Introduction

Summary

Our book is devoted to the structure of the general solution of the Einstein equations with a cosmological singularity. We cover Einstein-matter systems in four and higher space-time dimensions.

Under the terminology “cosmological singularity,” we mean a singularity in time, i.e., a spacelike singularity on a “submanifold” that can be viewed as the limit of a family of regular spacelike hypersurfaces forming (locally) a Gaussian foliation, such that the curvature invariants together with invariant characteristics of matter fields diverge as one tends to this submanifold.

The nonlinearities of the Einstein equations are notably known to prevent the construction of an exact general solution. From this perspective, the BKL work which describes the asymptotic general behavior of the gravitational field in four space-time dimensions as one approaches a spacelike singularity, is quite unique and exceptional. The central attainment of the BKL theory is the analysis of the delicate relationship between the time derivatives and the spatial gradients in the gravitational field equations near the singularity. The main technical idea of the BKL approach consists in identifying among the huge number of spatial gradients, those terms that are of the same importance as the time derivatives. In the vicinity of the singularity, these terms are in no way negligible. They act during the whole course of evolution up to the singularity, and it is actually due to these spatial gradients that oscillations do arise.

A remarkable simplifying feature nevertheless emerges as one tends to the singularity. This is the fact that the spatial gradient terms that must be retained in the dynamical equations of motion can be asymptotically represented as the products of some functions of the undifferentiated (in space) “scale factors” (which represent how distances along independent spatial directions evolve with time) by some slowly varying coefficients containing spacelike derivatives. This nontrivial separation springing up in the vicinity of the singularity leads to gravitational equations of motion which effectively reduce to a system of ordinary differential equations in time for the scale factors – one such system at each point of 3-space – because in the leading approximation, all relevant coefficients containing spacelike derivatives enter these equations solely as external (albeit, dynamically crucial) time-independent parameters.