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.

Stochasticity of the Oscillatory Regime

In the asymptotic vicinity of the singularity, the oscillatory regime described above acquires a stochastic (chaotic) character.

Of course, any system of differential equations, no matter how complicated it is, has one and only one definite solution for each set of initial data. From this point of view we have no place for stochasticity. When we are speaking about stochastic behavior of the solution of differential equations we mean a quite different situation, namely the case when the initial data are not known exactly but are distributed in accordance to some statistical law. Together with the initial data, all characteristics of the solution at any subsequent time will also be distributed in some way.

In the framework of such a statistical approach there are two essentially different types of behavior. The first corresponds to the non-chaotic systems for which the initial distribution being picked around some points in the phase space generates an evolution which remains picked around the corresponding trajectories (i.e., usual classical solutions) starting from these points. The systems of the second type are chaotic in the sense that, for them, any initial distribution is going to spread out over the whole phase space in the course of the evolution, independently of how sharply localized the distribution was at the initial instant (some authors say that such systems have a source of stochasticity). Moreover, in the chaotic case, the final asymptotic regime which arises at a late time has a universal character completely independent of the form of the initial distribution. We stress that it is this last property (known as mixing) that represents the basic feature of chaos. For complicated nonlinear systems of differential equations, the chaotic behavior (if it is present) should be considered as a rather useful property because it gives the qualitative character of the general solution. Indeed for such a system only approximate forms of the general solution can be obtained and all parameters in such forms have no exact sense; they are unavoidably uncertain and can be interpreted as statistically distributed.