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

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private study only. The thesis may not be reproduced elsewhere without

the permission of the Author.

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 1973

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 38

'I'ho ED scnlnr-tcnsor theory 38

V.'.lcu~~ oolutions 50

Stand.'.lrd BD cosnol og,J 58

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 ^Is 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

123

-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 the souree must be covariantly conserved and 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

11charge" 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 ¹¹bound"

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 gravita- 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 back- 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 ¹¹massive 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 the typists who have helped in the completion of the thesis.

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 space- t ime manifold a Riemann ::::tructure ,nd gave "meaning" to the gravitational field in termP. of curvature through his ~ravi- 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)

=

,

^2.2

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

xa( 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 accelera- 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 histori- cal origins, i n addition to the treatment given in Chapt- 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

length as well as a different direction and so for 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; . l..l are the 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

above, with the idea that atomic standards of length and

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

posi t ion. If the coefficient of pr oportional it y between (; and A is assumed real and put equal t o ~

2C

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-

47 •

^{Such a figure, howeve}r, does not exclude the geometric int erpretation of the electromagnetic field in terms of the Weyl space-time if, for example, Wcyl ¹s 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 dis- pl acement al lows the comparison of lengths taken at separated points. Without get ti ng involved i n prob- 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"''1~-

1• . -~-.,.., ro

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 action (

6) r lyd

^{4x )} will depend

and with Tl ·v replaced by g

1-L llV

on the Lorentz metric,

"\.l.Y ,

this action when added to

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 :¹rc.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.

\

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 1938 (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 nca ) number , t, of the order 10

40

, 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 with epoch then other dimensionless numbers may be re- quired to vary with epoch in order to keep the relations between themo

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 Dirac¹s 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 gravi- 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\(\~))

he and others developed the theory as a four-dimensional

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 Chapter Four while passin~ mention is made here to Caloi

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 scalar- 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 General Relativity". In this way the scalar-tensor and

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

(1948)

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 geo- 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 theories of gravitation ar e to be on equal footing as gravitation theories.