2.3 Decoupling of spatial points close to a spacelike singularity
So far we have only redefined the variables without making any approximation. We now start the
discussion of the BKL-limit, which investigates the leading behavior of the fields as
(
).
Although the more recent “derivations” of the BKL-limit treat both elements at once [43
, 44
, 45
, 48
], it
appears useful – especially for rigorous justifications – to separate two aspects of the BKL
conjecture.
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.
2.3.1 Spatially homogeneous models
In spatially homogeneous models, the fields depend only on time in invariant frames, e.g., for the metric
where the invariant forms fulfill
Here,
the
are the structure constants of the spatial homogeneity group. Similarly, for a
-form and a
-form,
The Hamiltonian constraint yielding the field equations in the spatially homogeneous
context
is obtained by substituting the form of the fields in the general Hamiltonian constraint and contains, of
course, no explicit spatial gradients since the fields are homogeneous. Note, however, that the structure
constants
contain implicit spatial gradients. The Hamiltonian can now be decomposed as before and
reads
where
,
and
, which do not involve spatial gradients, are unchanged and where
disappears since
. The potential
is given by [61
]
where the linear forms
(with
distinct) read
and where “more” stands for the terms in the first sum that arise upon taking
or
. The
structure constants in the Iwasawa frame (with respect to the coframe in Equation (2.30)) are related to
the structure constants
through
and depend therefore on the dynamical variables. Similarly, the potential
becomes
where the field strengths
reduce to the “
” terms in
and depend on the potentials
and the off-diagonal Iwasawa variables.
2.3.2 The ultralocal Hamiltonian
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
“
”.
Comments
- 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.