Confirming previous heuristic analysesà la Belinskii-Khalatnikov-Lifshitz, it is rigorously proven that certain "subcritical" Einstein-matter systems exhibit a monotone, generalized Kasner behaviour in the vicinity of a spacelike singularity. The D−dimensional coupled Einstein-dilaton-p-form system is subcritical if the dilaton couplings of the p-forms belong to some dimension dependent open neighbourhood of zero [1], while pure gravity is subcritical if D ≥ 11 [13]. Our proof relies, like the recent theorem [15] dealing with the (always subcritical [14]) Einstein-dilaton system, on the use of Fuchsian techniques, which enable one to construct local, analytic solutions to the full set of equations of motion. The solutions constructed are "general" in the sense that they depend on the maximal expected number of free functions.

In numerical studies of Gowdy spacetimes evidence has been found for the development of localized features (`spikes') involving large gradients near the singularity. The rigorous mathematical results available up to now did not cover this kind of situation. In this work we show the existence of large classes of Gowdy spacetimes exhibiting features of the kind discovered numerically. These spacetimes are constructed by applying certain transformations to previously known spacetimes without spikes. It is possible to control the behaviour of the Kretschmann scalar near the singularity in detail. This curvature invariant is found to blow up in a way which is non-uniform near the spike in some cases. When this happens it demonstrates that the spike is a geometrically invariant feature and not an artefact of the choice of variables used to parametrize the metric. We also identify another class of spikes which are artefacts. The spikes produced by our method are compared with the results of numerical and heuristic analyses of the same situation.

A longstanding conjecture by Belinskii, Khalatnikov and Lifshitz that the singularity in generic gravitational collapse is spacelike, local and oscillatory is explored analytically and numerically in spatially inhomogeneous cosmological space-times. With a convenient choice of variables, it can be seen analytically how nonlinear terms in Einstein's equations control the approach to the singularity and cause oscillatory behavior. The analytic picture requires the drastic assumption that each spatial point evolves toward the singularity as an independent spatially homogeneous universe. In every case, detailed numerical simulations of the full Einstein evolution equations support this assumption.

Numerical investigation of a class of inhomogeneous cosmological spacetimes shows evidence that at a generic point in space the evolution toward the initial singularity is asymptotically that of a spatially homogeneous spacetime with Mixmaster behavior. This supports a long-standing conjecture due to Belinskii et al. on the nature of the generic singularity in Einstein's equations.Comment: 4 pages plus 4 figures. A sentence has been deleted. Accepted for publication in PR

The global structure of solutions of the Einstein equations coupled to the Vlasov equation is investigated in the presence of a twodimensional symmetry group. It is shown that there exist global CMC and areal time foliations. The proof is based on long-time existence theorems for the partial differential equations resulting from the Einstein-Vlasov system when conformal or areal coordinates are introduced.

We use qualitative arguments combined with numerical simulations to argue that, in the approach to the singularity in a vacuum solution of Einstein's equations with T 2 isometry, the evolution at a generic point in space is an endless succession of Kasner epochs, punctuated by bounces in which either a curvature term or a twist term becomes important in the evolution equations for a brief time. Both curvature bounces and twist bounces may be understood within the context of local mixmaster dynamics although the latter have never been seen before in spatially inhomogeneous cosmological spacetimes. 1 However, it is not necessarily expected that the evolution converges to a single such sequence of Kasners. It may be that the evolution always eventually diverges from any one such sequence and another sequence, which again follows the Kasner map, becomes a better approximation.2 In this paper we mean by the term "observer" a timelike path with constant spatial coordinates. We assume that a foliation and threading have been chosen. Whether results of the sort discussed here will be seen by inequivalent sets of observers is not yet generally known. However, this does seem to be true at least in certain cases [7].solutions have been reported elsewhere [10]. Here, we discuss the behavior near the singularity for T 2 symmetric vacuum spacetimes.Since U (1) is a subgroup of T 2 = U (1) × U (1), the T 2 symmetric vacuum spacetimes are a subfamily of the U (1) symmetric vacuum spacetimes. 3 One may then ask why it is useful to study the T 2 symmetric family directly. The reason is that, since the equations for the T 2 symmetric family are considerably simpler (1+1 PDEs rather than 2+1 PDEs) the numerical studies can be done significantly more accurately. Hence, the studies are more accurate for the T 2 symmetric family, and the behavior of the bounces seen by the observers can be monitored more carefully. The MCP analysis has been carried out in great detail in this simpler case. We report this study here both to present the detailed picture it gives of the dynamics in these spacetimes, and also, we hope, as an aid to obtaining rigorous results about the dynamics.We define the T 2 symmetric family in Sec. II, noting the relationship between this family and others, such as the Gowdy [11] and the Kasner spacetimes. Also in Sec. II, we discuss the areal function and coordinates, recalling results which justify their use for T 2 symmetric solutions and writing out the field equations. In Sec. III, we set up the MCP treatment of the evolution equations for the T 2 symmetric spacetimes and use it to argue that oscillatory behavior occurs. We recall that in setting up the MCP form of a given set of evolution equations, one presumes that at each spatial point, the fields evolve to Kasner epoch values (not necessarily at the same time for all spatial points); one then substitutes these Kasner-like values of the fields into the right hand side of the evolution equations, and attempts to infer how the various terms in these equations sh...

We study the behavior of spiky features in Gowdy spacetimes. Spikes with velocity initially high are, generally, driven to low velocity. Let n be any integer greater than or equal to 1. If the initial velocity of an upward pointing spike is between 4n-3 and 4n-1 the spike persists with final velocity between 1 and 2, while if the initial velocity is between 4n-1 and 4n+1, the spiky feature eventually disappears. For downward pointing spikes the analogous rule is that spikes with initial velocity between 4n-4 and 4n-2 persist with final velocity between 0 and 1, while spikes with initial velocity between 4n-2 and 4n eventually disappear.Comment: discussion of constraints added. Accepted for publication in Phys. Rev.

We use qualitative arguments combined with numerical simulations to argue that, in the approach to the singularity in a vacuum solution of Einstein's equations with T 2 isometry, the evolution at a generic point in space is an endless succession of Kasner epochs, punctuated by bounces in which either a curvature term or a twist term becomes important in the evolution equations for a brief time. Both curvature bounces and twist bounces may be understood within the context of local mixmaster dynamics although the latter have never been seen before in spatially inhomogeneous cosmological spacetimes. 04.20.Dw,98.80.Dr