### 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