Problem of the physical singularity in the general ... - Springer Link

PROBLEM

OF

THE

GENERAL

THEORY

PHYSICAL

SINGULARITY

IN THE

OF RELATIVITY

M. E . G e r t s e n s h t e i n , a n d M. L . F i l ' c h e n k o v

L.

Kh.

Ingel',

UDC523.11

w The question of the singular state t = 0, f r o m which the "big bang" c o s m i c expansion c o m m e n c e s , is a v e r y i m p o r t a n t p r o b l e m of c o n t e m p o r a r y cosmogony [1]. The question a r i s e s of whether this p r o b l e m can be solved within the f r a m e w o r k of c o n t e m p o r a r y physics [2] or whether a new p h y s i c s , for example, a quantum t h e o r y of gravitation, must be employed [3]. We will a s s u m e that such a new physics is r e q u i r e d for d e n sities p > pg, where [1, 3, 4] pg = c S / ~ i G 2 = 5 " 1 0 93 g / c m 3. Two viewpoints a r e possible: 1) The universe was c r e a t e d f r o m a singular state p > pg, to wMch the concepts of space and t i m e a r e not applicable [1, 4]; 2) the solutions of the g e n e r a l t h e o r y of relativity a r e continuous through the singularity [5, 6, 7], i . e . , the expansion was preceded by a contraction. In the p r e s e n t study we will p r e s e n t t h e o r e t i c a l and experimental evidence in favor of the second viewpoint. We may a s s u m e that t h e r e a r e two c l a s s e s of h i g h - s y m m e t r y solutions which a r e continuous through the singularity [5, 6, 7]: 1) FriedmannVs solution[8]; 2) K a s n e r ' s v a c u u m solution [8-10]). This can be proved with use of either the integrals of the g e n e r a l - r e l a t i v i t y equations of motion [5], the variation principle [6], o r the principle of c o r r e s p o n d e n c e with Newtonian mechanics in the limit as c 2 ~ ~. It is not n e c e s s a r y to speak of the birth of the universe f r o m a singular state, and the following formulations of both viewpoints a r e p o s sible: 1) The singularity was so hot and dense that everything o c c u r r i n g before the singularity at t < 0 is c o m pletely lost and thus our study must c o m m e n c e f r o m the moment t = + 0; 2) infinite singularity t e m p e r a t u r e and density a r e p r o p e r t i e s of h i g h - s y m m e t r y solutions not contained in the g e n e r a l solution, and such events did not in fact o c c u r . The u n i v e r s e contains t r a c e s of events which o c c u r r e d at t < 0, and we must d e t e r m i n e how to detect and interpret these. The t i m e interval in question is then -- oo < t < + ~, as in Newtonian physics. w Physical singularities within the f r a m e w o r k of g e n e r a l relativity were f i r s t studied by Lifshits and Khalatnikov [9, 19]. As the zeroth approximation the K a s n e r m e t r i c was used: 3 d s ~- =

~t

d ~- - : ~ } ~ I ~',~-(dx')-': I / - - ~ = I~ I" ; c ----1.

(1)

Since the m e t r i c is continuous, we write I T I 9 In a vacuum the exponents p~ satisfy the conditions

=1;

I z

>0,

>0,

(2)

and, t h e r e f o r e , the m e t r i c f o r m s a o n e - p a r a m e t e r family. F o r an anisotropic m e t r i c t h e r e is no basis for an isotropic equation of state -- the dependence of p r e s s u r e p on energy density e. T h e r e f o r e , we take the equation of state in the f o r m [11, 12] P. -----~. ~, 7 ~ 0 . We a s s u m e the constants To to be specified and consider the s e l f - c o n s i s t e n t p r o b l e m only for a vacuum m e t r i c [13]. The equations of the g e n e r a l t h e o r y of relativity r e d u c e to the f o r m [11]

All-Union S c i e n t i f i c - R e s e a r c h Institute of Opticophysical M e a s u r e m e n t s . T r a n s l a t e d f r o m Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 7-12, July, 1978. Original a r t i c l e submitted June 26, 1977.

0038-5697/78/2107- 0841 $07.50 9 19 79 Plenum Publishing Corporation

841

(3)

where e = E / t 2 f o r a F r i e d m m m - t y p e solution. We have four equations for the six quantities g~ and 3'a, so that m e t r i c (3) f o r m s a t w o - p a r a m e t e r family of solutions which includes the isotropic Friedmarm solution 2 8 (t+-,,) '

2 3 ( 1 + - ; ) -~

(4)

in c o n t r a s t to the o n e - p a r a m e t e r K a s n e r family in a vacuum. F r o m Eq. (3), we obtain f o r E > 0

a

~

~

a

F o r l a r g e - s c a l e m o t i o n s the equation of state of an u l t r a r e l a t i v i s t i c gas is of g r e a t e s t interest. anisotropic state

In the

~ 7~ -- l , Eqs. (3) and (4) simplify:

F r o m the condition s > 1 we obtain Pa > 0; c o m p r e s s i o n o c c u r s along all a x e s , while in the vacuum solution (E = 0) expansion o c c u r s along one axis, the physical sense of which for an a t t r a c t i v e field such as gravitation is not c l e a r . In the t w o - p a r a m e t e r solution, singularities of the density c u r v a t u r e t e n s o r of the m a t e r i a l are ~ T - 2 so that all conclusions with r e s p e c t to chemical composition a r e the s a m e as in F r i e d m a n n , s solution. w An entire s e r i e s of cosmogonic r e s u l t s of the g e n e r a l t h e o r y of relativity, such a s , for example, F r i e d m a n n ' s solutionofthe homogeneous u n i v e r s e , can be obtained f r o m Newtonian m e c h a n i c s . To do this, we consider motion near an a r b i t r a r y particle of m a t t e r which is chosen as the origin of the coordinate s y s t e m [8, 14]. We employ the m e t r i c (1). We note that m e t r i c (1) is invariant for the shift [7, 8]: x~

T h e r e o r e , any d e s i r e d point may be the center.

(7)

x~+c%

In Eq. (2) we p e r f o r m a t r a n s f o r m a t i o n of v a r i a b l e s :

(8)

In the new coordinates l a , T, the m e t r i c takes the f o r m ds*" = (1 -- 7)2) d z~. ~ dF -t- 2"o~ dl ~ d z , 3

dP=~(dl92;

]

g= V-g~l;

7)"=~v~

u~l

--

7) 2

7&

(9)

u

We s t r e s s that the determinant of the m e t r i c t e n s o r does not have singularities at r = 0, and both conditions of Landau's t h e o r e m (g0a = 0, g00 = 1) a r e not fulfilled. F o r a fixed p a r t i c l e of m a t e r i a l , x~ = const, so that the quantity v~ has the sense of a velocity. At the center x ~ = 0 ; l s--O; v ~ O

for ".r

(10)

and n e a r the c e n t e r v