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

Chapter Two. B:-tclcground 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

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

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

lo

_{> 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 supposed that the .transported vector has a different

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 pr ob-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

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

:-partial dt:ri vati ves - • covariant derivatives.

one can always introduce at a point a local coordinate 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.

In contrast, the assumption of the

follow-ing chapters is that one wants t he scalar field to be an

integral part of t he description of gravitation for the purpose of giving position dependence to the gravitational

const ant, inertial mass or just to offer new models of the

Universe and therefore t he scalar-tensor and Einstein's

Chapter Three

Having introduced some of t he motivation that has been given for includi ng a scalar field in the description of gravitation, the natural procedure is to obtain an appropriate scalar-tensor gravitation theory as a modification of General Relativity. The basic equations of this "new" theory would generally follow from an action principle and so the relevant class of gravitation theories to which the "new11 _{theory would belong, would be the class} of Lagrangian - based scalar-tensor gravitation theori es.

Before looking at this class of theories in more detai l it is convenient to first establish some de -finitions and concept s by looking at some general pr operties of a broader class of physical theories and then specialis -ing t o t he class of gravitation theorieso For this pu r-pose some of the definitions and concepts given by Anderson (1), Trautman (2) and developed by Thorne, Lee and Lightman

(3)

ar e summarised under t he headings

3.1 Space-time theories 3.2 Gravitation theories

3.1

Space-time theories

The basic element of these theories is the four dimensional space-time manifold which assumes that physical events can in some way, be associated with a continuum of points and that the points "fit together" sufficiently smoothly to form a four dimensional

differentiable manifold, M.

The manifold

mapping group

(MMG)

(3)

is the

a

~ (P) transforms to a new coordinate system given by

I

c

u(....,'

~(

-1 )

X J.:) = X /\. p ,

P

E

M

A geometric object ( field) (2) is a correspondence

which associates with every point F E Mand every system

f X".a 2

of local coordinates t 5 around P a set of N r eal

numbers ( the components of t he geometric obj~ct ) to~ether

( I I I )

with a rule which det ermines _{y 1_.,y2_} , •• ,yF e:ivon

by-y: (P.,if x· Q'./ )

1

)

• ( Y1/ ;, Y2/ ,,•••,Y:T / )

i n t erm.s of ( Yv _Y2, •••,Yp ) and the- values at P of the

functions :.:,nd t lteir part ial derivati ves which r elate the

I f

.,,a

l

r

a, 1 coordinate syatGms l~ and tX >•

A space-time theory then,

(3)

,

is a theory t hat

possesses a mathematical r epr esent ation constructed from

a four dimensional spac0-time manifold and from geometric

objects defi ned on that manifol d. Tho geometric objeets

of a particular representation are called its varinbles

and the equations which t he variabl es must sati sfy are

call ed the physical laws of t he representation.

A kinematically possible trajectory ( kpt ) ( 1,

3)

of a particular r epr esent ation of a space-time theory is

any set of values for the com~onents of al l the variables

in any coordinate systemo

A dynamically possible trajectory ( dpt )

(3)

is any kpt that satisfies all the physical laws of the re-presentation.

A covariance group of a representation

(3)

is a

group G •.:hioh ( i) maps kpt of

-c

~

ie

re~.:."G~cn~c.:cion in~o kp ~

(ii) maps dpt into dpt

.

An internal covariance group is a covariance group that involves no diffeomorphi sms of the space- t ime ont o itself, i n contrast to an external covariance group which is a covariance group and al so A subgr oup of MMG. The complet e covariance group i s t he lar gest covariance

group of the representation and referring to

(

3)

t he effect of a particular group element G is characterised

as fol lows :

Suppose G consi st s of a diffeomorphi sm hand

an int ernal transformation Hand write

G = ( h, H )

,

where i t is under st ood that if G i s &n exter nal transforma

-t ion, His the i dentity transformati on, and i f G is an in -t ernal transformation then h i s the i denti ty mapping.

I f y is a geometric object ( eq. 3.2,

3

-

3)

and ~ comronent is written i n funct ional notation as

,

then the set of functions

defines a kpt. Under h this kpt maps into

3

.

5

I rxa

where l is defined by eq. 3.1 and under H, y~ trans -forms into a new geometric object

I

y

U

y

The net effect of G on the kpt ( eq.

3

.6)

is

I

G: YA (P,

fx

a

l)

f

•

.

y' "/P,

f

xa

J) •

The "changes in

y''

which

I

6yA

(P

,

l

xa

J)

characterise Gare de£ined by

I

cl

'

= y

A'

(P,

lx

(X

l}-yll. (h-

1

P

,

{x

J)

I

= y'

Al

evaluated at xa• (P)

Y.\

I

c·.r2.J.u.ated at xa =

x

a'

(P)

H

3

.8

I

I f

5yA

(P,fx

a

l)

= at al l P and for al l _3.10

coordi nate systems then G i s called a symmetry

transformation of the geometric object,

y

,

and the set

of al l such elements G for~ the symmetry group of this object.

If a space-time theory has a represent ation

for which MMG is a covariance gr oup then the variables of

this repr esentation can be classified according to three

types - confi n0d, absolute and dynamical .

The confined veriables

(3)

are t hose that do

not constitute the basis of a faithful r ealizat ion of Mr,,ro

e.g. universal constant s.

An unconfined variable, B, i s t E:rrnt:;d an

absolute or a dynamical variabl e by the followi ng t est

(3)

(i) Choose an arbitrary dpt and l et

BA

(x

a

)

be the functi ons which describe the components of B for this dpt .

(ii) Define the equivalLno~ class of a dpt

(iii)

as the set of dpt which map i nto the

given dpt for some clement of the com

-plete covariance group of the repr esenta

-tion •

Check to see if the same functions B

(

xa)·

·

·

A

appear in each equivalence class. If

they do for every equivalence class and

for every choice of the arbitrary initial

dpt then Bis an absolute variable. If

they do not for some particular choice

of the ini t ial dpt and for some particular equivalence class then B is a dynamical variable

coupled by the physical laws to the r emaining

variables of th0 representatioil and (ii) its variables can be eliminated from the representation without a lter-ing the structure of the equivalence classes of dpt and

without destroying the covariance of the representation under MMG.

Finally, for a represcntntion of a given

space-time theory the physicul laws can be classified

into four sets

(3)

( i)

(ii)

boundnry condi t ions these laws

which involve onl y confined variables

prior geometric constrni nts those

laws which involve absolute ( and

perhaps confined) variables, but

not dynamical var iables

(iii) decomposition equations those

and (iv)

As examples

which express a dynamical variable

al gebraically in terms of other variables

dynamical l aws

laws•

al l other physical

If the electromagnetic gauge group is ignored; the complete covariance group of General Relativity is MMG and al l the unconfined var iables are dynamical and contain no absolute parts. On the other hand, for Jordan's theory the complete covariance group is the

3.2 Gravitation Theories

Scalo.r-tensor gravitation theories belong to

the general class of space-time theories and, as in General

Relativity, they seek to combine gravitational and

non-gravitational laws by means of an equivalence principle.

Thus gravitation is described by a set of gravitational

fields i ncluding the metric t ensor,

gµv

,

and i t is required

that in the local Lorentz frame of ~ _oµv all non-gravitational

laws go over to their standar d special relativistic forms.

Some of the ideas i nvolved here have been made

mor e precise in a foundation analysis of gravitation theories

given by Thorne, Lee and Lightrnan,

(3

).

They distinguish between gravitational and non-gravitational phenomena by

r egarding gravitationul phenomena as either prior geometric

effects or effects generat ed by mass-energy and they further

clarify this by introducing the concept of a local test-ex

periment . For the purpose of formulati ng the equivalence

principles, the essential features of such an experiment

ar e that

(

i)

(ii)

i t is performed anywhere in space-time

it is performed with freely-fal l ing

apparatus

and (iii) it is performed over a region of

space-time, sufficiently small for

the inhomogeneities in al l external

fields to be irrelevant.

A special kind of local test experiment is the

local, non-gravitational, test experiment

(3)

which is

perform-ed in a region of space-time with a nearly uniform gravita

-tional potential throughout it, as calculated using Newton's

theory, and which if repeated with successively smaller

mass-energies in the r egion, that leave the characteristics

of the various parts - e.g. charge, angular momentu etc.

-unchanged, giv~an experimental r esult which does not

Dicke's

Weak

Equivalence

Principle

(

WEP

)

(

3

,

4)

states that if a test particle is placed at an initial event in space-time and is given an i ni t ial velocity ther e, then its subsequent world line will be independent of its internal structure and composition. Following

(3)

a test particle is taken to be an uncharged body with sufficiently smal l "self-gravitational ener gy" as calculated by Newton I_s

theory and with sufficiently smal l size to guarantee that any test of

WEP

is a local, non-gravitational test experiment. Given that

WE

P

is valid, the world lines of test particles ar e a preferred family of curves in space-time - with a unique curve in a given direction t hrough each given event.

that

Einst ei n Equivalence Principle ( 1EP ),

(3)

stutes

(i)

and (ii)

i✓ EP is vali d

the outcome of any local, non-cravitat ional t est experiment is independent of where

and when in the universe it is performed and independent of the velocity of the freely-falling apparatus.

Dicke's Strong Equivalence Principle ( SEP ) (

3

,

4 ) states that

(i)

and (ii)

l;JEP is valid

the outcome of any local, gravitational or non-gravitational, test experiment is

independent of where and when in the

uni-verse i t is performed and independent of the velocity of the freely-falling apparatus.

The scalar-tensor gravitation theories vi.date SEP because the relevant scalar fields introduce preferred location effects, while they satisfy* EEP and thus possess metric r epresentations (

3 ),

in which a metric is defined

on the space- t ime and the world lines of test particles

are the geodeoics of that metric. Also, for a metric

representation, the metric involved in EEP is called the

physical metric, while the other gravitational fields,

that is (3) unconfined, r elevant variebles of the representa

-tion which in the absence of gravity ( e.g. as in the analysis

of a local, non-gravitational experiment ) reduce to

con-stant, or absolute or irrelevant variables, are called

auxillary gravitational fields .

Two further general proper t ies of scal a

r-tensor gravitation theories ar 0 that

(i) MMG is ~ covariance group

and (ii) for a particular representat ion the

dynamical laws ar e assumed t c follow

from an action i ntegral which is made stationary with respect to var iations

of al l dynamical variables.

action principle written as

,

With the

the Lagranginn de~sity, ( a scalar density of weight ₊ 1 ·~)

can be spl i t into two parts

I

where the gravi tational par!, (3) , is the largest part

'

which contains only gravitational fields, and .t HG is the

non-gravitati onal part which will be oft en referred to as

representing the matter fields or matter "content" in a

spac0-time.

This representation is said to be universally

coupled (3) i f

( i)

·

<k.

HG contains a second rank symmetric

tensor,

iJr

11

v , of the same signature

as the Lorentz metric, as the only

gravitational field

This restriction on t i s sufficient ( e.g.(1)) to

guarantee that any transform ( by an element of MMG)

and

(ii)

(iii)

in the limit as gravity is "made absent"

ili becomes a Ri emann - flat second rank

'l'µv

symmetric tensor, TJ~tv and whenever such

a 'llµv total

rc=plac8s

,:,

_{'l'µv ' ~}

r

_HG becomes the special relativistic Lagrancian

the prediction for the outcome of any

local non-gravitat ional experiment remains

unchanged when *µv in the region of the

expcrimen t is r eplaced by a RiemRnn-fla t

second rank symmetric tensor.

This concludes, more or less, a gl ossary of terms

begun in Sec~ion 3.1. It consists of definitions and

con-cepts that have been borrowed largely as formulated in a very

recent paper by ThLrne, Lee and Lig~tman and adapted to pro

-vide what t he writer thinks is a relevant context for the

scalar-tensor theories ns gravitation theories.

3.3

The Scalar-Tensor Model of Gravitation

In this model gravitRtion is described in terms

of two fields defined on the space-time manifold - a metric

tensor field,. eµv and an always positive scalar field, ¢,

The most general

J:

G- , ( t o within n div(::rgence term ) that

gives dynamical laws of no higher than second differential

order and with the second derivatives appearing linearly

has the form (5)

3

.

·

13

where

f

₁ and

f

2 are arbitrary functions of the scalar field,

but may be reduced to constants by suitable recalibrations of

the variables ~ _oµv and~ respectively.

't' f

₃

is an arbitrary

function of the scalar field. R is the Riemann curvature

scalar.

On writing the non-gravitational part of the

Lagrangian density (

l)

as

L

NG

=

~

N"G-(X,gµv,cf>

)

23

the variable X.collectively stands for al l the non-gravita

-tional fiel ds and wher e for si mplicity it is assumed that

no higher than first derivatives of the dynamical variables

appear, the action principle for the model becomes

6j

[(

r

₁₍

¢

)R +

f /

¢

)

g

µ

v

¢

,µ

<t>

,v

+

r

3

(

¢

))C

c

+

i

i'-TG

]

a

4

x

=

o.

3

.14

For the model to obey EEP restrictions need t o be put on

,-,

t he functional dependence of cLTTG on the gravitational 1'!' :r

fiel ds • As this is so far a purely formal element of

t he model these r estrict ions wi l l involve a choice of

uni ts so that WEP hol ds and an interpretation of cLNG

so that EEP holds. In particular , as the non-gr avita

-t ional laws fol low from eq.

3

.14

in t he absence of gravity,

for variations of the non-gravitational fiel ds,

t

1~ in the

absence of gravity must be t he total special relativistic

.Lagrangian whi ch is denoted by

c:L

NG

(

X

,

11

µ

)

•

The natural choice of units is based on atomi c

standar ds - e.g. as Ohanian, (6) suggests, one takes a

neutral, massive spin-zero meron and defines the unit of

l ength (t ime) as the Compton wavel ength of the meson

and the unit of mass as the meson mass. Usi ng an arg u-ment due to Fierz,

(

?),

he shows that for t his choice of units the free meson falls along the geodesics of

g

µv

•

This would also apply to any localised system which could

be treated as a test particle.

Bergmann,

(5)

using a quite general argument

has shown that, unless £ ,J:-:G- has no scalar field dependence the motion of a test particle is indeterminate. In a simplified form due to Ohanian~

(6)

one first derives the

four differential i dentities for the matter field. Rather

than using the field equations as in

(6),

these follow more

easily from the fact that infinitesmal coordinate tran sfor-mations are symmetry transformations of the non-gra

vita-tional part of the action •

For variations of the variables and their first derivatives

which vanish on the boundary of the r egion of integration

eq. 3.15 can be written

where the Hamiltonian derivatives are defined, as e.g.

6

f'

=

a

£

-

a

(a

r

)

- '<--NG

7J

NG

'J

C(

7J

""'-i>TC-bg

µ

v

G

µv

x

G

µv,

a

~

,:.

r

NG = -""'

a

1·· NG _ -

a (

o: \ --

a

ti},1J NG )

6¢ 0¢ .

ox

,a¢

_,

a

The Euler-Lagrange equations of motion for 'X

and so

r

r-c

-

.,..

.

-

--

!,

/ j

-°·-,£NG

6c

µv

+

~

J

\:c-

u(_') 'j;

a.

··x -·

o

J

_o

b

µv

;_;

~~

-Writi ng the coordi nate transformation ss

I

J

•

•

a v'. .• c:: '

I

<-:a

I

/< ·1

X= X + s - '-"

3.

,9

eq. 3.10 gives = -l: -F "'•.t•V ··'v. µ

'

,

'

=

-

-¢

1-;,"\_ , Ct

where the covariant derivative "

the Christoffel symbols

_•

'

" is with r espect to

Eqs. 3.20, 3.21 in 3.18 give on integration by parts and

for arbitrary l;a

µ

or equivalently T

v;µ

where

•

In the external fields

g

µv

and

¢

,

a locally freely

falling frame can be introduced in which eq. 3.23 becomes

for a sufficiently small system and with its self-gravita

-tional field assumed negligible,

3

.

22

4-2

5

,

where

£

NG refers to the system and 'l'µv now denotes the

special relativistic energy-momentum tensor of the system.

with

gives

Integrating over

V

f

r

oV

3

7. 0 c:;

p

_

=

T

d

x

,

eq. J•~~ cLef-n.

V

d n

-· o~ d."'!'.

the volume of the system and

•

Thus the system will experience c:.n acceler ation relative

to the local fre:elJ fc,llinc frar:ie ur:less )(~~' v §_~l'TG d \ = 0

8~c)

approximat ion involved

or because of the

/

~L11,G d

3

x =

&cf/

"

0 •

This restriction holds if the gravit~tional

( \ ,, !\

fields enter into • uccording to d.. - i

(

X

f

(r!

,

)

r

,

• )

,.,_HG C, NG - \'.i.,NG ' l.~ y 0µv , where fl is an arbitrary function of t he scalc:.r field,

-~

as for example postulated for the scalar-tensor model con

-sidered by Wagoner,

(8

).

A metric recalibration can r e

-duce

or f

I+ to a constant and in this r epresentation eq.

3

.2

7

3

.

28

is satisfied.

For WEP to hold then, one can take

cL

NG to have no scalar-field dependence, and for

EEP

to hold one can

take

:LNG

to be

£,

N

G(X,

gµ

)

,

the minimally coupled

total special relativistic Lagrangian density for the

system.

i.e

.

J'...NG

(X,11µ)

•

I,NG(X,

g

µ)

as Tlµv • lsµv •

In this way a metric and at the same time universally coupled

representation can be arrived at with the action principle

given by

f

µv · ('

4

6

[(:r

_1(~)R

+

f

₂₍

¢

)

g

~

,µ

¢

,v

+:r

3

(

¢

))Cg

+

~NG-

(X

,

gµ

v)]

d

x=O

3

.29

Because

¢

can be recalibrated, independently of gµv ,

to reduce

:r

₂ to a constant this representation has effectively

two arbitrary functions and an arbitrary (dimensionless)

The main problem for the scalar-tensor model i s to

select a satisfactory theory by eliminating some of these arbitrary features in an acceptable way.

e.g. the Brans-Dicke t heory is obtained

f _{rom t e} h _me t _{ric represen} . t t _{a ion} . b _{y pu} t t _ing . -"' ,.~ _p , -1

.l

1 == ·r,.L 2 = - •X,) and which gives a simple wave equation as the field equation for

~o

To conclude t his section, mention i s made of a paper by Hart,

(9),

in which a scalar-tensor model is given by t he -action principle, eq. 3.14, with

£NG= f

5

(~6~G(X,gµ) , where f

5

is an arbitrary function

of the scalar field. From the differential identities for

J:'C'- (i • .:::. eq.

3

.

22 wit~ J.NG- replaced by

J\

:

.

)

the scalar-tensor conservation laws ar e derived i n a way

analogous to General Relativity. If f

5

is n constant

( as one needs for a metric or universally coupled r e -present ation) and/or a dimensi onless function it is found

that the total scalar-tensor conserved quantities have appropriat e units of ener gy or momentum.

Hence, as

¢

generally has dimensions of

some power of G , in order to have conserved quantities

that ar e meaningful unit-wise, at least, it is sufficient

to take f i:-: as a dimensionless constant , in which case the

'.)

scalar-tensor model considered is given by eq. 3029.

Using Noether's theorem ( e.g. Trautmann, (2) , )

which associates a differential conservation law with an

infinitesmal coordinate transformat ion, one can obtain

differential scalar-tensor conservation laws,

(9),

that

depend on the choice of ~a in the coordinate transforma

-tion eq. 3.19. Because of the infinite order of MMG

there is a corresponding infinity of conserved single index

quantities ( such as Komar' s vector ) and if the physically

important quantities are generated by infinitesmal coordinate transformations which are symmetry transformations of the

gravitational fields then from eq.

3.20

and eq. 3.21

e

=

og

µv

= -i; - _l;v;µ

µ;V

]

3

.30

_C(

27

Killing vectors with components which .satisfy the latter equation, Hart calls r estricted Killing vectors and an

outstanding probl em he suggests, is whether or not the

Killing vectors are actually r estrict ed for al l grav i-tational fields. Also in this connection he suggests

that the relationship between the restricted Killing

vectors and the scalar-tensor conservation laws needs

further study.

3

.4

Experiment al Tests and the Scalar-Tensor Model

of Gravitation

The metric representati on ( eq.

3

.2

9)

of the

scalar t ensor model can by n r ecalibration of the metric

and the scalar field variables be transformed into the

r epresentation with an action principle

where f

6 and f 7 arE.: arbitrary functions of the

scalar field and n =

±

1 • For this representation

Wagoner,

(8)

,

has considered the linearized we&k-field

limit in relation to t he solar-system experiments and arrived at two possible r estrictions :

or

(i) the locally measured gravitational

constant is the same as Newton's

gravitation~l constant, and

r

6

gives rise to a massive short range

scalar field

( ii) the locally measured gravitational constant depends on the scalar

field o.nd f

6 corresponds to a

"cosmologic':ll term".

theories ( see Chapter Five) which give t he same

predictions for the solar-system experiments as

General Relativity.

The second case leads to predictions for

the light deflection and perehelion shift observations,

which depend on the first and possibly second-order

terms i n the expansion for f

7

However, depending

on the sign of n , the model does not seem to be

i nconsistent with pr esent observat ions and certainly

no severe r estrictions ar e placed on the for m of the

action principle for the model.

~ prediction of t he scalar-tensor model does

lead t o a violat ion of WEP. Ohanian,

(

6)

,

has shown

that for a massive system t he inertial and gravitational

masses di ffer by a t er~ which is of the order of the

self-gravitational energy. This r esult holds also, c.ntl

in particular, for the Brans-Dicke t heory, (10.). ( 1Ck),

but because of the size of the violat ion there is no confli ct wi th the Eotv6s - Dicke experiments. Thus

the violation of WEP seems t o be a matter of principl e

and as such adds against the scalar-tensor model of

gravitation.

29

3.5

Conformal Invariance and the Scalar-tensor Model

Previously, i n section

3.

3

,

it was shown

that the scalar-tensor model possesses a metric ( and

universally coupl e• ) r epresentation which describes a family of scalar-tensor theories based on an action principle given by eq.

~29

.

After recalibration of the

scal ar-field variable this action principle can be put

into a form characterised by two arbi tr ary functions f 1

and f

3

of the scalar fi eld 0nd an arbitrary constant '{_, , i .e.

oj

ar/

¢

)(

R

-

'{_,(,°J-2

rJJ

,

µ9\

v

gf.+v)+f3(¢)]Cg~JG(X,gµv) L:ti1· X = 0

3

.32

and since observation does not rul e out the exi stence of

the cosmologi cal t erm, f

i¢)

or the presence of

f /

c)

)

i.he problem of removi ng at l east some of the Rrbitrariness from the dynamical laws remains.

For this pur pose then, it is int er esti ng to look at t he r estr ictions placed on the functions

f

₁

(

0

)

and

and

when the conform&l group defined by

gµ.v • gµ,v = 11.gµv, for A an arbi·~rary positi•,re

function of position:

is postulated to be an internal covariance group.

If the group is first restricted to be a

symmetry group of the gravitational part of the action princip2p then

where barred variables r epresent transformed variables, re-quires that, (11)

r/¢)

a

¢'°

1

r

3

(¢)

a

<J>2Tl

'1)2 = -

£

'{_,

and the transform&t ion of ¢

1

,I.. -;;, "\

11

,

~

'i.J • \ j J = f \ \JJ .

·co

~)e

Thus the Lagrangian density for the free gravitational

fields becomes

where and 11.

0

is an arbitrar y constant.

From the form

oft

G one can recognise the gravitati onal

part of Jordan's Lagrangian density ( Gee Chapter Five ),

if 11. is put equal to zero and ~ i s considered to be

0

arbitrary, and in particular if

n·

is put equal to unity

.f

G is the gravi tut ional part of the Brans-Dicke Lagrangian

~

density.

( The idea of usi ng conformal transformat ions to forQally

identify the Brans-Dicke and Jordan theories is also looked

at in Chapter Five ).

Returni ng to the conformal ly invariant Lagrangian

density

lg

,

(

eq.

3

.

39 )

,

one not es that it is conformally equival ent to

J,-

=

(

R

+ "-

)-.f.:.

g

G o

and so with c(_I\TG

=~TG

(

X

,

g

µ)

_,

= ¢Tl g _~µv,

one has t he Lagrangian

density for Einstein's fiel d equations with a cosmological

constant. This choice ofal NG implies that in the original

representation one takes the matter Lagrangian density l NG

.T}

to be minimally coupled to the "metric11 _{cp gµv • For a}

different theory one can t ake

tl

NG to be minimally coupled 7.

to gµv and with T} = 1, ( ~ - - ~ ) for simplicity, the action

principle for the metric ( and universally coupled )

re-presentation is

of[(~R

+i

¢

-\p,µ

~

\..,g

.

µv+ \

0

q,2)-,.Cg+¾

T

G(X,

g

µ)J

cf

x

=

0

The field equations ar~, for arbitrary variations of the

variables gµv and¢ , and their first derivatives which

[e

.

g

.

11]

Gµv +

22

c

r2(cp

µ¢ V -

½

P' d, d,, a\ A--- 1 (,,

_-- oµv• C(r I - y ~•ll• V

-1 , ' ,. ,

2

-

2 r/,j ;( - 1 -- '

3-

+

¢ ¢ 'I-- - 3¢ iJ,p

+

2).. c;,, = O

2

,

o:

0

where µv

g ¢

_;

_µ

_;

, by'clef'u.

V

Contracting eq. 3.41 and comparing with eq. 3.42 implies for consistency that,

( 11)

Hence in the metric r epr esentation, for the gravi -tational fields, g

µv and ¢ , which enter i cto a conformally invariant

<LG

to couple consistently with matter , the trace of the energy-momentum t ensor must vanish. This condition is in fact the ccndi tion for

J:

I-JG- to be car.formal ly invar iant ( see eq. eq _3.l1--7 ) "

Anderson, (12), iant if w = -

i

Thus in particular, as shown exp hci t ly by the Brans -Di cke theory is conformally invar -and. ~(,a = 0.

C(

To all ow the gravitat ional fi elds to coupl e with massive systems ( Tau

j

O ), one approach is to ext end the complete covariance group of the scalar-tensor model by in -cluding the conform.J.l group and

let

a scalar field interaction appear in

~

NCs • Thus the .x:t i on principle is written

oj

[(cp

R

+ ~ -1¢,µ¢, ,;;µv

+

A-0

1

/

J~

+l.:'JG(X~:;µvcp)]d~-x =

o

3 .41+

and beca~se the additional covar iance group elements are

specified in terms of a singl e position dependent function, the field equations satisfy an additional Bianchi-type identity. This fol lows for example from

6I:£NGd

4

x =

o

,

for the infinitesmal conformal transformation

which gives

whert as usual

o

5µv

= Bµv

o

)...

6¢

= -¢ •A. •

[ e •₆• 11 ] Ta a

•

•

01_{Hanlon and Tupper in their paper,}

(11)

,

_{show that the} covariant Lagrangian densities for massive fields of non-zero spin could be put into a conformally invariant form

if one postulat es a scalar-field interaction with mass as

which is in keeping with a conclusion of

(13)

t hat without

this mass transformation there is difficulty i n making

non-gravitational laws and equations of motion conformally in-var iant.

Effectivel y ( for simple cases anyway) the

same r esult occur s if one takes following Anderson

(12)

J:NG = £.NG ( X, q') gµ) ,

where the mass transformation has been accounted for in the coefficient of gµv o A metric r epresentation is obtained

by choosing}. =

(

r

1 in thE: transformation eq.

3

-

33

and the

action principle ( eq.

3

.

44

)

,

in the barred form is then

essential ly no different from the onG for Gener2l Relativity

( with a cosmologi cal constant ).

Although postulating tht conformal ~roup as a covariance group seems to bt too strong a condition on the

scalar-tensor model to give an interesting scalar-tensor

theory, it is perhaps still appropriate to add a few general

remarks. Under the postulate, the complete covariance

group of the scalar-tensor model is the direct product of riiMG

with the conformal group. The conformal group introduces

irrelevant parts into the dynamical variables Bµv and ¢ and a simpl e way to covariantly eliminate these part s is to carry out the transformation eq.

3

.

33

for some pr edetermined

}. • In the barred form the dynamical variables become

- 1

relevant variables and in particular, for the choice }.=¢

the scalar field variable is made completely irrelevant.

This enforces the result shown above t hat conformal invariance

can lead to a purely t ensor theory of gravitation.

Interestingly, a rather philosophical objection

that could be raised against the scalar-tensor model