2.2 Hamiltonian description
We assume that there is a spacelike singularity at a finite distance in proper time. We adopt a spacetime
slicing adapted to the singularity, which “occurs” on a slice of constant time. We build the slicing from the
singularity by taking pseudo-Gaussian coordinates defined by
and
, where
is the
lapse and
is the shift [48
]. Here,
. Thus, in some spacetime patch, the metric
reads
where the local volume
collapses at each spatial point as
, in such a way that the proper
time
remains finite (and tends conventionally to
). Here we have assumed the
singularity to occur in the past, as in the original BKL analysis, but a similar discussion holds for future
spacelike singularities.
2.2.1 Action in canonical form
In the Hamiltonian description of the dynamics, the canonical variables are the spatial metric components
, the dilaton
, the spatial
-form components
and their respective conjugate momenta
,
and
. The Hamiltonian action in the pseudo-Gaussian gauge is given by
where the Hamiltonian is
In addition to imposing the coordinate conditions
and
, we have also set the temporal
components of the
-forms equal to zero (“temporal gauge”).
The dynamical equations of motion are obtained by varying the above action w.r.t. the canonical
variables. Moreover, there are constraints on the dynamical variables, which are
Here we have set
where the subscript
denotes the spatially covariant derivative. These constraints are
preserved by the dynamical evolution and need to be imposed only at one “initial” time, say at
.
2.2.2 Iwasawa change of variables
In order to study the dynamical behavior of the fields as
(
) and to exhibit the billiard
picture, it is particularly convenient to perform the Iwasawa decomposition of the spatial metric. Let
be the matrix with entries
. We set
where
is an upper triangular matrix with
’s on the diagonal (
,
for
) and
is a diagonal matrix with positive elements, which we parametrize as
Both
and
depend on the spacetime coordinates. The spatial metric
becomes
with
The variables
of the Iwasawa decomposition give the (logarithmic) scale factors in the new, orthogonal,
basis. The variables
characterize the change of basis that diagonalizes the metric and hence they
parametrize the off-diagonal components of the original
.
We extend the transformation Equation (2.8) in configuration space to a canonical transformation in
phase space through the formula
Since the scale factors and the off-diagonal variables play very distinct roles in the asymptotic behavior,
we split off the Hamiltonian as a sum of a kinetic term for the scale factors (including the dilaton),
plus the rest, denoted by
, which will act as a potential for the scale factors. The Hamiltonian then
becomes
The kinetic term
is quadratic in the momenta conjugate to the scale factors and defines the inverse of
a metric in the space of the scale factors. Explicitly, this metric reads
Since the metric coefficients do not depend on the scale factors, that metric in the space of scale factors is
flat, and, moreover, it is of Lorentzian signature. A conformal transformation where all scale factors are
scaled by the same number (
) defines a timelike direction. It will be convenient in the
following to collectively denote all the scale factors (the
’s and the dilaton
) as
, i.e.,
.
The analysis is further simplified if we take for new
-form variables the components of the
-forms
in the Iwasawa basis of the
’s,
and again extend this configuration space transformation to a point canonical transformation in phase
space,
using the formula
, which reads
Note that the scale factor variables are unaffected, while the momenta
conjugate to
get redefined by terms involving
,
and
since the components
of the
-forms in the Iwasawa basis involve the
’s. On the other hand, the new
-form momenta,
i.e., the components of the electric field
in the basis
are simply given by
In terms of the new variables, the electromagnetic potentials become
Here,
are the electric linear forms
(the indices
are all distinct because
is completely antisymmetric) while
are the
components of the magnetic field
in the basis
,
and
are the magnetic linear forms
One sometimes rewrites
as
, where
is the set complementary
to
, e.g.,
The exterior derivative
of
in the non-holonomic frame
involves of course the structure
coefficients
in that frame, i.e.,
where
is here the frame derivative. Similarly, the potential
reads
where
is
and