5.1 The BKL picture

The ideas of BKL are often referred to as “the BKL conjecture”. This terminology is unfortunate, since it would seem to indicate that there is a mathematically precise statement that can be proven or disproven. In reality, the ideas are too vague to warrant the name “conjecture”. Nevertheless, they have played an important role on an intuitive level, and therefore deserve to be mentioned. The rough idea is as follows:

There are some caveats. First, the statements concern generic spacetimes; there are exceptions; see Taub–NUT. Second, the matter content might be important for special classes of matter models. For instance, in the case of a stiff fluid or a scalar field, the matter should play a dominant role. The statement that solutions should exhibit oscillatory behavior also depends on the matter model; stiff fluids and scalar fields are expected to suppress it. Furthermore, symmetry might prevent the appearance of oscillations.

The first statement on the list above can be ensured under general circumstances; a combination of Hawking’s theorem [60, Theorem 55A, p. 431], an energy condition and the existence of a Cauchy hypersurface satisfying suitable assumptions concerning the mean curvature will do. The statement of causal disconnectedness is, however, unsatisfactory in that it depends not only on the foliation, but also on a choice of diffeomorphism. It would be preferable to have a geometric condition, which is even independent of the foliation. However, to our knowledge there is no such definition; see the introduction of [44Jump To The Next Citation Point] for a further discussion. The last three statements are clearly very vague.

The general framework has been developed significantly since the work of BKL; see [28Jump To The Next Citation Point30Jump To The Next Citation Point47Jump To The Next Citation Point88Jump To The Next Citation Point] and references cited therein. We shall not describe these developments in detail, but let us mention that there are at least two somewhat different approaches. In the Hamiltonian approach, taken in [28Jump To The Next Citation Point30Jump To The Next Citation Point], billiards describe the asymptotic behavior (see also the work of Misner and Chitré; see [57, pp. 805–816] and references cited therein). In the dynamic systems approach, described in [47Jump To The Next Citation Point88Jump To The Next Citation Point], the solution is approximated asymptotically by a family of solutions to ordinary differential equations. In [3], the Gowdy spacetimes have been considered from this perspective.

Even though the two perspectives have differences, they have an essential assumption in common: causal disconnectedness in the direction toward the singularity. Furthermore, in both cases, the oscillatory spatially-homogeneous vacuum solution is of central importance. It is of interest to note that these two aspects are potentially contradictory; Misner’s original motivation in studying the Mixmaster Universe (Bianchi IX) [56] was the desire to demonstrate that there is no causal disconnectedness in the direction towards the singularity. In order for the pictures suggested in [2830] and [4788] to be consistent, causal disconnectedness should hold in the oscillatory spatially-homogeneous vacuum solutions. It is far from clear that this is the case. The reader interested in a discussion of the status of this question is referred to the introduction of [44].

Finally, let us mention that a different formulation of the BKL picture is given in [19, Conjecture 6.10, p. 58]; see also the references cited therein.

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