Scalar tensor theories of gravitation : a thesis presented in partial fulfilment of the requirements for the degree of Master of Science in Mathematics

A thesis pI·esEmted in partial fulfilment of the requirements for the degree

of

Master of Science

in NathematicG at

Massey University

Ian Paul Gibbs

Chnptor One. Introduction Chapter Two. B:-tclcground

page 3 14 14 19 22 ChaptGr 'rhrcG. Tho scab.r-tensor nod.ol of grnvitc.tion

3. 1

3.

2

3.

3

3.

4

3.

5

3.

6

Chapter Four

4.1

4

.

2

4

.

3

4

.

4

4.

5

1)..6

, n 'r • I

4

.

8

4.

9

4.10

Chaptor Five.

5. 1

5.2

5.3

Chapter Six. 6. 1

6.

2

6.

3

-

6.

4

6.5

606

Chaptor Seven. Appendix

Bibliography

Space-tino thoories Gravi t<1tion theorios

Tho sca.l:1r-tenscr nodol of gr,?..vi t:Ltion

Ezp-c:ri1.1ont:1l tosts ;1nd tho sc'J.l.'.lr-tensor IJ.oclel 27 Coni'orr:1al inv..,_rio.11c8 cmd tho scaJ ar-tonsor aodel 29 Doser' s scala.r-t;,,nsor theory

35

Tl10 Brans-Dicke sc.'.ll '.lr-tensor thoory

3

8

'I'ho ED scnlnr-tcnsor theory 38

V.'.lcu~~ oolutions 50

Stand.'.lrd BD cosnol og,J

5

8

Units tr:rnsforr:nti :.ns nnd tho BD theory

63

St'.'ln·.:lard BD cosr.:ology ( ctd) 6G

Hon-sbmu::..rc'. BD cos:,clcgy

69

Scnl-:J.r-n::i. t tcr fiold e-o.ugo fro_;doc 71 Nordstro.::-RGim::mer typo solutions 80

:wE

fcrnulation in the BD theory 82 BD Theory nnd Mach I_{s Principlo} 8!1-Spccial scalar-tensor theories

89

Gursey's theory

89

H:1ssi ve BD theoriGs 92

Othor scalar-tensor theories

94

The scalar-tensor nodol of gravitation (ctd)

99

Linoar connections 102

Weyl spo.ce-tine 104

Grnvitation in Weyl spo.ce-tine 107

Dirac's theory 113

Lyra space- time 117

Gravi tation in Lyra space-tine 120

Conclusion 126

-hapter One

INTRODUCTION

The problem the writer wishes to consider

here

,

is essentially one related to the classical field description of I~ature.

The framework of General Relativity provides

a theory for the geometry of the four dimensional

space-time manifold and at t he same time gives a description of

the gravitational field i n terms of the metric tensor,

while the electromagnetic field can be interpreted in

t erms of a particular second rank, skew-symmetric tensor

- - the covariant curl of a vector field defined on the

manifold. However the scalar field, the simplest

geo-metric object that could be defined on the manifold, does

not seem to be experimental ly evident when it is

interpret-ed as a third, classical long range field. I n spite of

this lack of experimental evidence and as there appears to

be no theoretical objection to the existence of such a long range field, the problem is to i ntroduce the scalar field

-into the classical scheme of things and to construct a

viable theory containing al l three long range fields.

It is interesting to compare the physical

descriptions involved with these fields. Both the

gravi-tational and the electromagnetic fields have gauge-like

degrees of freedom and before a situation could be

physic-ally relevant these degrees of freedom must be fixed

-for the g7avitational field by imposing coordinate condi-·

tions while for the electromagnetic field, after

co9rdin-ate ·conditions have been imposed the gauge of the field

potential must be chosen. As a consequence of these

gauge freedoms, ih order that the fields couple

consist-ently with matter sources, th_e energy momentum tensor of

electro-magnetic current density of the source must be conserved.

The scalar field on the other hand has no gauge-like

degree of fr~edom and consequently has no conserved

11_{charge" as a source. Thus for example, in contrast to}

the other two fields, no constraints exist by which the

scalar field could be separated int o a source 11_bound"

part and a free "wave11 _part.

In recent years the problem of incorporating

the scalar field i nto the description of gravitati on has

led t o the investigat ion of a speci al class of gravi

ta-tion theories - - the scalar-tensor gravitation theories.

With the previously mentioned probl em in

mind, the goals of this thesis are

(i ) to review work that has been done on

these theories and

(ii) to discuss them i n a way that compares

them to the theory of gravitation given i n General Relativity.

Chapter Two basically gives an historical ba

ck-ground and introduces mor e specific motives for considering

the scalar fi eld as a fundamental physical fieldo

Chapter Three consider s the important class

of scalar-tensor gravitation t heories based on a Riemann

space-time and Chapter Four continues this t heme by looking

at the "most developed" and perhaps simpl est member - the

Brans-Dicke t heory.

For completeness the 11_{massive Brans-Dicke"}

theories and some special scalar-tensor theories are looked

at briefly in Chapter Five.

Chapter Six retnrns to the scalar-tensor model

of gravitation developed in Chapter Three and looks at the

implications for the model in more general space-times.

ACKNOWLEDGEMENTS

I wish to express

my

gratitude to Dean Halford for

supervising work for this thesis. Also I would like to thank

Chapter Two

BACKGROUND

The electromagnetic end gravitational fields were described in t he introduction aG classical long range fields. These fields are r esponsible for forces that fall off inversely proportional t o the square of the dis -tance apart of the interacting bodies (sources); in contrast to short range forces which show an exponential hehaviour. Einstein (1916) ( 1 ), attributed to the spa ce-t ime manifold a Riemann ::::tructure ,nd gave "meaning" to the gravitational field in termP. of curvature through his ~ra vi-tational fi eld ~qu~tio~n

G-µv

,

wher e G i s Newton's gravitatio~~l constant.

The notation establi shed here is used in most sections. Units of length and time ar& chosen such t hat

c 5 _{1, although with this understanding some formulae may-still} contai-n c • Greek indices range over the VD.lue-s

l

o

_{> 11}:₁•_{, 3}} , the coordinates xJ and xi.,

(t

-

_{- 1 ' ,2 , 3 ) ,}

ar e assumed time-like and space-like respectively and the signature of the space- t ime metric, g!'.\/ is _

+

+

+

The Ri emann and Ricci tensors have the respecti~e forms

The close relation between the Riemann curvature tensor and gravitational effects is further illustrated,

for example, in the equations of geodesic deviation, (2)

define t he paths of a D

pair of Driighbouri11g, freely falling particles and.

I),r

denot es the absolute derivat ive along the curve

x

a

(

T

).

A freely fal ling particl e is at rest i n a coor dinate

frame fal ling with it, whereas a pair of neighbouring

freely falling particles will show a relative acce

lera-tion given by eq. 2.2. Tc an observer travell ing with

the frame thi s mot ion wi l l indicate t he presence of a

gravitational fielio

The el ectromagnet ic field on the other hand,

appear s in this picture as a field "embedded" in space-time,

the geometry of which i s det ermi ned by gravitation. A

resol ut ion of this difference i n the rol es of the two

l ong range fiel ds was pr oposed by Weyl (1918) , (

3

,

4 ).

Howev,,r, along with _,ther et tempts at unificat ion it was

general ly consider ed tu be physical ly unsatisfactory, and

so except for some special r eferences, the electromag

-neti c fi eld i s i ncl uded i n the source side ( i oeo the

r ight hand side of eq. 2.1 ) of Ei nst ei n's fi el d equa

-t ions or of these equations in any subsequent ly modified

formo

Basic t o Weyl' s approach was a general isation

of Ri emann space-time - the \Jeyl space-time, for short.

This space-time has been revived quite recently by some

authors ( e.g. Ross, Lord, and 0-mote,

(

5, 6, 7) )

as a

framework for scalar-tensor gravitat ion theories and for

this reason i t deserves a few comments about its his

tori-cal origins, i n addition to the treatment given in Chap

t-er Sixo

Curvature in Ri emann space-time can be related

t o the idea of the parallel displacement of a vector - the

transport of a vector by paral lel di splacement around a

closed curve resul ting in the final direction of the

vector being different from i ts i ni tial direction. Weyl

space-time, unless t wo points are infinitesmally close together lengths at these points can only be compared

with respect to a path joining t hem. b,'~ause a deter

-mination of length at one point leads to only a first

order approximation to a determination of length at neighbouring points one must set up, arbitrarily, a

standard of length at each point and with lengths r e

-ferred to this local standard a definite number can be

given for t he length of a vector at a point. If a

vector which has length,J,at a point wi th coordinates xc'

is paral lelly displaced to the point with coor dinates

xa .,_ <Sr''. then its change_ of length \veyl gave t c be 61::: =

where r; _{are the}

. l..l component s of u vector field.

For parall el di splacement around a smal l

closed curve t he totc:..l change of length of the transported

vector turns out to be

'

2.1+

where

&d

~

describes the element of ar ea enclosed by the

curve, and

Weyl set tJ

. 11

A and so f,,, ll I""''

field tensor,

f ~

!-'A r\ ·, 11 _{I ,}'\ _,,

proportional to thE electromagnetic potential is made proportional to the el ectromagnetic

~~ Thus the electromagnetic potential determines by eq. 2.3 the behaviour of length on parallel displacement and the electric and magnetic fields find ex

-pression in the derived tensor, f!Jj\. This tensor can be

shown to be independent of the initial choice of length standard, which is a necessary condition if i t is to be physically meaningful.

A difficulty of the theory was an apparent con-flict between eq. 2.3 and the interpretation given

time appear absolute and i ndependent of space-time ::,

posi t ion. If the coefficient of pr oportional it y between

2

C

(; and A is assumed real and put equal t o ~₀ (

· µ ll e i s

the charge of an el ectron and C is dimbnsionl ess ) then a

recent experiment,

(8)

,

places an upper bound on

C

of 10-

4

7

•

Such a figure, however, does not exclude the geometric int erpretation of the electromagnetic field in terms of the Weyl space-time if, for example, Wcyl 1_s origi nal idea of equivalent initial length standards is rnodi fi ed to give speci~l status t o atomic standards.

Aside from i ntroduci nG the Weyl space-time these comment s emphasize a feat ur e of length standards in Riemann space-time, wher e.once they ar e defined i n terms of atomic standards at a point , paral lel d is-pl acement al lows the comparison of lengths taken at separated points. Without get ti ng involved i n pro b-l ems of measurement we shal l just assume that on this basis, the physi cal descriptions of atomic systems ar e i ndependent of space-time posi t ion and that by using t hese syst ems the space-time i nt erval measured between neighbouri ng event s is given by

- !.l. V ~-l VdX" c1.x j =

wher e gµv i s identi fi ed wi th the gravitational field variable appearing in eq. 2.1.

With this brief and rat her indi rect look at some of the ideas i nvolved in the Riemann space-time we r eturn t o look at Einstein's field equations and his descript ion of gravitat ion i n order to pr ovide a back-ground before intr oduci ng the scalar field.

Because the dynamical variable of the

gravitational field in G~neral Relativity is t he rnetric t en~or, it plays an important geometry-det ermi ning role i n space-time and the field equatiuns can be understood to coupl e the g~ometry of space-time to matter. Thus the only physical constraint imposed by these equations

,:, so \ !ndei1 :?:,,,1'.:'.llel lli::J ~1J.::i.concnt e, vector ne.ilYCG.ilIB , r"'l\1",,,.-1.,~ _{.,.;~·~'1 ,..,o~·IV-lin--?· ·}1_{·n r+:ru·1i("'! c::--:::--Y'lr1;":'1}

on the nature of matter is that its energy-r.1omentum

tensor has zero divergence.

Implicit in dll of the discussion so far has

been the division between the gravitational field and

matter. In his discussi on of the action princi pl e

formulat ion of hi s field equations, Ei nstein

(9

),

stated

an assumption to ensure that this distinction carried

over to the action principle - - that is, the Lagrangian

density could be divided into two parts, one of which

refers t o the gravi tational field and contai ns only the

metric tensor and its deriv&tiveso The approprir1t e

density for this part is the Ri emann scalar den•ity and

apart from a cosmological t erm the r esulting act ion for

th0 free gravitational field is unique in givinc field

equat ions which ar e linear i n the second derivatives of

th~ metric and which in the weak-field limit give the

Newtonian case.

In spite of this success in giving empty

space- t ime field equations that are uni que modulo a

cosmol ogical term, the action principl e without fur ther

assumptions,does not offer much insight into the nature

of t he ener gy-moment um tensor. So the action principle

remains an important method for constructing field equa

-tions.

In order to make progress later on, much use

is made of the Princi12le of Mir,i mal Cou12line;, ( 0 . g.

10

which Anderson notes is not an essential part of General

)

'

RE:lativity. If a material system is considered in

Spec-ial Relativity as a set, X of matter fields th-:m H:; t~l::!r

-Lagrange equations of mot ion, i n some inertial frame, will

follow from an action principle

r

~

6j

~,rd'x

=

o

,

for suitable variations of the variables X • The

r

action (

6)

lyd

4x ) will depend

and with Tl ·v replaced by g

1-L llV

on the Lorentz metric,

"\.l.Y ,

the free gravitationJfield action gives the required

. 1

actio~ for the gravitational and matter field equations. denotes the matter or nongra.vitational part of the full Lagrangian density ( i.e. with the principle

assumed,

£.

, NG

is the mi nimally coupled

½vi:

)

then the energy-momentum t ensor of t he material system is de-fined in terms of the system' s :1_rc.spsnse11 _{to the metric}

field by

J

t

iJlo _{' _g,,y}\ _. rt;.

\

oxix

)

+

_•

_•

_•

2

.

8

This definition sti l l holds without appeal to the

Principl e of Mi nimal Coupling but i n this casn the

connec-tion described between / , HG and LU[ could not be supposed

t o hold.

An aspect of the field equations not ed here

then, is that the nature of the energy-moment um tensor is determi ned by criteria outside of General Relativity. To work within the framework of General Relativity leads to

a~ extreme position such as suggest ed by McCrea (11),

that t he Ei nstein tensor is to be interpreted or identified as an energy-momentum tensor and the central question is then which geometric constraints imposed by the field equations are physically meaningful. This rather formal approach has been developed a little by Harrison, (12),

in a way, to suggest that scalar-tensor gravitation theories are in fact derivative from Einstein's theory by suitable

interpretations of the energy-momentum tensor. However this view is a bit unorthodox and we shall r eturn to it later.

It was mentioned in the introduction that the

gravitational field, as described by Einstein1 has a gauge -like degree of freedom. This of course corresponds to

1 _{this minimalcoupling prescription is sometimes, when needed,} supplemented with the rule that

system, sometimes characterised as a locally freely

fal l ing system, i n which fer a sufficiently small

neighbourhood of the point the metric is the Lorentz

metrico By describing physics in this neighbourhood

in t erms of such a coordinate system the effects of the gravitational field are transformed away. Essent ially

it is this feature of General Relativity - that gravita

-tional or cosmological effects can be made t o vanish in

the small, which has been questioned and led to the

scalar-tensor gravitation theories.

In

1937

Dirac, (13), ( and

1

9

38

(14) ) ,

suggested that an expanding model of th8 Univ&rse not

only provided a cosmic time scale but also al lowed the

~ossibility that t he gravitat ional consta~t may vary with

this time.

to a unit of

one obtains a

Ey t aking the ratiu of the age of t he Universe

,

e

•

h

time fixed by atomic const ants \ e.c;. DOB or

n

ca

)

40

number , t, of the order 10 , and by tak

-ing the ratio of the gravitat ional force to the el ectric

force between typical ly charged-particles one obtair.s a

r~

n

2

_-40

dimensionless expression, · _--;2 , of the order 10 •

So for this epoch

~

and with m and o supposed constant this r elation becomes

G

'

which Dirac's hjpothesis implied, held for al l epochs.

In more general t erms Dirac's hypothesis, (14) , stated

that "any two of the very large dimensicnless numbers

occurring in Nature ar e connected by a simple mathematical relation in which t he coefficients are of the order of

magnitude unity" and as a consequence if a number varies

A feature of the cosmology which Dirac was

led to, is that fundamental significance need not be

given t o t hese numbers. For example, from eq.

2

.

9,

the

ratio of gravitation~l to electric forces is smal l because

the Universe is old. Although Dirac1_{s cosmology could}

almost be ruled out by present observations, the idea

re-mains t hat fundamental constants and in particular, the

gravitationQl constant, may not in fact be constant.

Some observable effects of a variable gravitational

con-stant have been discussed by Jordan,

(

15)

,

and by Dicke,

(16)

,

but because these effects are geophysical or

cos-mological, the systems involved are complex and the

numerical data avai lable i s i nsufficient as evidence for

variation of the gravitational constant. Recent results

by Shapiro, (17), using planetary radar systems, and

atomic clocks put an experimental limit on t he fractional

t ime variation of t he gravitational constant as

4

x 1

•

-

10/ year

und so the ide~ of a variab]e gravitational constant is st i l l

a conjecture which has not b~en est ablished by direct ob

-servation.

Einstein's equat ions appear at pr esent to

describe local gravitational effects quite adequately and

one could expect these equat ions to hold for t he Universe

ns a whole. But, since t he equat ions require t he gra

vi-tational constant, when measured in units defined by atomic

standards, to be constant they need to be modified if Dirac's

hypothesis is assumed t o be valid. A simple way to

intro-duce a variable gravitational constant into the field

equations is to make the gravitational constant a new local

scalar field variable depending on position in space-time.

Historically; Jordan

(1948)

was the first to use

this approach to incorporate a variabl8 gravitational constant

in a field theory of gravitation. He originally used the

five-dimensional r epresentation of General Relativity

de-veloped by KaJuza

(1921)

and Klein

(1926)

and later

(1955\(\~))

scalar-tensor formalism. These earlier references to

Jordan's theory are given more completely by Pauli,

(19) , and a comprehensive review of Jordan's theory is

given in an article by Brill,

(20)

.

The most widely

known theory of gravitation which i ncludes a variable

gravitational constant is the Brans-Dicke theory (1961)

which is formally, very closely related t0 Jordan's

theory. From 1961 onwards, t he existence of such a

long-range scalar field seemed feasible ( but perhaps

experimentally doubtful) and in t he writer's opinion

the most interesting developments to come from the Brans

-Dicke theory relate to the problem of constructing

dynamical laws involving the gravitational field variable,

the scalar field variabl e and matter field variables.

Finally, one notes that, to introduce t Le sco lar field

as n l ong range cosmological field for the purpose of

obtaining a variabl e gravitational constant is by no

means tho only way of giving expression to Dirac's

hypothesis.

In a pattern similar to that described above,

other authors have postulated a scalar field and

intro-duced scalar field terms into Einstein's field equations

in order to deduce from these equat ions preferable models

of the Universe. Hoyle's equations ( 1948) (21) implied

that matter was not conserved and gave a steady-state

model of the Universe. Here the scalar field was related

to the creation of matter, in contrast to the scalar field

postulated by Rosen (1969)

(22)

which had no interaction

with matter. Rosen's equations gave an oscillnting model

of the Universe,.

The Machian idea of a connection between local

physical laws and properties of the Universe as a whole

has already been partly met, with Dirac's hypothesis. In

an effort to explain inertia, the Brans-Dicke theory was

based more on Mach's Principle than on Dirac's hypothesis.

Some further references to these ideas are given in

and Firmani's (1970) (23) modi fied Brans-Dicke theory in

which radiation is given a more Machian property in

de-termining alone with matter, the inertia of a body.

However their theory is restrictive and applies only to

a homogeneous, iEftropic space-time in which the matter

content can be represented as a perfect fluid. Gursey's

(24) theory is Machian motivated in a different kind of

way and this theory is discussed in Chapter Five.

It is apparent that the scalar field

has been introduced into the General Relativistic

frame-work to incorporate many quite different physical

features which have been t hought desirable and found not

to follow from the usual interpretations of Einstein's

field equations~ The field equations of the

scalar-t ensor gravitation t heories that have been devised,

poss-ess cosmological solutions describing a variety of models

of the Universe. So with these rather general comments

summarizing (and substituting for) what could have been a

lengthy look at the individual theories, the relation be

-t ween t he scalar-tensor and Einstein's descriptions of

gravitation is taken up with reference to Harrison's

papers (12, 25).

Harrison, (12) , states that the scala

r-tensor f ield equations "constitute in f&ct a limited and

particular class of equations that derive from General

Relativity and are of lesser generality". He arrives

at t his view after showing that the forms of the action

principles of different scalar-tensor theories can be

transformed into each other and into the form of the

action principle for General Relativity, by recalibrations

( i.e. conformal or scaling transformations ) of the field

variables. Thus1 together with the observation noted

earlier that the physical nature of the energy-momentum

t ensor lies outside of the scope of the theory of General

Relativity1 a scalar-tensor gravitation theory seems to

be, (25), "a specialised application of the theory of

Einstein' s theories of gravitation do not have the same status as gravitation t heories. General Relativity be-comes in a sense a generic theory where one works in a Riemann space-time and postulat es field equations based on assumptions about the content of the energy-momentum tensor in Einstein's field equations. A classic example of this procedur e is given by McCrea

(

1951),

(26), who found that Hoyle's results

(19

4

8)

could be derived from Einstein' s field equations if negative stress was allowed in the energy-momentum t ensor of the Universe. Another example is implied by r emarks of Dirac

(

1938)

that, assum -ing the gravitational constant was variable with respect to atomic standards of measurement, Ei nstein' s equations should hold for units which ~ary appropriately with r espect to the atomic standards.

Perhaps t his view emphasizes the ge o-metrization of gravitation achi eved by General Relativity

and t he special importance placed on the int erpr etation of the Einst ei n tensor.