ( a t + 7 t) H--------------------------------------h . . . , (3.42)T

n . 4 / t 7 i / i + 7 o ! i / i „0 = q 4--------1---------- 1- . . . . (3.43)r r1



These lead immediately to the results

<7 = 0, /i = 0 7 , = —a j 7 2 = 0 . (3.44)

We see that q, and hence the Lagrange multiplier field W#, are associated with Pa-

papetrou’s unphysical solution where / = Cr2. By taking the Newtonian limit of the

solution obtained thus far, we can identify the constant 7 1 to be 7 1 = — 2AI where

M is the mass. Continuing the expansion to higher orders then leads to the following

form for the metric:

Unfortunately, there is no way to fix the constant / 2 using this method. However,

we do know that it has the dimensions of [Mass]2, and we also know that the only

dimensionful constant available is M . This means we can write / 2 = s M 2 where s

is a dimensionless constant. We later found out that the above expression for the

metric is in fact the large r expansion of the Wyman-Schwarzschild metric [42], as

first derived by Max Wyman in 1950.

3.2.2 The exact solution

The exact solution is obtained by making clever changes of variables in the field

equations. Introducing the variables

1 +2 M 3 A //| 12 M 2f 2

r 5r 5 5r6r (3.45)

£1

/

(3.46)

(3.47)

x = (3.48)



simplifies the field equations greatly. In terms of these variables the field equations

(3.35)..(3.38) become:

2.4" - (.4')‘- + (B ' f + .4 'ln (.r/7 ) = 0 (3.49)

(dry)" + ^(ln 7 )'ln(:r7 )' = 0 , (3.50)

q" + ^ 'I n ( x 7 )' + 2(i + C ) ^ l = 0 . (3.51).’ r

The first of these equations is redundant as (3.50) and the complex equation (3.51)

are together equivalent to three real equations for the three metric functions % a and

/ . Defining

(3.51) simplifies to read

A(r) = {y ' f - , (3.52)\ y

= 0 . <3.53)dv dv

where q + v — p and v = ln7 . This equation can be integrated once to yield

( £ ) ’ A + 4(i + C)ep = c i , (3.54)

where ct is a complex constant. We may write c t in the form ci = 4A/*( 1 4- is) where

M and s are real constants. Equation (3.50) tells us that

dX- = 0 . (3.55)

Asymptotically Minkowskian boundary conditions require that

/ —► const, as r —* oc, (3.56)


3.2. Wyman-Schwarzschild Metric 4 6

which in tu rn requires C = 0. Integrating (3.54) we find

4 Mv = ——:arcsinh

v^Te x p (-p / 2 ) (3.57)

Taking the real and imaginary parts of this equation we arrive a t the Wyman-

Schwarzschiid solution:

7

c*

- eM 2(i/)2e - ' ( 1 + a2)

(cosh (av) — cos (bv))2

(3.58)

(3.59)

/ =2 M 2e "(sinh(at') sin(6 ^) + s ( l — cosh(ai/) cos(bv))

(cosh(az/) — cos (bv))2(3.60)

where

a = vT + s2 + 1b =

V l + s 1 - 1

and v is given implicitly by the relation:

(3.61)

•ir =2 M 2e "(cosh (av) cos (bv) — 1 -t- ssinh(ai/) sm(bv))

(cosh(ai/) — cos (bv))2(3.62)

For 2M / r < 1 and .sA/2 / r 2 < 1 we find bv —> 0 and av —*• 0 so tha t (3.62) becomes

v '

v 1-T

1 , 1 - s 2 -1 + — V " + ___"— I/6 + . . .

12 360(3.63)

This equation can then be solved iteratively for v:

2 A/ 2 A/ 2 8 A/ 3 4 A/ ' 1 32M 5 s2 M 5 32M6 2 s 2 A/ 6v =

r- 3r:: r -1 tjr 5 1 5 t-5 3 j-6 + v r +•o ru(3.64)

R eproduced with perm ission o f the copyright owner. Further reproduction prohibited without perm ission.

3.2. Wyman-Schwarzschild Aretric 47

Using this expression in (3.5S)..(3.60) yields the following asymptotic form for the

Wyman-Schwarzschild metric:

2 M s2M* 4 s2il/67 “ + + ' ( 3 ' 6 5 )

/ 2 A/ 2s2 M [ 7s2 A/ 5 87s2M li \ ~ ‘ft — I 1 ----------1-----r —j--- 1------ :---1-----— 7 ---- h . . . I . (3.GG)

\_ r Or'1 9/-’ lor6 J

s M 2 2s M 3 6s M 11 - — + — + ' s ? r + ' " • (M T)

where the higher order terms in M / r include higher powers of .s also. We see th a t the

above expansion agrees exactly with our perturbative derivation in §3.2.1 if we make

the identification s = 3.s.

For experimental testing, it is often convenient to study the metric in isotropic

coordinates. The metric can be re-cast into the isotropic form,

ds2 = y{r)dil — d(r) (d f2 + f 2dQ2 + f 2 sin" flr/r/r) , ■ / ( f ) , (3.G8)

by performing the coordinate transformation

. r I M I 2M s2M l r = - I 1 ------- + 1 / 1 ----------+

2 1 r V r 18r‘,.-t

, 21A/1 + — h

0 r(3.G9)

The transformed metric functions are given for f > M /2 by

(1 - M /( 2 f ) ) 2 S - i f i ( 2M 67A e \(1 + A //(2f))! 90f5 V t 28r" 7 ’ ’’ ’

a/ A/V‘ s2MA ( 1 1 A/ 2 A/ 2 \

= 1 + — - - r n r 1 + — + - ^ + - , (3.71)V 2r ) 18r'* \ r r l )

- s M 2 / 2 M 8 M 2 7A/ 3 \^ — ( 1 + — + i f r + i o F + - j • (3-72)

We see that for large f the NGT corrections to the Schwarzschild solution are small


3.2. Wyman-Schwarzschikl Metric 48

for any |.s| < 1, and can be arbitrarily small if s is close to zero. Clearly, experimental

predictions such as the bending of light by the sun and the perihelion advance of

Mercury will only be minutely affected by having $ non-zero.


C hapter 4

A n alysis o f th e

W ym an-Schw arzschild Solution

In this chapter we will perform an in depth study of the Wyman-Sohwar/schild so

lution. A great deal can be learnt about a gravity theory by studying its spherically

symmetric vacuum solution. We will see th a t while there is dose agreement between

the W yman solution and the Schwarzschild solution for weak fields, the two solutions

differ dram atically once we reach the Schwarzschild radius.

4.1 T he W ym an G eom etry

In deriving the Wyman solution we suppressed all mention of hyperbolic complex

geometry and worked instead with the real functions nr, 7 and / . Restoring the

hyperbolic complex element ut and adopting a modified form for o, the Wyman metric

can be written as

g = j d t 0 dt — adv 0 dv — r2d0 0 dO — r l sin2# difr 0 r/r/> -f u; / sin# d.0 A d<\>. (4.1)

The hyperbolic complex vierbeins corresponding to (4.1) are given by

e? = (>/7 , 0 ,0 ,0 ), (4.2)

49


4.1. The Wyman Geometry 5 0

c“ = (Ot V ST0 , 0 ),

e*e = ( 0 , 0 , P , ~ ) ,2 p„ u f sin 0 .

c* = (0 . 0 , — —— , psin 0),

wherer2 + y/r* + f 2

The line element formed from is given by

dl2 = 7 dt2 — a d v 2 — r 2d 01 — r 1 sin2 0 d(f>2

The metric functions a and 7 can be compactly expressed as

7 = e2 A /V 1 + s-

r 2 (cosh(azv) — cos{bu)) ?

a =7 r 1

a / 2 ‘

Here r is defined by r* = \J rx + / - 1.

The causal structure is captured by the metrics

1 mnu

( 1 0

0 - a

0 0

VO 0

( 7 0

0 — a

0 0

VO 0

0

0

—r 2

0

0

0

- f ' / r 2

0

0

0

0

—r 2 siu2 0 j

0

0

0

r ’s u rf l/r 2 j

'I t is interesting that f shows up naturally in the metric = e“c^r/a6:

p = 7dt ® (it — adz/ ®dv — r~d0 ®d0 — r2 sin'0 d<f> ® dtp.


(4.3)

(4.4)

(4.5)

(4.6)

(4.7)

(4.8)

(4.9)

(4.10)

(4.11)

4.1. The Wyman Geometry 51

and

/T>7 / ( r ‘ - p ) 0 0

0

0 \

0

0

- r V C r 1 - p )

0

0V o 0 r 's i n*0 /r* /

Time-orthogonal hypersurfaces have proper spatial volume

(cosh(«//) — cos(6 //))--oc J-oo (a:

Circles of constant {£, r , (p) have proper circumference 'Inf, and surface's of constant,

{ t.r} have proper surface area 4/rr2. Since the proper surface1 area vanishes at r — 0 ,

we see that f = 0 is topologically a point.

Minimally coupled test particles will follow paths defmeel by

where u,L is the four-velocity of the particle. The non-vanishing symmetric connections

are given by

sin 0 cos 0,

(4.15)

(4.16)

(4.17)

(4.18)

(4.19)

(4.20)

(4.21)

(4.22)

(4.23)


4.2. Curvature and Redshifts 5 2

From (4.14) and (4.15)..(4.23) it is a simple m atter to show that radially directed test

particles can pass freely through the surface r = 0 , so the spacetime extends below

r = 0 .

The components of the Killing vectors. for the non-Riemannian geome

try of NGT are given by

+ 9au + W (K)f = 0 . (4.24)

For the metric (4.1) the Killing vector which is timelike a t spatial infinity is given by

< % = (?, 0 .0 .0 ) . (4.25)

The norm of this Killing vector is given by

(V (% = 7 - (4.26)

4 .2 C urvature and R edsh ifts

The deviation from Riemannian geometry is controlled by the size and slope of the

invariant skew potential

s ’ = S|H9M = -pqrji = F • (4-27)

At large r / M this quantity approaches

s2M'1 / 44 / \ /iooxS - - - & T ( 1 + — + ---) ' (4'28>

and the Wyman geometry is very' similar to the Schwarzschild geometry. In the strong

field regime 7 « 1 ( - 1/ » 1 ) and we find

/ ~ \ 2/(a—1)7 = ( ^ 7 ) ( l + s*)l/{2- 2a), (4.29)


4.2. Curvature and Redshifts

n (4.30)

(4.31)

/ ~ - 4 ■UV0" 0*' (sin(/n/) + scos{bu)) (4.32)

(4.33)

Both r 2 and / oscillate an infinite number of times as v —• —oo. In contrast, r is a

monotonic function of v. The zeros of r . r„, occur at

, (2/t — 1)v{r n) = ------- :------ arccosb \ / 1 4- s-2

and the zeros of / . occur at

K / n ) = -(2 n - 1 )

arcsiu 1*1

V I + S-+ 7T

where n is a positive integer. For small .s the tirst zeros occur at

27T* ( / , ) = ~ •si .s

Similarly, r '1 equals f 2 for the first time a t

7TV —

(4.34)

(4.35)

(4.36)

(4.37)

For economy of notation, we will refer to the first zero of r, r t , as r = 0 , even though

r has an infinite number of zeros. The areal radial coordinate, r, does not vanish

until v —*• —oo. For small I#I the surface r = 0 occurs at

(4.38)



so studying the metric in the neighbourhood of r = 0 corresponds to studying the

region just inside the Schwarzschild radius f = 2M. Taking |s| 1 we find the

invariant skew potential near r = 0 is given by

We see that S is of order one below the Schwarzschild radius and essentially zero

above r = 2 A/, with rapid 1 /r* fall-off at spatial infinity. The geometry of spacetime

at and below r = 2A/ is dominantly non-Riemannian.

Just, inside the Schwarzschild radius (r = 2M, r ~ 0) the Wyman metric is

described by

7 = 7u + ( { / ) + 0 ( ( r / M ) ' l , (4.40)

- = 470(1 y {s!)) ( j j ) V70(1 y ’ (s2>) ( j ^ W / A / ) * ) . (4.41)

/ = u * (4 - i f + + C V ) ) + M + * ' 1 + + 0 ( r 1) . (4.42)

where 7 0 is given by

7 o = exp ( ~ + 0 (« 2) ) • (4.43)

The above expansion was for |s | < 1, but expansions with |s| > 1 can also be found.

The expansions with |s | > 1 retain the same form as a series in r /A I , but the coeffi

cients change. For example, 7 0 becomes

7 0 = exp I -7T.'2 tts2

(4.44)

The most im portant feature of the solution can be seen from (4.42), where it is clear

that the limit |s | —► 0 does not recover the GR metric with / = 0. The product bv

approaches — 7t / 2 and cannot be made to vanish as |s| —► 0. The only way to have


4.2. Curvature and Redshifts

log(z)

60 •

40 -

50

20

30

10

0 ■Q

-2 -1 0 2log(s)

F ig u re 4.1: The redshift between r = 0 and r = oo is plotted as a function of s on a logarithmic scale. The circles are the exact, uumerical result, while the d.ished line is the analytic expression for small $ and the solid line is the analytic expression for large s.

/ = 0 globally is to put M = 0 whereby both the GR and NGT solutions reduce to

Minkowski space. Thus, the limit |.s| —*• 0 is uon-auulytie for 2 M / r > 1 and analytic

for 2 M / r < 1 .

The redshift between r = 0 (f ~ 2A/) and r = oo is given by

*o = - 1 ■ (4.45)V7o

If the skewness constant s is small, is given by

while if 6- is large, z0 is_given by

2 0 = expv\s\ - s ( l + 2 C - * ) 4- Q(s°y

2 t ts 2- I , (4.47)


4.2. Curvature and Redshifts 56

Both of these expansions give excellent approximations to the exact numerical

results in their respective regions of validity. The large s expansion is very accurate

in the range 1 < |.s| < oc, and the small s expansion is very good in the range

0 < |.s| < 0.5. Even in the intermediate region 0.5 < |s| < 1 both expansions are

good to within ~ 10%. The utility of these expressions is m ade clear in Figure 4.1.

The Schwarzschild radius is not a surface of infinite redshift in NGT. More

formally, the norm of the timelike killing vector, 7 , does not vanish for any finite

value of the radial coordinate f . In Figure 4.2. we display the norm of the timelike

Killing vector as found for the Schwarzschild and Wyman solutions near r = 2A/.

0.3

0.25

0.15

0.1 -

0.05

r/M0.5 1.5 2.5

F ig u re 4.2: The norm of the timelike Killing vector for N G T with s = 1 (solid line) and G R (dashed line).

While it is true that the Schwarzschild radius is no longer a surface of infinite

redshift in NGT, it is by no means a benign region of spacetime. The redshift still

skyrockets as we approach f = 2M . In addition the spacetime becomes highly curved.

The fate of a hapless spaceship encountering the surface f = 2M would not be pleasant

- it would be the spacefaring equivalent of a bug hitting the windshield of a speeding

car. This is very different to the situation in general relativity where one can pass

through the event horizon of a large black hole without even noticing a jolt. In GR


4.2. Curvature and Redshifts 57

the curvature at the horizon is of order 1 /A /2. For a very massive black hole the

curvature at the horizon can be very small. In NGT the curvature near f = 2A/

is exponentially larger by a factor of e2 lr'l*i for |s| «: 1. This is because / and its

derivatives become very large at and below f = 2M.

In the weak field region where 2A //r < 1 , the curvature is very similar to what

one finds in GR. For example, the generalised Kretschmanu scalar is given by

Just inside the Schwarzschild radius, (r = 2 A/, r % 0) all the curvature invariants

* 1.97 1.98 1.99 2 2.01 2.02 2.03 2.04 2.05r/M

F ig u re 4.3: The ratio IiÂ'ngt/ ^ gr) jus a functions of the areal radius f when

(4.48)

(4.49)

The other invariants have similar forms such as

_ 96APvpnv — r 9 (4.50)

3 = 1.0.



scale as For example, the Kretschmann scalar is given by

. (4.51).S'1 (1 -f* I 17T/ 2 -f- . . .) (2-K + 4 s + S27 r/4 + ..

A = — ------L- ;—----------- - exp 12 9 M 4 H

*>r~|s|il/ 2

The Kretschmann scalar is displayed graphically in Figure 4.3 as a logarithmic ratio

of A 'nct ancl A g r = 48M 2/ f 6 in the region near the Schwarzschild radius.

Near r = 0 the line element (4.7) takes the form

- 0 . •■>“ “ ' ( © - ” *• - 5 ^ « , ) • " - ) - « • '■(4.52)

and the curvature invariants diverge as

1 / A / \ 2(«+»/(“- 0

i p ( r ) ' (453 )

For s2 < 2400 the curvature singularity in NGT is actually more severe than it is in

GR2. A photon completes the journey from F = e « C l t o T = Oin the finite proper

time2 M ( e _ x

T ~ a (H - s 2 )l/(2«-2) \ 2 M J ? (4‘°4^

which proves tha t infalling m atter will encounter the singularity in a finite proper

time. However, before r = 0 is reached, a rocket with powerful enough engines will

always be able to escape to safety (given th a t it can survive the crushing tidal forces).

This is contrary to the situation found in general relativity where all trajectories inside

a black hole must term inate a t the central singularity.

The surface gravity diverges at f = 0 since

2 dr a ( r )-y rr=0

1_2 '~ — - (4.55)

If we were to make the same identifications used in general relativity, we would con

clude that the W yman solution is characterised by zero entropy (area) and infinite

*Our original suggestion [40} that the vacuum Wyman solution is non-singular was wrong.

R ep ro d u ced w ith p erm iss io n o f th e copyrigh t ow n er. Further reproduction prohibited w ithou t p erm issio n .

4.3. Causal Structure

tem perature. Clearly, such an identification is nonsense tvs the infinite curvature at

r = 0 invalidates any semi-classical reasoning. The possibility that the Wyman solu

tion radiates Hawking quanta would have to be explored using a complete quantum

theory of gravity. On the other hand, it does appear that information can be stored

in the Wyman solution. The no-hair theorems of general relativity do not apply.

For example, the Wyman solution can support static scalar hair as the scalar wave

equation1 1 cF‘F

0 . (4.56)

has the static solution

7 dt~ a dv1

$ — (ho + . (1.57)

The scalar field is regular everywhere except at its singular source, r = U. The infor

mation contained in m atter which encounters the singularity can remain encoded in

forms such as baryou-number hair. In this way, NGT might be free of the information

loss problems found in semi-classical general relativity.

The central singularity tha t lurks at the heart of the Wyman solution is not

naked as no signal can propagate away from r = 0. The reashift from f <C 1 to f — oo

is given by/9/\ f \ •/(“-■)

c ( f ) ~ (1 + . (4.58)

Thus, infinite redshift comes hand in hand with infinite curvature in NGT. This seems

to suggest tha t event horizons in NGT will require a quantum gravity description,

just as they do in string theory [50].

4.3 Causal S tructure

While we have found th a t r = 0 is a singular point, it is unclear that it lies in a

physical region of spacetime. Some very strange behaviour is to be anticipated just

below r = 2M. All three metrics used to describe rnassless NGT’s triple light-cone

structure become degenerate just inside the Schwarzschild surface. The first metric

to break down is 3rn ^ , which becomes degenerate when f 2 = r'1. For small s this


4.3. Causal Structure 60

occurs a t

(4.59)

The other two causal metrics become degenerate shortly thereafter at r = 0, which

corresponds to

In the limit s —> 0, all three metrics become degenerate simultaneously a t r = 2M.

Although the Schwarzschild surface is not an event horizon in NGT, it is the outer

boundary of what we might term a chronology horizon.

We have already seen that radial trajectories pass freely through the infinite

sequence of chronology horizons which surround r = 0. Signals can propagate freely

from r = e to r = oo for any e > 0. However, it appears th a t close _i time-like

curves can occur for particle trajectories whose causal structure is described by im tlv

or 2 ^/n/- Only the radial direction remains spacelike throughout spacetime. Below

r 2 = 0 the cyclic coordinates 0 and 0 become time-like, and particles may follow

closed time-like trajectories. In general relativity, closed time-like curves are usually

associated with solutions possessing non-zero angular momentum. In NGT, the role

of angular momentum is played by the non-vanishing torsion, r ^ j .

At this stage it should be emphasised th a t we are describing a pure vacuum

solution. It is unclear that the globally vacuum Wyman solution will ever arise as

a physical solution in NGT. In general relativity, the Hawking-Penrose singularity

theorems, and associated no-hair theorems, tell us th a t the vacuum Schwarzschild

solution has to be taken seriously. These theorems do not apply to NGT. Moreover,

the vacuum solution tells us that a finite radial pressure can always support a fluid

element a t the surface of an NGT star, no m atter how compact the s tar. This is

not true in general relativity. Perhaps a version of Hawking’s chronology protection

conjecture holds true in NGT. For static spherically symmetric systems in NGT,

chronology protection would require all fluid bodies to have radii greater than 2 M ( 1 —

(4.60)

|s |tt/16).


C hapter 5

E lectrovac Solutions

In this chapter we consider the NGT generalisation of the Reissner-Nordstrom charged

black hole [51]. In general relativity the metric for a charged, non-rotating black hole

is given by

ds2 = ^1 — ^ dt2 - dr2 - r2d02 - r2 sin'2 Od<l>2 . (5.1)

T he electric field is given by

F,o = § . (5.2)Ti

T he Riessner-Nordstrom black hole has an outer event horion at

r = M + \]M'> + Q2 , (5.3)

so long as M > Q. For Q > M the Riessner-Nordstrom solution describes a naked

singularity. Solutions with Q — M are referred to as extremal.

The NGT-79 generalisation of the Riessner-Nordstrom solution was found by

M ann [30]. Like the Wyman solution, we will find the Mann solution to be free of

event horizons of finite size. The origin will again be a point of infinite curvature and

redshift where quantities such as the electromagnetic energy density diverge. One nice

feature we discovered about the Mann solution concerns the total energy contained

in the electromagnetic field. In general relativity this quantity is impossible to define

61


5.1. The Mann Metric________________________________________ 62

globally duo to the even horizon. In NGT we find the nice result th a t the total energy

contained in the electromagnetic field is finite

5.1 T h e M ann M etric

For static, spherically symmetric spacetimes the electromagnetic field has com

ponents:

Fio = E(r), iv) = £?(r)sin0, (5.4)

all other components being zero. From (2.86) it follows th a t and from the

equation:

Fû,(t 4" F-Jcr t 4" Fa^ u — 0 , (5-5)

we find th a t B(r) is a constant, which corresponds to the magnetic charge. We shall

assume in accordance with Maxwell’s theory tha t the magnetic charge is zero. We

have_E

0 7 — w

The determinant of the glw is given by

^ l0 = (5-6)

sfFg = sin0(«7 - u»2 ) 1 / 2 ( / ? 2 + / 2) 1/2. (5.7)

The solution to Eq. (2.61) is

2 1'IQ' 7

W = 0 i + p + ’ (0,8)

where Z2 is a constant of integration which is identified with the NGT charge. The

Maxwell equation

- K y / ^ g M g ^ F ^ ) = 0 , (5.9)

has the solution:Qp2 \ _ Q py/a j — w*

p1 + K1 0 ) - (p2 + K 2 /.l) ’- ( ? ) ( (5.10)


5.1. The Mann Metric 153

where Q is the electric charge of a particle and

2 J2 ,p — J + / •

For I2 = 0, we have

E = - /on P

The field equations for the static spherically symmetric case take the form:

' } B ' - (3 A1 + '

2 a

3 ( E l 2+ 2 k Q

p- \ w

l a

I ( El2Y ( Ql2 2kS 'p2 \ w ) ~ \ p 2 -f k2B )

-.4" + ±(lna)'.4'- i p ' ) 2 + (B')2] - iln (7t/)" + iin (7t/)'h>(^j)'

((1 - COP')2 4- (B')21 + i(ln 7 )'li._7_2 a

+

Here, .4, B and U are defined by

.4 = lnp, B = tan 1 ,

(5.12)

(5.13)

(5.14)

(5.15)

.(5.10)

(5 . 17)


5.1. The Mann Metric 64

U =11 + p1

= 1 -

won'

C is a constant and A' — dA/dr.

It is convenient to use the notation [42, 44]:

x = — , y — 7 U, exp (q) = exp (A + tB) = / + id.Q

Then the field equations can then be written in the form:

2 A" - (A ' ) 2 + { B ' f + A'ln(z /y) = 0,

where

(lnv/)" + ^(lny)'ln(xy)' = - F ,I X

q + k i n W + 2(i + = ( g - 1 F .2 x \ x J x

Let us define:

G = —8 /cexp (—q)

Mr) = (y'f

Ql2p2 + K*l*

Q l2 i 2

p2 + k'T1

Then, (5.22) can be written as

,rf2p , dXdp .. , .7z* + I h d z + + )exp^ ~ 2 GcxP(P)»

where q + z = p and 2 = Iny. An integral of (3.25) is given by

g ) ’ A + 4(i + C)eXp(p) = / 2 0 * 3 ® * + c„

(5.18)

(5.19)

(5.20)

(5.21)

(5.22)

(5.23)

(5.24)

(5.25)

(5.26)

(5.27)


5.1. The Mann Metric £ 5

where ci is a complex constant. Equation (5.21) requires that

dX , , „

— = 4exp(c)F. (5.2S)

We shall consider the solution for which cj = Ao(l + is) when' Au and s are real

constants. Asymptotically Minkowskian boundary conditions require that

/ —► const, as r —* oo, (5.29)

which in turn requires C — 0. This also ensures that we obtain the Reissner-

Nordstrom solution when / = I1 = 0.

When Q = 0 . the above equations reduce to the Vanstone solution [-14]:

/ + iP —

iAo4//

(1 + t.s)cschJ[ \y T + fs / 2 1 n7/]. (5.30)

= ( r + p j »• <*=— ^ — • < - !1>

- - % (5-32)V ^0

where Ao = 4A/ 2 and y is an arbitrary function of r . If we set the NGT charge I,'1 to

zero and make the coordinate choice p — r 2, the Vanstone solution goes over to the

Wyman solution described in the previous chapters.

For Q ^ 0 and I2 = 0. (5.23) and (5.24) simplify to

G = 0 , F = Q 2 - co n st., (5.33)

and (5.28) can be solved to yield

A = \Q l exp(c). (5.34)


5 .1. The Mann Metric 6 6

From (5.27), we then find

arcsinh e x p (- ,; / 2 ) y ^ a r c ta n h y j l + 4Q2y/X0 + const.. (5.35)

The various constant must be set from the Newtonian limit and by choosing realistic

boundary conditions. For r —► oc these must include:

a —► 1 , 7 —► 1 , ft —► r2. (5.36)

Choosing ft — r2 we find

y = 7 = exp(i/), a = (7 / ) 2 ( / 2 + r<) 7(47 Q2 + A0) ’

(5.37)

and

An/ = I — 1 (coshi>a — costpb) [s(l - coshâcos^6) + sinhâsini^], (5.38)

27

where

'/’« = 2 a ^ a r c s i n h - a r c s i n h ^ = â(« «-► 6 ), (5 .3 9 )

and

Moreover, we have

Jvr+Z+I,

Ao = 4(A/ - Q 2).

(5.40)

(5.41)

T he function u is given implicitly by

7*

2exp(t/)(cosh#a - c o s ^ )2— = [ssinht/>0 s in ^ - (1 - coshâcos^)]. (5.42)A n


5.2. Analysis o f the Solutions 67

The areal radial coordinate r is given by

* = ^ = (5.,3)v COSll t'a — COS’Vb

The solution for the extremal case Q — M cannot be obtained from the above

expressions, and must be derived separately. The extremal solution reads:

r t W r 4 ,r my = 7 = exp(t'), ft = ■ - , (5.44)■i t Q-

and

where

q 2

/ = -— (1 — cosh^cos't/.*)(cosht/; - cos^ ) ~ 2 * (5.45)2 7

6*0 "r 2 — ——sinhtysiri{/;(coshf/; — c o s , (5.4G)

2 7

Here the areal coordinate f is given by

r'- = - - ( c o s h i / / — cost/;) 1.-(cosht/; — cost/;) • (5.48)2 7

5.2 A n alysis of th e Solutions

We are only able to invert (5.42) analytically for r / M < 1 , r /Q < 1 and 2 M /r <

l , Q / r < 1 and we must resort to numerical methods to establish the intermediate

behavior. We find for 2 M /r < 1 , Q / r < 1 and 0 < s M 2/ r 2 < 1 that the metric takes

the near Rcissncr-Nordstrorn form:

7 . t - H + g + W - S LT ( » + H H z S . + . . .) , ,5.49,r r 2 15r4 \ r r l )

2M Q1 x - Q»)» U , 7M | Xr r 2 9 r 4 \ r Jft =

- 1

(5.50)

R ep ro d u ced with p erm issio n o f th e copyrigh t ow n er. Further reproduction prohibited w ith out p erm issio n .


/ = s (M 2 - (?) - +[ 1 2 M 6 A/ 2 — Q~3 3r

(5.51)

where the higher order terms in M /r and Q /r include higher powers of $ also. We

observe; tha t for small enough .s* the NGT corrections to the Reissuer-Nordstrom

predictions can be made arbitrarily close to the experimental predictions of Einstein-

Maxwell theory.

The r coordinate has its first zero just inside the Reissner-Nordstrom radius

' /ov = M -F x/.'V/ 2 + Q-. For Q <g; M we have f fijV — 2A/ + ryQ2/ M . while to leading

order in s the surface r = 0 occurs at

We can develop expansions around r = 0 where r / M < 1. r /Q < 1 and 0 < s <

1 ,(2 /A/ < 1 (similar expansions exist for — 1 < s < 0). The leading terms are:

f ( r = 0 ) = 2M - 2 - + (5.52)

(5.53)

(5.55)

(5.54)

where 7 0 is given by

(5.56)

For r near zero, 0 < s < 1 and (M — Q) small, we get

(5.57)

(5 .5 8 )


5.2. Analysis o f che Solutions 6 9

f = Q " f 1 ~ T + T ( 1 " h 2 >+ •••) + ° \ \ m

Here, we have

= ( A /Q 2 Q -) exP ( - ; - + ° ( G s ,

For extremal solutions (Q = M), the Reissner-Nordstrom radius is at r

The first zero of the r coordinate occurs just inside fB,v at

r ( r = 0) = ^ « 0.8S5 Q .i/cosll 7T + 1

to leading order in |.s|. Near r — 0 we find

,s + 0 '

' + \ \ q

< t = SK-yto + l) ( r Y + 0 l ( r ' "(coslltr — l) (v ^ -f TCyflf \ Q ) \

/ =Q2^+,r/ f +2 ( I + coslur)

Note that the above expression are exact for anv s > 0.

Let us consider the electric field obtained from (5.12). We have

f 2 + r '1 = ^ e x p ( - 2 i/)(l + s 2 )(eoshi/>„ - cosik Y ' 1,

and(7 W + r*)

4 [m2 4- ( 7 - 1)Q2)’

This leads to the result:

2(m2 + ( 7 — \ )Q 2) ll2

(5.59)

(5.60)

If jV = Q-

(5.6 L)

(5.62)

(5.63)

(5.64)

(5.65)

(5.66)

(5.67)



Using (5.4) and (5.6), we can calculate the invariant quantity:

<f» = (5.6S)

We find that

The above result follows directly from electric flux conservation, and is identical

to the GR expression using an areal radial coordinate. The electromagnetic field

diverges a t the origin in accordance with Gauss’ law. If we define the dual tensor:

*F,U' — then it follows tha t the other electromagnetic invariant, *F(U/Ftu/

equals zero.

Let us calculate the energy density of a charged body between f = 0 and

r = oo. We have

We see tha t the NGT curvature invariants will diverge at least as fast as l / fA at

f = 0. As we found for the W yman solution, this singularity is not naked as the

redshift also diverges at f = 0. Integrating (5.70) over a volume in spherical polar

coordinates gives

cLr J y/rj}T°0drd6d(f>dydy

2 Jo (AyQ22 Jo (4yQ2 + Aq) 1/ 2

(5.71)

For the case when A0 = 4(A/ 2 — Q2). we find that

(5.72)



For M 2 > Q. this becomes

For the extrem al case Q = M , we get

We have obtained the remarkable result that the total electromagnetic field energy

for a spherically symmetric charged particle is finite. In the past, it was suggested

that electrons had a finite size in order to overcome the problem of divergences in

field theory. This proposal suffered from the problem that the Coulomb repulsive

force would blow the particle apart. Here the finiteness of the particle energy is not

achieved by giving it a finite size, but by chaugiug the geometry near the origin. A

correct description of an electron must be based on a (piantum field theory, so we

cannot expect a classical NGT description of the electron to be realistic. In natural

units Q M for the electron, whereas in M ann’s solution Q < Af.


C hapter 6

T he Short R ange W ym an S o lu tion

The previous chapters dealt with static, spherically symmetric solutions to massless

NCIT-79. Since this theory is now thought to be inconsistent, it is im portant to

investigate how the results we discussed are changed by the introduction of a mass.

The main changes we will see are the loss of the g\tr\ metric component, and a far more

rapid, Yukawa-like fall-off for the remaining skew field, g\o0\ ■ Both of these changes

are to be expected. The </[*,.] component was sourced by the NGT charge in NGT-79.

making its demise likely in NGT-95 where NGT charge is no longer conserved. In

massive NGT-79 we have now discarded NGT charge all together, so we know from

the outset that </[frj = 0. The behaviour of the reumiuiug skew field, g ^ , turns out

to be rather special for spherically symmetric systems. The relations

= 0 . ( ^ 1- ^ = 1) . (6 . 1 )

are identities, valid for arbitrary g ^ . In the context of NGT-95 this tells us tha t

H ';1 = 0. Thus, NGT-95 and massive NGT-79 share the same solution for static

spherically symmetric systems. This is one of the rare situations where NGT-95 has

a smooth limit as /i —► 0 (this limit is always sm ooth in NGT-79). Thus, if the skew

field has a very long range compared to the Schwarzschild radius, ie.

2/iM « 1 , (6 .2 )

72


6.1. Field Equations 7 3

then the massive solution will exactly recover the Wyman solution a t short, distances.

This means that the massive solution will also be free of finite area event horizons so

long as f iM is small. Infact, we will see that finite area event horizons are avoided for

any value of no m atter how large. In our derivation of the solution we will he

using the field equations of NGT-95. but the solution we arrive at is also a solution

to massive NGT-79.

6.1 F ield E quations

\

We return now to consider the most general, static spherically symmetric metric for

NGT:1 7 (r) (i’(r) 0 Q

-iu (r ) — a(r) 0 0

0 0 -: i(r) / ( r ) s i n 0

0 0 —/(r)s in fl - ,d(r) sin~ 0 j

The radial coordinate can be chosen such that fi(r) = r2 without loss of generality.

The [fr] component of (2.G9) gives

(6.3)

0 = I ■ ^ — — + ~ / r - P { w /* , n , 7) ] w .rc\ I (6.4)

The function P is a complicated nonlinear differential equation which is at lejist

quadratic in w and / . One solution to (6.4) is given by

r>/ 2 , 2 \ hl(7 /« ) ' 1P(w , f , 0 , 7 ) = + - / / • (6.5)

However, an analysis of this equation in the limit r —* oc reveals tha t the resulting

solution cannot produce an asymptotically flat space time. This means that the only

physically interesting solution to (6.4) is the purely algebraic solution w(r) = 0. In

the original formulation of NGT, w was proportional to the NGT charge.

W ith the simplification w = 0, the remaining field equations can be combined


6.2. The Fur-Field Metric__________________________________________________ 74

to yield the following three equations for a, 7 and / :

/,* + A’) + ( £ ± P * ) ' - ( ^ ) * * + (< ^) (,ow). = 0 ,

tr _ * +[n£±ms _ w±n\ M0,+f (logQ7). it61a J \ a J \ a.

(6.7)4 r2 - t f p 2 /- (r> + p ) A > \ , (.4 ' ) 2 - (S ' ) 2 - 2A" n

J + ------------ = 0 , (6 .8 )

wliere

A - ^ log(r‘‘ + f 2) , B = arcta n (r2/ / ) . (6.9)

W yman [42] found an exact solution to these equations in the context of unified

field theory when fi = 0. Unfortunately. W yman’s solution cannot easily be gener

alised, since endowing the skew field with a mass fundamentally alters the structure

of the field equations as /i is dimensionful. This complication has prevented us from

finding an exact solution to the field equations. Instead, we have been able to find

asymptotic expansions for the metric functions which provide a fairly complete pic

ture of the spacetime structure. The approximate solutions are valid in three different

regions, the far-field (M /r ~ l/(/xr) <g; 1 ), the mid-field (2 M / r ~ fir < 1 ), and the

near-field ( r / M ~ fir <S 1 ). These three regions will now be considered in turn.

6.2 T h e Far-Field M etric

When the skew fields are small and slowly varying, the NGT vacuum field equations

can be recast as Einstein’s equations with an effective m atter source, along with an

equation of motion for the skew field. The effective energy-momentum tensor is due to

a non-minimally coupled, non-linear antisymmetric field. By solving the equation of

motion for the skew field on a fixed G R background, and then resolving the Einstein

equations w ith the skew field as an additional source term, an iterative solution to

the full NGT equations can be found.

The skew fields’ equation of motion and energy-momentum tensor can be cal

R ep ro d u ced w ith p erm iss io n o f th e cop yrigh t ow ner. Further reproduction prohibited w ith out p erm issio n .

6.2. The Far-Field Metric I S

culated from the linearized Lagrangian

A = i V 'aW + i(2rt + - iir„u-'‘

(6.10)Here h ^v] is the hnearized skew field (tensor potential) and F„„A = is the

associated held strength. All metric quantities and operations R, VfJ) refer to

the GR background. The skew potential /q^j obeys the linearized equation of motion:

^P'nutr + = 0 , (6.11)

and W(l is given by the purely algebraic relation

= - 2 V xhwx] . (6.12)

The energy-momentum tensor is derived from the relation

f T ^ S g ^ y f ^ d ‘lx =

with

= \ ( T (V „(«S„) + V 0 ( i 5 „s ) - V „ ( ^ 0„)) . (6.14)

For static spherically symmetric spacetimes, the back-reaction of the skew field

on the original Schwarzschild background spacetime can be calculated be solving the

following equations:

a " 1 = 1 _ 2 — - — f J f t f 2 dr . (6.15)r r Jra

and

7 = a - 1 exp | J* — ^ ) 87rfa(f) </f j . (6.16)

In order to derive an explicit form for the equation of motion for / must first

be solved. The linearized “quation of motion (6.11) on a Schwarzschild background

\ f ^ g d lx , (6.13)


6.2. The Far-Field Metric

takes the form

76

1 /* 2jV/> .. 2 / 1 3M , / 2 2 8 iV/. ,r (1 -------- ) /" - 2 r ( 1 -------- ) / ' - r(fi r + ----- ) / = 0 . (6.17)r r r

To leading order in M / r this equation is solved by

/ = ( l + /zr + “ T [ 2 + (ire2tirEi{ 1 ,2fir){fii - 1 )] j , (6.18)

where Ei is the exponential integral function

roc gEi{n ,x ) — / —^~dt . (6.19)

J i 1

The constant s M 2/3 is fixed from the exact W yman solution by taking the limit

fi —► 0. The next order in M / r cannot be found in terms of known functions, so it is

not profitable to continue without making approximations in fir also.

The large r solution for / can be found as an expansion in the dimensionless

quantities M / r and 1 /{fir). First, an expansion for Ei m ust be found. Integrating

Ei by parts yields the identity

xEi(n, x)ex — 1 — nexEi{n + l ,x ) , (6.20)

from which the following expansion can be derived

E i( l , i ) e * = f ; ~ 1)! . (6.21)n — 1 x

This can then be used to express (6.18) as a power series:

8‘ 10 4 5 94 - 1 -------1-----------------------1-----------L

fir {fir)2 (fir)3 {f ir ) 4

(6 .22)

This equation can now be inserted into (6.17) to find the higher order corrections in

R ep ro d u ced w ith p erm iss io n o f th e cop yrigh t ow n er. Further reproduction prohibited w ithou t p erm issio n .

6.2. The Far-Field Metric II

M/r:

/ =s M 2

exp < —fir 13A/ 2 5A/ 3 35A/ 4

2 r2

i i 1 iV/x 1 1 H + —fir r

4 r 3

1 5+ - — +

24r‘1

(fir) -ItMfir

A/ 2

r-5 13

+ - — +

2 4 fir 2 {fir)2 8 (p r ) 3

398 8 /ir 32(/xr)2

A/ 3 [15 39 1 A/ 4 r 1 95116 1 16/rr 1.128 j ) ■

(6.23)

The above equation is valid up to order e‘l where e = M / r ~ 1 /( f ir) . Nonlinear

corrections to the linearized equation of motion for / need not be considered when

1 /(f ir) is small since they are exponentially suppressed relative to the linear terms.

Using the above expression for / , the various components of the energy-

momentum tensor can be calculated. For example, the energy density and radial

pressure are given to order e5 by

5 fiAr 4-

18

2 fir

13 fi: 18

3A/ 2 _ 5A/^ _ 35 A/1! 1 2r2 4r3 24r‘l J

5 MfiA 1

+ 18

+ ( 5 M V + 1 1 M f? + r - i12 9 4

/95 Mf i 2 25 A /V 4 // 1 5 1 A /V V -.- ,+ l 36 + 36 + 3 + 72 J r

(6.24)

and

f rr ,r M r

82M*exp

8w (fir)2 11 1 fi3

36

| - 2 fir

r - 3 , » , V r + ( 12 + 9

3 A/ 2 5 A/ 3 35 A/ ' 1

2 r 2 4r3 24r4

—i

+ p ^ + ^ + I 4 £ £ ) r - . ] . (, , ,

W ith the energy-momentum tensor in hand, it is now a simple m atter to calculate rv


6.3. The Mid-Field Metric 78

and 7 using equations (6.15). (6.16):

2 M _____r ( f ir)2ft,Kl

4

o “ ‘ = 1s2M 4fi4

x +13

+ +

—2 fir

M

13 A/ 2 5 A/ 3 35 A/ 1

2 r 2 4r3 24H37 A/ 2 5 A/

+ r r : —rrr + i —:—t +36 f i r 36 fi2r2 9 /z3 r 3 4 r 3 /z2 144 r'/z 2 8 r 4 /z3 9 /z4 r 4

1 -

2 zV/ _ g2 A /Vr (fir)2>lXl

1 1

exp < —2 /zr3 A/ 2 5 A/ 3 J5A/ 4

2 r 2 4r3 24r4

+M

12 fi2r2 8 /z:ir ;i 72 r 3 /z2

(6.26)

(6.27)

From these expressions it. is clear tha t the Schwarzschild metric is rapidly recovered

at distances exceeding the range, /z-1 , of the skew field.

6.3 T he M id-F ield M etric

If the mass of the skew field is very small, the range of the skew field can be larger than

the observable universe and the solution is essentially unchanged from the massless

case for all practical purposes. In general, if the mass of the gravitational source

is large compared to the range of the skew field, then f iM <§; 1 and it is possible

to develop approximate solutions where fir and M / r are both small parameters.

The leading terms in the expression for the skew field can again be found from the

linearized equation of motion (6.17), although nonlinear corrections will have to be

included to accurately solve for higher order terms. The leading M / r term can be

read off directly from (6.18) since this equation is valid for any value of fir:

/ = s M 2( [ l - i ( / z r ) 2 + . . . [ 2 + (ln 2 + 7 , - 2 )(/zr) + ...]^ . (6.28)

The constant 7 e is Euler’s number. Using this equation as a starting point, all higher

orders in A = M / r ~ fir can be found by simultaneously solving for a , 7 and

/ using the full field equations. This method of solution is more direct than the

Einstein plus m atter treatment, and is the easiest way to go beyond the first order


6.3. The Mid-Field Metric 2 9

0.12

0 .0 8

0 .0 4

0 20 6 04 0 8 0 100r ! M

F ig u re 6 . 1 : The numerical evaluation of the skew field (solid line) is compared to the large fir (dashed line) approximation of Eqn.(3.21) and the small fir (dotted line) approximation of Equ.(3.27) for the choice .s' = 0 . 1 , f i~l = 50A7.

back-reaction calculation. The downside is that the familiar Einstein form of the

calculation becomes obscured.

Proceeding using this method, the skew field is found to be given to order <V‘

by

/ = ~ ~ ( l - + \^ ( f i r f - i(/xr)'* + y [ ‘2 + (In 2 + 7 ,. - 2 )(///*) - ( //r ) 2

f l , . « / xil M'1 r i 8^ 2 (In 2 + 7 e — 2 ) + - J (fir) + — [, — + 2 (ln 2 + 7 ,. - 2 )(/«■) - - ( / t r ) J

H La 3

— ~~(/rr) 2 In(/zr) 0

A/ 3 [32 18,, „ 1 A/ 1 '80 2 .s24— T" — + — (In 2 + 7 ,. - 2)(fir)

. 0 0H-----~

r 1 7 636.29)

To order the metric functions 7 and a are given by

2 A7 s 2 A/ ' 1 ( M 7 = 1 --------- + r i

5 A7

4 A/ 2

lo r lo r 2

2A/ 2

I , , 2 17A/:I3G(,‘r) + M rT

- r T - ( /z r ) 2 + t ^ ( 1u 2 + 7 , - 2 )(/rr) o4r Io r2 ,

(6.30)

a - 1 , 2 A/ .s' 2 A/ ' 11 ----------1----- 5—r r 1

2 7A7 87A729 + 9 r + 4or2

2 + 7 * ~ 2)(/tr)


6.3. The Mid-Field Metric 80

191M 3 M . 2 14M2 , . . . -“ 7r ( / i r H ------=—(ln2 + 7e -2 ) (A ir ) . (6.31)45r:! 2 r

It; is interesting that, to leading order, the energy momentum tensor due to the skew

field takes the form ^Trr = —3^X/ cc —iV /'/r6. which is exactly the same as that

arising from the vacuum polarization of quantum fields near the event horizon of a

black hole.

In Figure 6.1 a numerical solution for the skew field is compared to the two

approximations (6.23) and (6.29) for a choice of [i which ensures both c and 8 are

small parameters in some region. The initial conditions for the numerical evaluation

were set using (6.23) a t r = 200A/.

3 .5

2 .5

1 .5

0 .5

r / M

F ig u re 6 .2 : The skew field near r = 2M when s = 1 for both the massless case (dashed line) and for the massive case with a range fi~l = M (solid line).

To avoid having an event horizon, the skew field must get larye near r = 2M

in order to make an impact on the spacetime. From the results of Chapter 4 we know

this is always the case in the massless theory. We find this continues to be the case

when /i ^ 0 since no m atter how small the skew field is for r > 2M . it is always

large enough a t r = 2M to eliminate the horizon. One way of understanding this is

to consider the linearised equation of motion for the skew field (6.17) near r = 2M


6.3. The Mid-Field Metric

where

f " ~ M(r - 2M ) + 2 ( ^ ‘> A /2 + l ) / ) • (6 -3 2)

Outside of r = 2il/t / can be very small and slowly decreasing. Near r = 2M . f"

becomes very large which implies —/ ' becomes large also. This means th a t / must bo

rising precipitously as r — 2M is approached from outside. Once / and its derivatives

start to get large the linearised equation of motion is no longer valid. This in itself tells

us th a t the skew field is starting to have a marked effect on the spacetime geometry.

In Figure 6.2 a comparison is made between the massless and massive case. In

the massive case the skew field extinguishes rapidly outside of 2 M . while still having

a large value a t and below 2M. In Figure 6.3 the range of the skew field is taken to

be very short. Even when the range is a twentieth of the Schwarzschild radius, the

skew field still manages to take large values at r — 2M. In order t.o numerically study

very short ranged examples the amplitude of the skew field must be made very large.

3 .5

2 .5

1 .5

0 .5

3 50 2 t1

F ig u re 6.3: The skew field near r = 2A/ when the field is short and very short ranged. The solid line shows the skew field when .s = 1 with i range // 1 = M while* the dashed line has a = 1010 and a range of = 0.1 M.


6.4. The Near-Field Metric 82

6.4 T h e N ear-F ield M etric

Motivated by the form of the exact solution for n = 0 and by the numerical results,

it is easy to show tha t the metric near r = 0 takes the form:

*> / r \ " 00 / r \ " oc / r \ n1 = ( - g ) , a = Y . ( i g ) ’ / = E In ( j j ) • (6.33)

n= 0 / n= 2 71=0

Since the field equations are first order in a and 7 and second order in / , four initial

conditions must be provided. Choosing these to be 7 0 , a 2, f 0 and /•>, all the other

coefficients can be determined. For example, the next coefficient in each expansion is

a 2(4 + 8 / 2 / 0 0 :2 ~ 2 0 / / - /r/o a-2 ) ,rn i = m « • (6-34)

7o(4 - 4 /2 - / r / o 0 2 )

' 2 = w , • (6'3o)

e 4/ 2/002 - H2foOc-i - foSfj - 8h = ----------------- W o ----------------- (6'36)

In the massless case it is possible to determine the initial conditions from the exact

solution. This cannot be done when /i / 0 so we do not have an explicit solution

near r — 0. However, as — \J r 1 + / - tends to zero it makes sense to expect the

massive and massless theories to coalesce as the rapidly rising curvature overwhelms

the mass terms iu the field equations. Our uumerical investigations supported this

expectation.


C hapter 7

G ravitational W aves

7.1 Spherical System s

la general relativity, Birkhoff’s theorem guarantees tha t a spherically symmetric grav

itational field in empty space must be static, with a metric given by the Schwarzschild

metric (only in the region outside of r = 2M is the metric actually static). There is

no analog of Birkhoff’s theorem in NGT. If the skew held is massless (/t = 0 ), a spher

ically symmetric system can radiate monopole radiation. In finite range NGT there

is no monopole radiation, but the vacuum solution is time dependent (see also [54]).

Transforming to retarded time coordinates ( u,r,0,<'>) when; a — t r when

r —* oc, we find the time dependent, spherically symmetric metric takes the form

( \ Ywv2ii — UI tv

fJnv —0

0

r 2 , + u) w

0

0

0

0

0

—r 2

-/sinfl

0

0

/s i nfl

- r ’-’shrff

(7.1)

Taking /i = 0 . the static solution is given for large r in these coordinates by

, 2 A/ 2 .s'* ‘ + 9f‘ 19s2 A/r> - 45/“A/I = 1 --------- -1------- rr-i----- + + .

18r* 4 or5(7.2)

3 =91* - 2 s 2 A/ 1 13s2A/5

3Gr1 45rr>

63

(7.3)


7.1. Spherical Systems 84

(7.4)

(7.5)

The time dependent solution can be found by expanding Wfi . iu and / in inverse

powers of r, with the restriction that the fields fall-off at least as fast as 1 j r at T +.

The solution is controlled by the "news function’', n(u). which describes the radiative

character of the skew field. When n(u) = 0. all time dependence disappears and we

recover the static solution. The time dependent solution is given by

The mass .U, NGT charge /- and the skewness parameter .v change with time accord

ing to

2 AI 45 /r + Ss.\['20nn + ---------------------------h (7.6)

3 rr 2 sM 2n1 Gr'2 9 r3

(7.7)

(7.8)

•) —;•> •>i r r r~z—;--- h (7.9)

(7.10)

0U(P) = 0 . (7.11)

and

du( sM 2) = 3n . (7.12)


7.1. Spherical Systems 85

Averaged over a large number of oscillations, the change in the mass is given by

To see this result another way, consider the change in mass A M during t he interval

Ui to tif. such that the solution is static before «, and static after uj. but radiative

iubetweeu. Using 11(11,) = n(itj) = 0 we tiud

Since the mass decreases with time, the monopole radiation carries positive energy.

Thus, the radiative solution is well behaved.

Oulv in the infinite frequency limit, u,' = k — o c c a n the skew radiation reach I*

and cause the mass of the system to change. Physical skew IInet.nations with linite

frequency do not reach T r. so there is no monopole radiation in finite rang** NGT.

However, there is still no aualog of Birkhoffs theorem as the vacuum solution is only

non-radiative, not static. An important question, which remains to I e answered, is

whether the static solution is rendered unstable by time-dependent perturbations.

The analysis will be; difficult since the origin requires a non-perturbative description.

The simplest approach would be to look for exact, stationary, non-radiative solutions

in the limit /t — 0. If /t <SC A/-1, the ft — 0 solution will give a very good description

for 7 < 2M. The use of a non-radiative. stationary solu' on is justified since the

(7.id)

< 0 . (7.M)

When the skew field has a mass (i we find tr = t) and / is given bv

,>i(kr - u.'t)

r

with the restriction

= y j k i + f . (7.10)


7.2. Axi-Symmetric Systems 86

solution with /z ^ 0 is non-radiative.

7.2 A xi-S ym m etric System s

This section is largely based o r two papers w ritten in 1993 and 1994 by Cornish.

Moffat and Tatarski [37, 47]. The field equations we studied were those of massless

NGT-79 since the new' version of NGT, NGT-95, had not yet coine into existence.

At the time, we were only concerned about the behaviour of the degrees of freedom

which couple to NGT charge. For this reason, wre chose to set the magnetic1 degrees

of freedom j equal to zero in our analysis. These days we would be inclined to do

just the opposite and drop the electric degrees of freedom (since the NGT charge has

since been abandoned) and study the behaviour of the magnetic components of j.

A brief analysis of the magnetic sector will be presented at the end of this section for

completeness.

In General Relativity (GR) gravitational radiation from bounded sources has

been studied not only through the linearized theory but also with the use of exact

solutions. The la tter was done for the general case of a bounded source in asym ptot

ically flat spacetime [48]. It was found tha t confining the arguments to the axially

symmetric case did not cause any essential loss of generality. Since even the rele

vant GR calculations are very tedious and the level of technical difficulty in the case

of NGT increases considerably, we limit ourselves to the axi-symmetric case. The

GR gravitational waves from isolated axially symmetric, reflexion symmetric systems

wcre studied in detail by Bondi et. e i [49]. O ur treatinem of the axi-symmetric

NGT ease will closely parallel this work.

Our physical picture is that of a system initially and finally quiescent, but

which undergoes a non-static, radiative phase. By studying such “sandwich waves",

we ensure that any decrease in the system’s energy measured at 1 + can be attributed

to a positive flux of energy emitted during the radiative phase, rather than to some

'This terminology is borrowed from unified field theory where </[,,„] was taken to describe the electromagnetic field strength F^v


7.2. Axi-Symmetric Systems 8 7

unphvsical condition at I ~ . In order to have a well defined notion of energy at J , we

have to demand that the metric a t I transform under the Bondi-Metzner-Sachs group

[48]. Once these conditions are enforced, it is up to the Held equations to determine

whether or not the theory is physically sensible. By requiring a well defined notion of

energy at J +, we are excluding the possibility of bad asymptotic behaviour for o

priori. Our analysis is however able to say whether or not the skew excitations with

good fall-off carry positive energy. If the skew-waves carry negative energy, we would

find th a t the mass of the system increases with time.

From the time when NGT was first introduced [4], then* have been few ana

lytic solutions of the field equations published. The exact solutions known to date

include the spherically symmetric vacuum case [4] (see also Chapter 3), the spherically

symmetric interior case [5G, 57] (see also Chapter 8 ) and Bianchi type I cosmological

solutions with and without m atter [58. 59]. This, at least in part, follows from the fact,

th a t deriving NGT field equations relevant for particular erases of interest is not, <is

technically simple as may be suggested by its superficial similarity to the correspond

ing GR situations. Firstly, since the underlying geometry is uon-Riemannian, neither

the fundamental metric tensor <y;,„ nor the affine connection is symmetric. This does

not constitute a serious problem for the choice of the form of since we can al

ways assume that its nonsymmetric part takes ou the isometrics of the svtn met de

part, which in turn has a well defined GR limit. On the other hand, calculating the

connection coefficients proves to be a tedious and time consuming exercise, indepen

dent of the method chosen. Secoudly, the resultant formulae for the uonsymtuetrie

connection coefficients are extremely complicated for all but the simplest forms of

the metric, thus becoming unwieldy to use in tin* derivation of still more complicated

field equations.

7.2.1 The A xi-sym m etric M etric

In analogy with GR. the simplest NGT field due to a bounded source is spherically

symmetric. However, a spherically symmetric source cannot excite the dipole or

quadrapole radiation we wish to study. Following Bondi et a i [49] we shall consider


i .2. Axi-S}Tnmetric Systems 88

the next simplest case, namely, a field which was initially static and eventually be

comes such, but undergoes an intermediate wave emitting period. Also, spacetime is

assumed to be axially symmetric and reflexion-symmetric at all times. Because 0 1

the complexity of the field equations, we are forced to use the method of expansion

to examine the problem. This approach, namely expanding in negative powers of a

radial coordinate, was also used in the GR analysis [49) and naturally suits a wave

problem.

Due to the physical picture sketched above and to the fact that wo are in

terested in the asymptotic behaviour of the field at null infinity. I . (in an arbitrary

direction from our isolated source) polar coordinates x° = //. x = (r.O.o) are the n a t

ural choice. The "retarded time” u = t — r has the property that the liypersurfaces

u — constant are light-like. Detailed discussion of the coordinate systems permissible

for investigation of outgoing gravitational waves from isolated systems can be found

in [48. 49].

The covariant GR metric tensor corresponding to the situation described above

is: V r ~ le2li — U2r2e2'/ r2ii Ur2c2'} 0 ^

e2ti 0 0 0

U r2c27 0 - r ' c *7 0

\ 0 0 0 —r 2e " 27 sin2 0 j

with U, V, /3, 7 being functions of u, r and 0 was first given in [60].

For any metric in polar coordinates, form conditions must be imposed in the

neighbourhood of the polar axis, sin 0 = 0, to ensure regularity. In the case under

consideration we have that, as sin 0 -+ 0 , K 6. U/ sin 0 . 7 / sin2 0 each is a function of

cos 9 regular at cos 8 = ± 1 .

In order to find the NGT generalization of the metric tensor (7.17) we require

tha t the symmetric part of the NGT metric tensor be formally the same as the G R

metric tensor. We then impose the spacetime symmetries of the symmetric metric

onto the antisymmetric sector. This is achieved by enforcing = 0 , where the

Killing vector field, is obtained from g ^ j = 0. The solution to this equation

Uliv — (7.17)



for the metric (7.17) yields the single Killing vector field £(1( = ^ d a = sin2 0do.

Imposing £j?n <j\i>.u, = 0 yields:

i v !J[ti3\dv£(i) 9[^ f \ ^n^ ( i ) — d . ( i . 18)

This equation gives coffin = 0. but does not exclude any antisymmetric components.

This is markedly different from the static spherically symmetric case where the above

procedure excludes four of the six antisymmetric components. Without further sim

plification. the NGT calculation would involve ten independent functions and ten

independent non-linear differential equations. This would constitute a huge increase

in complexity from the system of four equations and functions found in the GR case.

To make the problem tractable, we need to make further simplifying assump

tions to eliminate some of the antisymmetric degrees of freedom. To accomplish this

we note that the imposition of axi-symmetry splits the antisymmetric field equations

into two sets of three independent equations. (This can be seen directly from the

block-diagonal form of the G R metric). The first set explicitly involves the three

skew functions r/01- . /yl°2) , f /1-!;

= 0 (/i = 0 . 1 . 2 ) .

% i , 2 i = 0 . (7.19)

These four equations are not independent due to the one identity:

= 0 (/*. ^ = 0 ,1 ,2 ) . (7.20)

T he second set of four equations explicitly involves the three skew functions <7^ :

( ^ 1 , - »•

= 0 . (7-21)



These four equations are also not independent due to the one identity

' ; w = 0 . (7.221

We may now choose to work with either set of three equations and three functions,

noting tha t eliminating one set of three functions simultaneously eliminates the three

corresponding equations. We begin by considering the three functions . ; / >2t and

tfd-l as they couple directly to the NGT charge. We shall return to consider the

magnetic components . r -u i and at the end of the chapter.

7.2.2 The Electric Sector

W ith = 0 the NGT generalization of the metric tensor (7.17) is:

/ y r 1 e2Zi _ y l r l e'li (.i.i +u? -j- \

e — u! 0 rr

\

Ur2e21 — A

0

—a

0

- r r 7

0

0

0

0

■r't'"'1 sin* 0

(7.23)

where u), A and a are functions of u, r and 0. The eoatravariant metric tensor is given

bv:

■1iiU

( - a ' - v r ^ A g01

gio T(; i a y _ x2e~2l)A

where

,,02

12

920 921

9

■{exti - u 2) A c 2'’

0

0

(!

0

— r ~2 c27 sin - 2 0 /(7.24)

.4 =

9 =

=

r 2 sin2 Q9

r 2 stn2 0 [ - r 2 (ew - (w - a U )2) + aew - 2', (2A - a - ) } ,T

[r2(uj — aU — e2/J) + a A e ' ^ A ,


7.2. Axi-Sym m etric Systems 91

g02 = ( e V - u f c e - ^ A ,

g12 = [Ur2{a — a; - e2S) -r e23~2l{ \ — a V r ~ l) -h Aa;e_2 'r].4.

gP* = (/^[(u.a, A) -+ (-u ;, -<r, - A)].

The analysis determining the form of the functions U, V'. 3. 7 . a;, A. a in our case

is a natural extension of that given in detail by Bondi et al. [49] for finding the fo rm s

of U, V. 3 , 7 . The requirement that the field contain only outgoing radiation at large

distances from the source gives the form of 7:

/ ( M ) , g(u,0) ,

7 = H 3----- 1- -r r J

Demanding that the solution have the correct static limit (or equilibrium configu

ration) leads to the following forms for U. V’ (3 and 7 (unless otherwise stated, all

coefficients in the general expansions are functions of both u and 6):

(7.25)

(7.26)

(7.27)

(7.28)

The skew functions, u/, A and er, are constrained by the requirement that the

spacetime is asymptotically Lorentzian and admits inhomogeneous orthochronous

Lorentz transformations (the Bondi-Metzner-Sachs group). This requirement de

mands th a t g ^ g ^ —► 1 /r 2 as r —*• 0 0 . In our present coordinates this condition is

satisfied if ui, A and a have the following forms:

u =b \ U,- + - ( + . . . .

V =

r r l

r - 2M + — + ..

p =

rBi Bi — + — ■ + . . . ,

7 =

r r l c C - | c 3r + ^3 1 - + . .

B\ Li u — ----- F - j + . . . , (7.29)r r £

A = Vo H— ~ + . . . , r (7.30)

<t — So H-------F . . . .r (7.31)



The functions M, cr, C. >o, and L-, are all functions of integration. Bondi et. al.

refer to c as the “news function” as it controls the form of the gravitational radiation

in the symmetric sector. In an analogous way. Y0 is the “news function” for the

antisymmetric sector. Consistent with these identifications, we shall see that the

solution reduces to the static, non-radiative case when both c anil VJ, are set to zero.

We begin our analysis of the field equations by considering the simplest set of

field equations - the skew divergence equations (v / - ? # ^ ) .* = 0. The 0 component

of this set becomes

o = s„.„ + + . . . . (7.32)/•

while the u component gives

0 = ( I t sin 6 + (Sosin 9).e) + ((5 -- ~ 25°c) Sm ° ]'° + - . . . (7.33)r

The first of these equations tells us that S0 is time independent and thus can play

no part in the radiative phase of the system’s evolution. Indeed, it is inconsistent to

allow such terms if one wishes to describe a system which undergoes (quasi) periodic

motion. Moreover, if we restrict our analysis to one of the quiescent periods (i.e.

dropping the u dependence), we find So — 0 so we set So = 0 always. The first term

in (7.33) then tells us that L\ = 0. while the second term demands S\ — 0 for the

solution to be regular on the polar axis. The remaining, r, component, yields:

0 = Vo cot 6 + Vo — h i iU + . . . . (/ .34)

At this stage of the calculation, it is not profitable to continue to work with

the skew divergence equations, as the next orders also contain unknown coefficients

from the symmetric functions. Somewhat surprisingly, however, we are already in a

position to calculate the NGT charge of the body, and to prove tha t it is conserved.

The NGT charge, I2, is defined by the Gaussian surface integral

12 S h J ^ ^ 9[u% d 3x = L2 sin 9 dd = < L2 > , (7.35)

R ep ro d u ced with p erm iss io n o f th e cop yrigh t ow ner. Further reproduction prohibited w ith out p erm issio n .


where the biackets < > denote the angular average. The charge is conserved since

/2“ = U S i P 9 d 0 = \ sin d)-e (W==0- f7-36)

This follows from the fact tha t Vo must be regular on the polar axis. We note th a t

our solution allows situations where the total XGT monopole charge of the body. I2.

vanishes, while the dipole and higher moments are non-vanishing, i.e. < L > > = 0 b u t

< L2 cos 6 > ^ 0 , < L-2 cos 6 sin 6 0 etc.

We now turn our attention to the set of field equations = 0. The affine

connections that we require to construct these generalized Ricci tensor components

are obtained by solving the system of 64 equations (2.60). Rather than provide an

exhaustive list of the R^„) = 0 equations, we shall simply exhibit the components to

the order that we require to obtain the solution. We begin with the (rr) component

which demands:0 = 2 B , + £ + « , 6 f t +

V /

this gives B x = £ 3 = 0 and Bo = —<?/£. The (r0) component then gives

0 = — + — t + C/ + 2cCOt 6 + . . . . (7.38)r r

which yields U\ = 0 and U2 = — (ctg + 2 c cot 9). Inserting these expressions into the

(uu) component results in the important condition

A/u = - c u 2 + ^(cgg + 3c 0 cot 9 - 2 c) u . (7.39)

We can now use the (ur) components of R^v) = 0 at next order to solve for V), bu t

first we must choose how we wish to write the function of integration contained in

t/3 . Following Bondi et. al. we shall write t /3 as

t / 3 = 2N + 3cc 0 + 4c2 cot 6 , (7.40)


j.2 . Axi-Symmetric Systems U4

where .V is the additional function of integration2. Now the (nr) equation at next

order.

0 = sin* 9 ^llc* — 5(cjj)* -t- 4c*c „ — 6 cc ^

+L *3 sin 9 cos 6 — 8 c2 — 19cr # cos 9 sin 9

+ sin* 9 [pzja ~ + ‘-1 1) • (7.41)

can be solved to give

I 1 = —-Vy — I V COt 9 + f' o" + lcc y cot 9

+ ~ c 2( l + 8 co t* 9) 4- Vq* . (7.42)

It is interesting to note tha t 11 contains the first explicit difference between the

symmetric functions found for G R and these found for NGT. Substituting the above

results into the (99) and (u9) components of = 0 produces the following auxiliary

conditions on the multipole moments C, N:

4C„ = 2 c2 c u + 2cAJ + N co t9 — iVj — V, 2 , (7.43)

—3iVu = Mfi + 3ccug 4- 4cc„ cot 9 + c ucg . (7.44)

We see tha t the quadrupole moment of a source will decrease more rapidly in NGT

th an in GR due to the T02 term.

For completeness, we shall now return to the antisymmetric sector where the

skew divergence equation can be used to obtain the additional relations:

Z,3 = 2S2,<j + S 2 cot 9 , (7.45)

S 2,u = Yl - 2 c Y 0 . (7.46)

2The reason that U 3 is written in this way rather than as C/3 = N , and that the second function of integration in 7 is written as C — ^c3 rather than as C, is that we can then identify N and C asthe mass dipole and quadrupole moments of the source, respectively


7.2. Axi-Sym m etric Systems 95

The only remaining antisymmetric field equation.

•ftf01.2| = -%)Ij,2 + -Rfl2].0 + jR[20|,I = 0, (T.47)

gives to lowest order:

(L -2 .0 + Vi + S2.u)>u = 0 . (V.48)

This equation yields the additional relation:

L ‘i ,o == 2d'o ~ 21 1 . ( i .49)

To demonstrate the physical interpretation of M and L>, we consider the static

limit. We can scale the functions c and i o for either one of the stat'C periods so that

c = >o = 0 (forsaking the 6 dependence of c limits us here to a static spherically'

symmetric system). We now remove the terms containing the functions N and C

since they correspond to multipole moments. Since there is no radiation during the

static period, N = C = 0. The metric (7.23) tends now to its static spherically

symmetric limit:

9oo 1 - 2M ‘ +

17(ot) —

r/-t

1 + - L- 9 r 4 ’

<7[oi] =. £r 2’

17( 02 ) = 5 [ 02 ] = 0 ,

922 = - r 2,

9x\ = —r 2 sin2 Q,

r 4 r 5 (7.50)

(7.51)

(7.52)

(7.53)

(7.54)

(7.55)

where by A/,, I; we denote the static limit of M and 12, respectively.

Now a coordinate transformation from the retarded time u to the usual time

coordinate t = u + r converts the above metric into the NGT static spherically


i .2. Axi-Symmetric Systems 96

symmetric metric [55]:

- idr1 - r1 (dO1 + sitrtW o') , (7.56)

and

Thus, the static spherically symmetric limit, d /,, of the “mass aspect" M(it.O) can

only be interpreted as the mass of the system. Similarly, the static spherically sym

metric limit, /;. of the ^charge aspect” £•>(«. 0) is identically the NGT charge of the

system as shown by equations (7.35) and (7.36).

If we define the mass m{u) of the system as the mean value of M(u, 0) over

the sphere:

then c(u.O) completely determines the time evolution of the mass rn(n). Integrating

(7.39) and noticing th a t the second term does not contribute to the integral due to

the condition that c be regular on the polar axis, we get

Since we discussed here systems whose initial and final states are static, the physical

interpretation of m(u) as the mass of the system is unambiguous. Analogously to the

GR case the main result is as follows: The mass of an axially symmetric NGT system

is constant only if the system remains static. If the system evolves in time (emits

waves), the mass decreases monotonically.

7.2.3 T he M agnetic Sector

If we discard the N G T charge we are left with the three magnetic components

ff[0 3p 5 [i3] and gpy- To ensure proper asymptotic behaviour at T + we require the

o (7.57)

(7.58)


7.2. Axi-Symmetric System s 97

skew functions are of the iorrn

Him] — A -h D2/ r ~ . . . . (7.59)

<J{ i:*i = A + A / r -t-. . . , (7.60)

//[23i = r*"\ + £•>-)-... • (7.61)

when' A , A , F, are each functions of u and 0. At lowest order the skew divergence

equation. ( \/-Tl!l'L'i‘'1),u = 0 . gives E Uu = 0 => £ \ = 0 . The next term gives F.t) =

E >,«. The equations = 0 then give

F _ F , ( (A - F , . fl)s in 0 ),* r 2 " - F, + -- ■ u m )

The angular average of the above expression yields < F>.u > = < F >. The rate of

mass loss is increased by an amount

! ^ i2,u + ^ F F .u u (7.63)

over the GR result. Here Fi(u.O) plays the role of the magnetic news function n(u)

introduced in §7.1. Averaged over several oscillations, the total rate of mass loss is

given bv

< rn u > = < 2(cn)2 + (Fi ,u ) 2 > (7.64)

The skew contribution corresponds to a positive flux of monopole radiation, in ac

cordance with the spherically symmetric analysis of §7.1. When the skew fields are

endowed with a mass the monopole flux disappears and we are left with the usual

G R result.


C hapter 8

Interior Solutions

So far in this thesis we have been considering vacuum solutions to the NGT field

equations. In this chapter we explain how couplings to m atter are able to give rise

to the integration constants and .s that appeared in the vacuum solution. We will

model the m atter by a perfect fluid energy momentum tensor along with an additional

coupling to the vorticity of the fluid. The complete energy momentum tensor is taken

to be

T'“/ = {p + p)u»uv - pgtUf + 4 *7 * ' " ' " ^ , (8 . 1)

where au is the fluid's four-velocity and p and p are the usual internal energy density

and pressure, respectively. The constant k has the dimensions of length in natu

ral units. The generalised Biauehi identities, which result, from the diffeomorphism

invariance of NGT. give rise to the m atter re sp o n se equations

+ (Jttp J + r " * = o •

(8 .2 )

The static spherically symmetric interior case was studied for mass less N'GT-

79 by Savaria [56. 57]. In Savaria’s study the magnetic skew field ij\o,t,\ was set equal

to zero. Even with this truncation, the analysis was more complicated than the usual

G R case due to the inclusion of additional source terms arising from the NGT charge.

In what follow's, m atter will be taken to be NGT charge neutral so that S tL = 0 and

98


8.1. Equilibrium Equations 99

[2(r) = 0. We will also assume that the range of the skew field, 1 /p., is very much

larger than the size of the astrophysics! bodies we are studying. The field equations

we will use are those of NGT-79, but the results will hold equally well for NGT-95 as

1 1), vanishes in spherical symmetry.

8.1 E quilibrium E quations

The non-vanishing components of the energy momentum tensor are:

Tt t = fr , . (8.3)

Trr = pet . (8.4)

Too = pi'1. (8.5)

Tvt = pr1 sin2 9 , (8 .6 )

7[O0 j/(s in 0) = -pf + n p ^ rX 7 / . (8.7)y/Ot

The field equations can be most compactly expressed in terms of the quantities:

.4 = i log(r1 + /'-’) , B = arctan(r2/ / ) . (8 .8 )

The usual G R expressions are recovered wrhen .4 = 21ogr and B = tt/2 . With

w(r) = 0. the field equation (2.61) is automatically satisfied. The (t t ), ( r r ) . (00) and

(0d>) components of the field equations yield

2nRtt/ l = (log?)" - \ (log7 )' ^log + A' (log7 )'

= 8 tm (p + 3p) - 1 6 x « :p -;^ ^ L ^ (lo g 7 )' , (8.9)

2R rr = —2.4" + -4'(log « )' - ((.4 ' ) 2 + (R ')2) - (log 7 )" + \ (log 7 )' ^log ^ j

= 8 <ra (p-p) + 16/TKp—==j====y(log 7 ) ' , (8 .1 0 )



f ”= 4 ~r2 ip - p) -h 8-Kp-j==L.r (tog 'A' . <8. It)

v / r ' - r / - v'rt

D , . a ( r-B' + / .4 'V D, ( f B ' ~ r 2A' \ / r f f + / , l ' V ,f l« , / sm * = , + f - - B I — U ( ™ W . . M

r 1 •>= -4 r r f ( p - p) 4- $znp r ; (log '•)' + ^H*i*,*|/siu0 (8.12)

v/r -f /* v ° 1 '

The constant q comes from the Lagrange multiplier tield l l '0 = 3</cosd/2, and prime

denotes derivatives with respect to r. One of these equations may he replaced bv the

r component of the m atter response equation (8 .2 ). which simplifies to read

P = " ( P + p)(log7)' - . /-= (log 7 )' - (S. 13)2 v /r 1 + f-y/tx

When k = 0 this relation is identical to its GR counterpart. The agreement with GR

follows from NGT obeying the weak equivalence principle when all direct couplings

between gûj and m atter are dropped. When k ^ 0 or S-, / 0 additional equivalence

principle violating terms come into play.

The set of equations (8.9)-(S. 12) are best studied as three useful combinations.

The first combination is / times (8.11) plus r~ times (8.12). which yields

2 . / ( r ‘ + m V / V + / 2)\ ,/ + "r + ( 2 a - ) + 7 7 —

= S~K(ir~ r 1 2 (log7 )‘ . (H.M)s/a

The second combination is / times (8.12) minus r 2 times (8.11). which yields

= 4 7 r(r'‘ + f 2)(p - p) . (8.15)

The third useful combination is (8.10) plus (8.9) minus 4« / ( r 4 + f 2) times (8.15)



which si vcs

r 2 _ f(} J ( {rA + p )A. y (Ar 2 _ {b,)2 _ 2A„H------------- = LOTTp . (8.16)rA + f 2 r4 + f '1 \ a J 2a

These three equations, along with the conservation equation (8.13), provide the most

useful, complete set of field equations describing hydrostatic equilibrium in inassless

NGT-79.

Having the constant q non-zero leads to a jump discontinuity in the curvature

invariant at the edge of the structure. This follows from the fact that

Wq = 0 for the exterior Wyman solution [42, 40], while IV# = 3gcos P/2 in the

interior. Consequently, the matching of these solutions a t the edge of the structure

demands q = 0 .

The complexity of the field equations makes it difficult to combine them into a

simple, fundamental hydrostatic equilibrium equation. However, considerable insight

into the nature of the interior solutions can be gained by considering power-series

solutions about the origin. Here the origin f = 0 coincides with r = 0. We begin by

expanding about Minkowski space near r — f = 0 :

P — Po + Pit* + P2f2 + . . . (8.17)

P — Pa + P i t + p i r 2 + . . . (8.18)

7 = 7o(l + ~t \r + y i r 1 + . . . ) (8.19)

a = ft0 -F aqr + o^r2 + . . . (8 .2 0 )

/ = / j r 3 + / 4 r 4 + . . . . (8 .2 1 )

Substituting these expansions into (8.13-8.16) yields

/ = \ / p f r 3 ^ 1 - ~ r ~ _ Kpa(po + 3p0) - ( ^ P o - jrirpo + g P /) J r 2 + . . . j (.8 .2 2 )

P = Po ~ [^ (P o + Po)(Po + 3p0) + 47rv/p7Kpo(po + 3p0)j r2 + . . . , (8.23)

7 = 7o ( l + | i r (po + 3p0) r 2 + . . . j , (8.24)



/Sir ~ \ •»Q — 1 + ~1~ 4 ^ J r ~ + • • • • (8-25)

The energy density p/ is contributed by the skew field / . The skew field increases the

mass of a structure above that of a structure with a similar m atter distribution. p(r),

in GR. It is convenient to parameterise p / as some fraction of the central density:

Pf = epo • (8.2G)

We will see tha t large departures from the usual GR results occur once e reaches

£crit ~ 0.1 (A //R). The fact that the critical value for e scales with the surface redshift

zs = M /R is in keeping with the general physical picture of NGT. Large departures

from GR occur when the redshift is large. This relationship is made more precise by

considering the matching of the interior solutions to the exterior Wyman metric.

8.1.1 M atching Conditions

The boundary between the interior and exterior solutions occurs at a radius defined

by p(R) = 0. Formally matching the solutions a t r — R is not very enlightening,

since 7 is only known as an implicit function of r in the Wyman solution. More can

be learnt by matching onto a small M /r expansion of the Wyman metric, where the

metric functions take the near-Schwarzschild form:

t 2 M a2 A/ 5 4 s2M«7 - 1 - — + ^ + ^ r + - ' .

ot = + + ^ + (8.28)V r Or1 9r5 4orG I

s M 2 2sM 3 6s M 4 3r "** 5r2

These expansions are only valid for 2M /r < 1 , which is not a serious restriction seeing

as most structures have 2M /R 1.

One unusual feature of the matching a t r = R is that the matching of the metric



functions does not ensure the matching of the gradients of the metric functions. In

G R, any pressure and density profile which smoothly approaches zero at the edge

of the structure leads, via the field equations, to a smooth matching of a ' and 7 '

once a and 7 have been matched. This in turn ensures th a t all curvature invariants

smoothly match at r = R , which is the physically im portant condition. In NGT, it is

not enough to match the metric functions. If only the functions are matched, jumps

can occur in the derivatives. For example, the jum p in 7 ' is given by the relation

4R 3 = [ ( / ') ’) 2 / , (8.30)

where square brackets denote the jump in the enclosed quantity at r — R. In the GR

limit, f — 0 and the above equation yields [7 ' ] = 0. In NGT we must match a , 7 , /

and f to ensure there are no discontinuities.

To get a feel for how stars behave in NGT, let us adopt a crude model of a con

s tan t density s tar with a surface redshift M / R £ 0.1. In addition we will assume f ( r )

can be modelled by the first two terms in the expansion (8.22). These approximations

should provide a crude approximation to objects such as neutron stars. Working to

first order in M / R and matching / , / ' and a at r — R reveals:

A<jr

M = — p0R 3 , (8.31)

and

- T B - ( 8 ' 3 2 )

The above equations explain how the parameters s and M of the vacuum solution

are related to a physical m atter source.

To leading order in M / R the skew contribution to the energy density is

p/ = (t ;)! s2a/2(p“+3?o)! ■ (8 3 3 )



W riting this as p f = e p0 we find

This equation tells us tha t for compact objects with M / R ~ 0.1 we expect to see

~ 10% departures from the G R results when |s| reaches about 5. In terms of the

coupling constant k this translates into k/ R « 1 , it. k of order the radius of the

object. For a neutron star this tells us that the coupling constant k must be of

order 106 cm for an interesting departure from general relativity to occur. If the

coupling constant k is universal, we can conclude tha t NGT and GR will have identical

predictions for anything other than high redshift objects such as neutron stars. On

the other hand, as a star undergoes gravitational collapse and M / R becomes large

we expect m ajor departures from the predictions of general relativity. To study the

strong field regime properly a complete numerical study needs to be undertaken.

Since the corrections to the GR expression for the mass of the structure are

always positive, the gravitational mass of a body in NGT will exceed that of a body

with a similar density distribution in GR. This increase in mass is primarily due to a

decrease in the binding energy, which occurs as a result of the repulsive contributions

to the gravitational force coming from the skew sector. If k is too large, the repulsive

contributions from the skew sector can cause a structure to become unbound. More

moderate values of k serve to stabilise a structure relative to its GR counterpart.

8.1.2 Binding Energy

The internal energy of a body is defined in the usual way to be E — M —m ^ N , where

m.N is the rest mass of the IVth constituent and N is the conserved particle number

for this constituent:

N = J y/~~9 J% drd9d((> = J 4ir ^ /a (r)(r 4 + / ( r ) 2) n (r )d r , (8.35)



and n ( r ) is the proper number density for this constituent. The total internal energy

can then be broken up into its thermal and gravitational components as E = T + V'.

The thermal energy is given in terms of the proper internal material energy density,

c(r) = p(r) — Tntvn(r), by

T = j * 4 - y a ( r ) ( r > + /( r ) 2 ) e(r)dr , (8.36)

while the gravitational potential energy I ’ is given by

V' = M - f 4tt ^ a ( r ) ( r l -F / ( r ) 2) p(r)dr . (8.37)J o

The second term in the above expression remains unchanged, to leading order, from

its value given by GR while M is always greater than in GR. Thus, the gravitational

binding energy, SI = —V , is always smaller in NGT than it would be for the same

mass distribution in GR. To leading order, the binding energy is given by

n = n CR_ 239A/.,’ ,A A 3288 ( £ ) - • . <*»>

which explains why a large value of s causes structures to become unbound. For

polytrope equations of state:

p = a p {l+l/,n) , (8.39)

the Newtonian expression for the binding energy becomes

Q N ew t. = 3 _ , ( 8 .4 0 )5 — n R

which can be combined with (8.38) to give an expression for the value of s for which

the structure becomes unbound:

! 96 \ 1 /2 / R,23(5- n ) , ( I ) ■


8.1. Equilibrium Equations 1 0 6

This in turn gives an approximate upper bound on k:

The largest mass W hite Dwarfs are well approximated by a n = 3 polytrope, and

have M / R « 4 x 10~l, which bounds s to be below = 4 x 103. This sets an upper

bound on k of = 4 x 10u cm. The major source of error in (8.41) comes from the

approximate expression for the binding energy, which can be off bv a factor of two for

White Dwarfs. The second term in (8.38) is essentially exact for W hite Dwarfs and

agrees with the numerical results within 0.1%. A plot of the dependence of S 1 /M on

s is displayed in Figure 8.1 for a White Dwarf with M /R = 2 x I0 _r>.

3 "

2.5

1.5

0.5

5000025000

F ig u re 8.1: (fi/A / x 105) as a function of the skewness constant » for a White Dwarf with central density po = 1.45 x 10- 1 3 km-2 . The lower line, (a), is the exact numerical result while the upper line, (b), is the analytic result employing the Newtonian approximation for the binding energy.

The largest mass Neutron stars can be approximated by a n — 3 /2 polytrope,

and have M /R « 0.1, which gives the far more stringent bound on .v of = 11.

This value of Soo will be roughly 30% too large due to the approximate value used for

the binding energy. The bound on « from Neutron stars is very much tighter than

from W hite Dwarfs and is given by (8.42) to be «oo = 2 x 1 0 f> cm. Since Neutron


8.1. Equilibrium Equations 1 0 7

stars arc known to exist, a tighter bound on k can be obtained by requiring th a t the

repulsive force causes a t most a 20% reduction in the binding energy. This then, gives

a maximum allowed value of «;CI.it = 10G cm. The best testing ground for N G T will

be Neutron stars and “black hole candidates" (to use the parlance of GR) such as

G'ygnus X-l. If k is a universal constant, the skewness parameter for the Cygnus X-l

candidate would be bounded above by s \_ i ~ 0 .1 , making it a very dark s tar with a

surface redshift around z ~ 10l. Supermassive black hole candidates such as the one

purported to reside in the gakixy M87 will have very small values of s and very large

surface redshifts making them difficult to distinguish from GR’s black holes. The

coupling constant k will be universal if NGT obeys the weak equivalence principle.

Even if n is large enough to cause a 20% reduction in the binding energy,

numerical evaluation shows that the density profile is almost identical to that in GR.

with the difference in p(r) never exceeding 0.1% anywhere in the structure. These

studies were done using the equation of state

state goes over to a n = oo polytrope. Choosing a central density of p0 = 0 . 2 pc and

taking s = 0 yields a Neutron star with mass M = 0.0782, radius R = 0.8325 and

binding energy QGR = 0.0074 in units of p~l/2. From (8.41), these values lead to

the prediction that s = 8.2 will cause a 20% reduction in the binding energy. The

numerical results showed a 20% change was caused by having s — 7.6, and gave rise

to a Neutron s tar with mass M — 0.0798 and radius R = 0.8322. Even for relativistic

situations, such as Neutron stars, the approximate solution for / given by (8.22) yields

remarkably good results.

If s is small, say |.sj < 1 , the skew fields of NGT have essentially no effect on

the description of any known astrophysical object, save black holes. Since both the

Chandrasckar and Oppenheimer-Volkoff limits on the masses of W hite Dwarfs and

Neutron stars rely on quasi-equilibrium reasoning, and since the equilibrium descrip

(8.43)

where pc = 1/(727t) km 2 is a critical density. Above this density the equation of

R ep ro d u ced w ith p erm iss io n o f th e cop yrigh t ow n er. Further reproduction prohibited w ithou t p erm issio n .


tion of these structures is unchanged for small s. the visual sequence of gravitational

collapse will occur. The departure of NGT from GR will occur as the collapse contin

ues into the strong field regime. A complete numerical study of the collapse process

in N G T remains to be done. W hat we can say for certain is that the skew fields of

NGT serve to stablise a star relative to its GR counterpart. In addition, the depar

ture becomes larger as the surface redshift of the object increases. The combination

of these facts suggest that catastrophic gravitational collapse to form a black hole

can be avoided in NGT. An alternative endpoint for gravitational collapse might be

a "Skew Star", supported not by m atter pressure, but rather by the skew fields. The

question then becomes, how could we tell such an object apart from a black hole?

The only unique observational test for black holes comes from studying the

massive objects believed to exist at the center of most galaxies. If these galactic

nuclei have masses above 1 0 sA/q, GR predicts that a star passing by will be torn

apart by tidal forces once it is within the horizon of the massive black hole, so the

star would vanish without a trace. On the other hand, if the galactic nuclei is some

massive, compact object supported by skew fields, the star’s death would be visible.


C hapter 9

Sum m ary and C onclusions

So whore do we stand? Did we succeed in showing black holes will not exist in NGT?

The short answer is uo. What we did find was a range of results th a t suggest NGT

will be free of black holes and singularities.

Our study of the vacuum Wyman solution revealed tha t a point mass in NGT

is not a black hole. In general relativity, a trapped surface forms around m atter that

collapses within its gravitational radius. The m atter is then crushed to a singular

point. If m atter can be forced inside its gravitational radius in NGT no trapped

surface forms, but the causal structure o f spacetime breaks down. But can matter

be concentrated inside its gravitational radius? In general relativity we know that

it can. In NGT we found several indications that this is not the case. The primary

reason is that when a concentration of m atter falls within its gravitational radius the

gravitational field becomes very large. Since NGT violates the strong equivalence

principle, the large gravitational fields are able to change the local microphvsics. An

interesting possibility, that has not been explored, is that the very form of the funda

mental energy momentum tensor used to describe the m atter will change in a strong

gravitatiomd field. In weak gravitational fields the local symmetry group of NGT

is approximately the Lorentz group 5 0 (3 ,1 ) and m atter can be expected to behave

much as it does on Earth. In strong gravitational fields the full GL(4, R ) local symme

try group of NGT must be taken into account and our description of the fundamental

forces may need revision. At a more mundane level, we found that a simple coupling

109


110

of vorticity to torsion was enough to increase the stability of compact objects such as

neutron stars. Moreover, the effect gets larger as structures get smaller, eventually

becoming non-perturbative when the m atter readies its gravitational radius.

In rnassless NGT we saw that time dependent, spherical collapse could give rise

to monopole radiation tha t carried off energy. This could also provided a mechanism

for a collapsing star to shed its skew fields, just as a star in JBD gravity could shed

its scalar field to form a Schwarzschild black hole. In massive NGT this mechanism

is not available, so it is not clear that a star could divest itself of torsion in order to

form a black hole. A definitive answer will require a sophisticated numerical study of

the highly non-linear partial differential equations th a t govern the collapse dynamics.

So we are left with inanv tantilising hints th a t NGT will be fret* of black holes

and singularities. We understand the basic mechanism by which total gravitational

collapse can be halted in NGT. We have also found evidence to suggest, that, these

mechanism will indeed come into play. Perhaps spacetime torsion will allow m atter

to spin free from the spacetime curvature that threatens to imprison it. That would

certainly be a interesting twist in gravity's tale.


C hapter 10

A ddendum

To obtain a succesful Hamiltonian formulation, the NGT-95 action (2.68) has to be

supplemented by the constraint [39]

C-c = y / - a a ^ u u\ v , ( io .i)

where A„ is a Lagrange multiplier, and v? is an arb itarv observers four velocity.

Variation of (10.1) with respect to AM yields the three constraint equations

gW u„ = 0 . (10.2)

It is these constraints th a t globally remove the electric components of g After

eliminating the Lagrange multiplier field the vacuum field equations read

Riv») d" = 0 > (10.3)

& ®ua — —g ^ ^ R + = 0 , (10.4)

and

9liv,a —" Q p u 9 p p ^ a u ~ 0 • (10.5)

111


112

Choosing the frame of reference id1 = (y /g" . 0 , 0 . 0 ), the constraint equations (10.2)

reduce to

= ( 10.6)

which eliminates the electric part of NGT-95. Clearly, any coupling to NGT charge

is out of the question when the constraint (10.1) is imposed. Since we have largely

dispensed with the NGT charge in this thesis, very few results are affect by the new

constraint. Most importantly, the W yman solution and its short-range generalisations

go through unchanged.


A p p en d ix A

N o ta tio n and C onventions

The signature is (+ , —. —, —) throughout. Greek indices run from 0 to 3, while Latin

indices run over the spatial indices 1 to 3. W hen writing line elements in spherical co

ordinate's (r ,Q .o ), the following shorthand notation for the angular part is frequently

used:

dQ2 = dB2 -t- sin2 Qdo2. (A .l)

Partial derivatives are denoted by the comma preceding the appropriate index

or the symbol of a variable, or by the symbols:

«. - <A.2a)

30 S £ ( A J b )

When it is unlikely to introduce confusion, the following notation is also used:

• _ d .4 ( f ,r ) t, _ d A ( t , r ) , A oX.4 = — , .4 = —g^ , < )

where A is an arbitrary function of t and r.

The symmetric and antisymmetric components of a two index tensor are de

noted by and A\pt,) respectively, where

= 2 (Ajw + A^n) , — 2 (A/u» Ayn) . (A.4)

113

R ep ro d u ced with p erm issio n o f th e cop yrigh t ow n er. Further reproduction prohibited w ithout p erm issio n .

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Documents

GRAVITY WITH A TWIST - University of Toronto T-Space · ported reference vector [2]. This thesis is devoted to the study of a theory of gravity that employs both curvature and torsion