Inhomogeneous cosmologies: New exact solutions and their evolution

Abstract

An algorithm is presented for determining all solutions of the Einstein field equations representing a perfect fluid with metric of the form ds2=dt2−e2αdz2 −e2β(dx2+dy2) and fluid flow vector u=∂/∂t. The entire class of solutions is then invariantly characterized. These new solutions generalize Szekeres’ inhomogeneous cosmological models containing dust. A subclass of these solutions is studied in detail and it is interesting that some of these models approach isotropy but not homogeneity for large cosmological times.

Singularity Theorems and Their Consequences

“…234) , if the three-spaces orthogonal to the velocity vector of the¯uid have curvat ure divergences less strong than that of their second fundam ental forms [59]. A good account of t he possible big-bang singularit ies and their propert ies in spherically symmetric and Szekeres models [209,206,207] can be found in [97,124,204] and references therein. To illust rate these point s and De® nit ions 3.5 and 3.6, let us consider two ® nal examples.…”

“…T hus, u = ± dt and the assumpt ion on the expansion is hj t 0 > 0. Notice t hat h is a funct ion of all four coordinat es, as there is no restriction concerning the symmetry group of the spacet ime Ð t his happens for exam ple in the Szekeres cosmologies [209,206,207,124 ]. T herefore, t he condit ion hj t 0 > 0 means h( t0 , x i ) > 0 for all x i in a suit able coordinat e system including t. W hen there is a three-dimensional symmetry group acting transit ively on spacelike hypersurfaces, the theorem is also proved and was carefully analysed in [68,71].…”

Complete classification of purely magnetic, nonrotating, nonaccelerating perfect fluids

“…For n 2 = n 3 the Petrov type is D, and it was further deduced in [43] that the corresponding models are spatially homogeneous and LRS III. Note from (97) that any PE model in A is of Petrov type D (with n 2 = n 3 = 0) and was proved [50] to belong to the Szekeres-Szafron family [51,52,53,54]. This family is here characterized as the PE subclass of A, whereas Lozanovski's family [37] forms precisely the PM type D subclass of A.…”

On the thermodynamical interpretation of perfect fluid solutions of the Einstein equations with no symmetry

“…Finally, Quevedo and Sussman [7] gave an example of the Szafron β = 0 model [8] that has no symmetry and allows for a thermodynamical scheme, and showed that the parabolic Szafron β = / 0 model does not allow for it unless it has a symmetry. Quevedo and Sussman [9] also analyzed the conditions for the existence of a thermodynamical scheme in the…”

“…The models of Stephani [2] and of Szafron [8] are so far the only known exact solutions of the Einstein equations that have no symmetry, can be considered to be cosmological models (because they generalize the Robertson -Walker class of solutions) and allow for nontrivial thermodynamics; see also Ref. [10].…”