The first aspect is that the spatial points decouple in the limit , in the sense that one can replace the Hamiltonian by an effective “ultralocal” Hamiltonian involving no spatial gradients and hence leading at each point to a set of dynamical equations that are ordinary differential equations with respect to time. The ultralocal effective Hamiltonian has a form similar to that of the Hamiltonian governing certain spatially homogeneous cosmological models, as we shall explain in this section.

The second aspect of the BKL-limit is to take the sharp wall limit of the ultralocal Hamiltonian. This leads directly to the billiard description, as will be discussed in Section 2.4.

In spatially homogeneous models, the fields depend only on time in invariant frames, e.g., for the metric

where the invariant forms fulfill

Let us now come back to the general, inhomogeneous case and express the dynamics in the frame where the ’s form a “generic” non-holonomic frame in space,

Here the ’s are in general space-dependent. In the non-holonomic frame, the exact Hamiltonian takes the form where the ultralocal part is given by Equations (2.32) and (2.33) with the relevant ’s, and where involves the spatial gradients of , , and .The first part of the BKL conjecture states that one can drop asymptotically; namely, the dynamics of a generic solution of the Einstein–-form-dilaton equations (not necessarily spatially homogeneous) is asymptotically determined, as one goes to the spatial singularity, by the ultralocal Hamiltonian

provided that the phase space constants are such that all exponentials in the above potentials do appear. In other words, the ’s must be chosen such that none of the coefficients of the exponentials, which involve and the fields, identically vanishes – as would be the case, for example, if since then the potentials and are equal to zero. This is always possible because the , even though independent of the dynamical variables, may in fact depend on and so are not required to fulfill relations “” analogous to the Bianchi identity since one has instead “”.

- As we shall see, the conditions on the ’s (that all exponentials in the potential should be present) can be considerably weakened. It is necessary that only the relevant exponentials (in the sense defined in Section 2.4) be present. Thus, one can correctly capture the asymptotic BKL behavior of a generic solution with fewer exponentials. In the case of eleven-dimensional supergravity the spatial curvature is asymptotically negligible with respect to the electromagnetic terms and one can in fact take a holonomic frame for which (and hence also ).
- The actual values of the (provided they fulfill the criterion given above or rather its weaker form just mentioned) turn out to be irrelevant in the BKL-limit because they can be absorbed through redefinitions. This is for instance why the Bianchi VIII and IX models, even though they correspond to different groups, can both be used to describe the BKL behavior in four spacetime dimensions.

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