Space-Times with Spherically Symmetric Hypersurfaces

When discussing spherically symmetric gravitational fields one usually assumes that the whole space-time is invariant under the 0 (3 ) group of transformations. In this paper, the Einstein field equations are investigated under the weaker assumption that only the 3-spaces t = const are 0 (3 ) symmetric. The following further assumptions are made: (1) The t lines are orthogonal to the spaces t = const. (2) The source in the field equations in a perfect fluid, or dust, or the A term, or the empty space. (3) With respect to the center of symmetry the fluid source may move only radially if at all. Under these assumptions one solution with a perfect fluid source, found previously by Stephani, is recovered and interpreted geometrically, and it is shown that it is the sole solution which is not spherically symmetric in the traditional sense. The paper ends with a general discussion of cosmological models whose 3-spaces t = const are the same as in the Robertson-Walker models. No new solutions were explicitly found, but it is shown that such models exist in which the sign of curvature is not fixed in time. w (1) : Introduction When discussing spher ica l ly s y m m e t r i c gravi ta t iona l f ields in general re la t iv i ty one usual ly assumes t h a t the whole o f spacet ime is spher ica l ly symmet r i c , i.e., t h a t the me t r i c is inva r i an t u n d e r the 0 ( 3 ) g roup o f t r a n s f o r m a t i o n s o f coordina tes . This a s sumpt ion seems m o r e res t r ic t ive t h a n necessary because w h a t one has in m i n d whi le do ing obse rva t ions is the g e o m e t r y o f the 3-spaces t = cons t w h a t e v e r the choice o f t ime is. The g e o m e t r y o f the spacet ime can t h e n be 1 Present address: Max-Planck Institute for Physics and A~trophysics, Karl-Schwarzschildstrasse 1, 8046 Garching, Munich, Germany. 1021 0001-7701[81/1100-102l$03.00/0 9 1981 Plenum Publishing Corporation 1022 KRASIr~S~ recognized by indirect investigations, and it might be interesting to see what conclusions can be drawn if we assume that only each of the 3-spaces t = const is 0(3) symmetric while for the whole space-time it is not necessarily so. In this paper I want to imagine a space-time as made of O(3)-symmetric three-dimensional spacelike hypersurfaces strung onto a timelike line orthogonal to them all, and to investigate the properties of such a space-time if the Einstein field equations are fulfilled. We shall also assume that the velocity field of matter, u s (if any matter is present), has only the u t and u r components, i.e., moves only radially if at all with respect to the line joining the centers of symmetry of the 3-spaces. This assumption is justified by the fact that transversal motions of matter could be easily revealed by any observer, and would thus constitute a too-obvious evidence for the lack of ordinary spherical symmetry in the space-time. However, one could go on without the assumption of purely radial motions and see what follows. This problem still awaits investigation. The most general source in the field equations considered here will be a perfect fluid, whose special cases (in the mathematical sense) are dustlike matter (pressure = 0), the A term ("pressure" = const de=f A ~ 0, "energy density" = -A), and the pure empty space (pressure = energy density --0). In the case of the A term and pure empty space no nonspherical solutions were found, i.e., the space-times considered here are forced by the field equations to be spherically symmetric in the well-known traditional sense. However, in the case of a nontrivial perfect fluid one solution is found which is not spherically symmetric as a space-time. The reason for its nonsphericity is found to be the curvature of the lines onto which the 3-spaces are strung (see Section 5). It is the solution found in 1967 by Stephani [1]. The physical difference between the space-time which is 0(3) symmetric as a whole and one that has only O(3)-symmetric hypersurfaces can be described in the following way. If we are given the 3-spaces t = const without any device to measure the time in different points, then we can reveal only the spherical symmetry of each 3-space by purely geometrical measurements. If, in addition, we attach a clock to each point of the space, then we can say: the space-time considered here is spherically symmetric as a whole if its 3-spaces t = const are spherically symmetric and all the clocks placed on one sphere go at the same rate. 1 The solution from Section 5 fits this definition. The plan of the paper is as follows. In Section 2 the problem is posed by writing a metric form concordant with all our assumptions. In Section 3 we discuss the case which is analogous to the solution of Nariai [2] of ordinary spherical symmetry, and in fact we only recover the Nariai solution. In Section 4, we discuss the case strictly analogous to the standard spherically symmetric spacetime and we find one nontraditional solution: the one of Stephani [1]. The geometrical properties of the Stephani solution are investigated in Section 5. In tThis statement is due to N. Salie. SPACE-TIMES WITH SPHERICALLY SYMMETRIC HYPERSURFACES 1023 Section 6 we discuss the most general space-time, obeying all the aforementioned assumptions, and we show that Sections 3 and 4 actually exhausted the problem. The final sections of the paper are devoted to an analogous problem with homogeneity. Here it is assumed that the 3-spaces t = const are homogeneous with respect to a three-parameter group acting transitively, but the whole spacetime is not necessarily invariant with respect to this group. The problem here is considerably more complicated, so it is assumed for simplicity that the 3-spaces are spherically symmetric in addition to being homogeneous, that they are strung onto a line which is orthogonal to them all, and, as before, that matter displays no transversal motions to the distinguished observers. One class of solutions, found previously by Stephani [1 ], was reobtained. Another class was also investigated, in which no new solutions were found explicitly. In both classes the geometry of each of the subspaces t = const is the same as in the Robertson-Walker (RW) metrics, but the curvature of the 3-spaces t = const is varying in time in a different way, so that it can change its sign. The present paper is one of the possible specifications of the idea of C. B. Collins [3], who proposed to investigate space-times having subspaces with definite symmetry groups. In fact, the paper was motivated by the recent beautiful and enlightening criticism of standard cosmology by G. F. R. Ellis, expressed particularly in [4]. The calculations for this paper were carried out with use of the symbolic formula-manipulation computer system ORTOCARTAN [5, 6]. w (2): Definition and Statement o f the Purpose We shall deal with such space-times, whose subspaces t = const are all spherically symmetric in the normal sense. Thus, it should be possible to choose the coordinates in the space-time so that in every space t = const the metric form is dsa 2 = ~(r) dr 2 + g(r)(dO 2 + sin 2 0 de 2) (1) where ~(r) and g(r) are arbitrary functions of the coordinate r [7], implicitly understood to be the values, at a fixed t, of some arbitrary functions of two variables, 3 '2 (t, r) and 5 2 (t, r), respectively. The most general such space-time has the metric form ds 2 = D 2 (t, r, O, 9) dt2 + 2al (t, r, O, 9) dt dr + 2a2 (t, r, tg, 9) dt dO + 2a 3 (t, r, O, 9) dt d~ 3'2 (t, r) dr 2 5 2 (t, r)(dO 2 + sin 2 0 d9 2 ) (2) where D, a l , a2, and a s are arbitrary functions of four variables. For simplicity, just to gain an insight into a new kind of geometry which seems not to have been considered before, 2 we shall assume throughout the paper that the spaces (1) 2An exhaustive search through the PhysicsAbstracts from the 1926 volume till the present volumes, conducted partially in connection with ReL 7, did not reveal any attempt of the kind considered in the present paper.