S O M E R E S U L T S IN T H E G E N E R A L S C A L A R T E N S O R T H E O R Y

T A R K E S H W A R S I N G H a n d T . S I N G H *

Department of Mathematics, Banaras Hindu University, Varanasi, India

(Received 17 October, 1983)

Abstract. Raychaudhuri-type equations are written for a cosmological model filled with a perfect fluid and obeying the equations of general scalar-tensor theory proposed by Nordtvedt. It is found that a non-rotating cosmological model may, in general, have a minimum volume. The Hawking-Penrose energy condition suggests that the singularity can be avoided in this theory.

1. Introduction

Einstein's field equations being not completely Machian in nature, Brans and Dicke (1961) introduced a scalar field into it to make things less relient on the absolute properties of space. But recent experimental evidence suggests that if the Brans-Dicke theory is to justified the value of the f~ in this theory should be very large. With this large value of f~, however, it is difficult to distinguish between the two theories, at least from their consequences. On the other hand, since there is no a priori reason to exclude the introduction of any long range scalar field in the evolution of the Universe, which might be quite important at some epoch, Nordtvedt (1970) modified the Brans-Dicke theory where ~ now becomes a function of scalar field ~ instead of being a constant.

Raychaudhuri (1955) discovered an important equation for arbitrary cosmological models with incoherent matter. This equation is very useful for making general pred- ictions about the evolution of cosmological models obeying Einstein's law of gravitation. The equations were generalized to the case of matter exerting pressure by Ehlers (1961) and to the steady-state theory of Hoyle and Narlikar (1962) by Raychaudhuri and Banerji (1964). Later Banerji (1974) has discussed Raychaudhuri-type equations for the gravitational theories of Brans-Dicke (1961) and of Hoyle and Narlikar (1962) in the general case of a Universe filled with a perfect fluid. Recently, Singh and Singh (1983) have obtained the Raychaudhuri-type equations in the modified Brans-Dicke cosmol- ogies (cf. Endo and Fukui, 1977; or Uehara and Kim, 1982).

It is worthwhile generalizing the results in some other theory such as scalar-tensor theory of gravitation. In this note we propose to write Raychaudhuri-type equation in the general-scalar tensor theory of Nordtvedt (1970). The Hawking-Penrose energy condition has also been discussed.

* Applied Mathematics Section, Institute of Technology, Banaras Hindu University, Varanasi, India.

Astrophysics and Space Science 102 (1984) 223-228. 0004-640X/84/1022-0223500.90. O 1984 by D. Reidel Publishing Company.

224 TARKESHWAR SINGH AND T. SINGH

2. Homogeneous Cosmological Model and Raychaudhuri-type Equation

The field equations in the general scalar-tensor theory of Nordtvedt are obtained from the variational principle

c5 f I167zY + ~bR + f~(q~)qSu~b~'l x / ( - g) d4x = 0 , (1)

where q~. = ~p, # - ~?(?/Ox ~', 0 being the strength of a scalar field. R is the scalar curvature and ~ is the matter Lagrangian.

The field equations are

R~,~ 1 8re T.~ - - ~g.~(p~,q5 ] - 2 g ~ R - ~ ~ [ ~)u~) v _ 1 o:

1 - - [ e . ; . - g . ~ ? ~ ] ( 2 )

and

~ ; \ = ( _ g ) - 1 / 2 [ ( _ g)1 /2 ~ ] , ~ _ 8g ~ ' T qS~q5 ~ , (3)

3 + 2f~ 3 + 2f~

where f~' = dfl/dq~ and semi-colon represents covariant derivative. T = T~, the con-

tracted form of the energy momentum tensor T.v. We now take the energy momentum tensor of a perfect fluid as

T~v = (p + p)u~u~ - p g . ~ , u~,u # = 1, p > 0 , p > 0. (4)

Since T~; ~ = 0 here as in Einstein's theory we have, in accordance with Banerji (1968),

ti" - p" ~ h"~ (5)

p + P

and

0 - P " (6) p + p

where/~" = u;~v u v, and in what follows, the dot will indicate the covariant derivative along the word line. h "v and 0 are the projection tensor and the scalar of expansion, respectively, defined as

h #~ = g~V - M'u v , 0 = u.",~. (7)

We may introduce l by the equation

10 i . (s) l

GENERAL SCALAR TENSOR THEORY 225

and if so, Equation (6) goes to the form

p i - 3 .

p + p I

Substituting from (4) into (3) we have

n' (p - 3p)

3 + 2f~ 3 + 2f~

We write Equation (10) in the form

[q)~(_g)l/2],c _ x / ~ g 3 + 2 f l

(9)

(10)

(11)

We now assume that our universe is spatially homogeneous. We then choose the time lines along the world lines of matter (co-moving coordinates) and define the homogen- eous varieties at the t-constant spaces. In view of this choice of the coordinate system

and spatial homogeneity, g44 is at most a function of t alone and can be reduced to unity by a suitable transformation of t.

Therefore, the line element is given by

ds 2 = dt 2 + 2g4; dt dx i + gik dx; dxk. (12)

The Greek indices stands for the number 1 to 4 and Latin indices for 1 to 3. On account of spatial homogeneity ~p; =p; =p; = 0 and because of co-moving coordinates (9 = dg)/dt. Then (11) reduces to the form

[g~4~)(_g)l/2];~_ x / ~ [87z(p- 3p) - cp~q)~f)']. (13) 3 + 2 f ~

The left-hand side will obviously be zero for a stationary universe. Thus, one has

8~z(p - 3p) = g~'V(?~,(pvO'. (14)

In Nordvedt's theory we have f~' v~ 0 and also g~'"c~,(pv v ~ 0 for a nonvanishing scalar field and thus one can conclude that (p - 3p) ~ 0. Hence, we have the result that the spatially homogeneous stationary perfect fluid cosmological models in Nordtvedt's general scalar-tensor theory cannot include the radiation-filled universe or the empty universe at the limit.

This result is in contrast to Banerji's (1974) result in Brans-Dicke theory, but is similar to Singh and Singh (1983) result in the Brans-Dicke theory with cosmological term. This result has been discussed by Banerjee and Santos (1981) also, but we have mentioned it for continuity and completeness of the results obtained in this paper.

From Equations (7) and (8) we have in this coordinate system

d ln(_g)t /2 d ln13 (15) dt dt

226 TARKESHWAR SINGH AND T. SINGH

Putting the constant of integration zero we may write

(_g)~/2 = l 3 " (16)

For incoherent dust, p = 0 and we have from (9)

pl 3 = f ( x i ) . (17)

But if we have a spatially-homogeneous universe with both p and p constant on the

homogeneous varieties, then

l = S ( x 4 ) W ( x i ) . ( 1 8 )

If, however, the rotation vanishes - i.e., the homogeneous varieties are orthogonal to the t-lines - then we can always reduce the gi4 to zero. Then from Equations (11) and

(18) we have

f S g ( p - 3p)S 3 f &2s3n' ~S 3= ~ + ~ d t - 3 + 2 ~ dr. (19)

The Equation (19) has been derived by Banerjee and Santos (1981) also, but we have mentioned it here only for the continuity of the results obtained in this paper.

Now we shall write Raychuadhuri-type equations using the field equations. From (2)

and (4) we have

,1+o, } { - - ~ }l R . v - p u~,u~ (3 + 2f~) g"~ + p uuu~+ (3 + 2 ~ ) g ~ +

+ n'g"~dd~,dp~g,v f~ - ~;;~ (20) 2q~(2f~ + 3) (p2 O.CPv

We recall a few general equations from Ellis (1971): namely,

Rl.~u~u v = ~) + 302 - / t ~ + 2(o .2 _ co2),

and

(21)

and

(02 1 ,uv = 2(oft v(0 "

h ~ ' R . v u v = -..--;h~'(~ - o.;~ + 20"") - ((0~ + a~)i#' , (22)

where 0 and h~ are given by Equation (7). o.~v and (0.v are shear and rotation tensors,

respectively, given by

�9 1

o.~,~ = u(~; o - u(~uo - 5h~,~ O,

(0~ = uE~,; vl - t/t~,uvl, a2 = 5a~vo.l ~v (23)

G E N E R A L SCALAR T E N S O R T H E O R Y 227

Instead of the rotation tensor, we introduce the rotation vector defined by

COc~ = i ~ , o : B , a v , , .,~ _ � 8 9 u . - - 2 ' 1 t413v~'.uv - , v

then 092 = - c%co =. As shown by Banerji (1968) we can write

(24)

R~u" = O, #(2g~# + �89 #) _ a ~ +1502.u ~ -

- / t ; ~ U c~ - 0-~/)/~ - 2 ( . 0 2 U ~ + t l J z v B e ( 6 0 ~ , # U v - 2 C O . U v f t B ) . (25)

From (20), (21), and (8) we can write

3) _ _ ~ t ~ + 2 ( a z - c o 2 ) = l

8~ r / 2 + n ~ 3(1 + n) ] - ~ - / t , ~ - 6 ) p + (3 + 2~) ~ j +

+ - - + (26) t_2a+3 ~ e

This is analogous to Raychaudhuri's equation for a Friedmann universe in general relativity written in notation of Ellis (1971). /~" in the above equation is given by Equation (5) and is zero for p = 0.

For incoherent matter (26) reduces to

3)" 2(co2-o'2) 8~(2+f~) ~ r . 2~' ~] ~ (27) 7 = q~(3 + 2f~) p + q5 [_2n + 3 q~

Unlike in Einstein's theory, the sign of liI depends on the sign and magnitude of the derivatives of 4. Therefore, we cannot conclude - as in Einstein's theory - that 1)l is always negative when co = 0. Thus a non-rotating cosmological model may, in general, have a minimum volume.

However, if we assume spatial homogeneity, then for a nonrotating dust-filled cosmol- ogical model we obtain from (19) in the coordinate system introduced earlier the equation

+ 3 }~ - 8rcp ~b2f~ ' (28) S 2 f ~ + 3 2 f ~ + 3

Hence, from (27) we obtain in this case

2S 2o.2 872(3 + ~) ~b2F 3f~' ~ ] 3~S - + - - (29)

S 3 + 2 f 2 P +~-1_2f~+3 S

The sign of SIS will be determined by magnitude of the derivatives of ~p. Using Equations (20), (25), and (26) we find that, in general,

2 2 8 T A R K E S H W A R S I N G H A N D T. S I N G H

2h~nO, n - a?~ - a~ni#~ - 2o-2u ~ + ~ ' " ~ ( % , , nuv - 2oouuvit~) =

= h LqSqS~l~(2~+ . (30)

We can also use Equation (22) and write instead the equivalent equation

/ x , , - - ; v - - 0"; v -F ~x ----

~e F. f 2f~' n

Now we consider the Hawking-Penrose energy condition (cf. Hawking and Ellis, 1973), for any time-like u" one must have R . ~ u " u ~ > O. Thus Equation (20) leads to

8rc[(2+fl) 3(l+fl) ] +2[ 2fl' 7] ~+q$"//V>-O" (p 3 + ~ P + ( 2 f ~ + 3 ) p + ~ 2 f*+3 ~ q5

This inequality may not be satisfied. Hence, the presence of the scalar field q5 may avert the occurrence of singularities.

References

Banerjee, A. and Santos, N. O.: 1981, Phys. Rev. D23, 2111. Banerji, S.: 1968, Pro& Theor. Phys. 39, 365. Banerji, S.: 1974, Phys. Rev. D9, 877. Brans, C. and Dicke, R. H.: 1961, Phys. Rev. 124, 925. Ehlers, J.: 1961, Abh. Math.-Namrwiss. K1. Akad. Wiss. Litt. Mainz 11,793. Ellis, G. F. R.: 1971, in R. K. Sachs (ed.), General Relativity and Cosmology, Academic Press, New York,

p. 127. Endo, M. and Fukui, T.: 1977, Gen. Rel. Gray. 8, 833. Hawking, S. W. and Ellis, G. F. R.: 1973, Large-Scale Structure of Space-Time, Cambridge University Press,

Cambridge, p. 95. Hoyle, F. and Narlikar, J. V.: 1962, Proc. Roy. Soc. A270, 334. Nordtvedt, K.: 1970, Astrophys. J. 161, 1059. Raychaudhuri, A. K.: 1955, Phys. Rev. 98, 1123. Raychaudhuri, A. K. and Banerji, S.: 1964, Z. Astrophys. 58, 187. Singh, T. and Singh, Tarkeshwar: 1983, Astrophys. Space Sci. 97, 127. Uehara, K. and Kim, C. W.: 1982, Phys. Rev. D26, 2575.