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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 N = √g-- and N i = 0, where N is the lapse and N i is the shift [48Jump To The Next Citation Point]. Here, g ≡ det(g ) ij. Thus, in some spacetime patch, the metric reads2
2 0 2 0 i i j ds = − g(dx ) + gij(x ,x )dx dx , (2.3 )
where the local volume g collapses at each spatial point as x0 → + ∞, in such a way that the proper time dT = − √g-dx0 remains finite (and tends conventionally to 0+). 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 gij, the dilaton φ, the spatial p-form components A (mp)1⋅⋅⋅mp and their respective conjugate momenta πij, πφ and πm1 ⋅⋅⋅mp (p). The Hamiltonian action in the pseudo-Gaussian gauge is given by

[ ] ∫ [∫ ( ∑ ) ] S g ,πij,φ,π ,A(p) ,πm1 ⋅⋅⋅mp = dx0 ddx πijg˙ + π φ˙+ πm1 ⋅⋅⋅mpA˙(p) − H , (2.4 ) ij φ m1 ⋅⋅⋅mp (p) ij φ (p) m1⋅⋅⋅mp p
where the Hamiltonian is
∫ d H = d xℋ, ℋ = K ′ + V ′, ∑ −λ(p)φ K ′ = πijπij − --1--(πii)2 + 1(π φ)2 + (p!)e-----πm1(p)⋅⋅⋅mp π(p)m1 ⋅⋅⋅mp, (2.5 ) d − 1 4 p 2 ∑ λ(p)φ V ′ = − Rg + gijg∂iφ ∂jφ + --e------g Fm(p)⋅⋅⋅m F (p)m1⋅⋅⋅mp+1. p 2(p + 1)! 1 p+1
In addition to imposing the coordinate conditions √-- N = g and N i = 0, we have also set the temporal components of the p-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

ℋ = 0 (“Hamiltonian constraint”), m ⋅⋅⋅mℋi = 0 (“momentum constraint”), (2.6 ) ϕ(p1) p−1 = 0 (“Gauss law” for each p-form, p > 0).
Here we have set
∑ ℋ = − 2πj + π ∂ φ + πm1 ⋅⋅⋅mpF (p) , i i |j φ i (p) im1⋅⋅⋅mp (2.7 ) m1⋅⋅⋅mp− 1 m1⋅⋅⋅mp−1mp p ϕ (p) = − p π(p) |mp,
where the subscript |m p 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 0 x = 0.

2.2.2 Iwasawa change of variables

In order to study the dynamical behavior of the fields as x0 → ∞ (g → 0) and to exhibit the billiard picture, it is particularly convenient to perform the Iwasawa decomposition of the spatial metric. Let g(x0,xi) be the matrix with entries gij(x0,xi). We set

g = 𝒩T 𝒜2 𝒩, (2.8 )
where 𝒩 is an upper triangular matrix with 1’s on the diagonal (𝒩ii = 1, 𝒩ij = 0 for i > j) and 𝒜 is a diagonal matrix with positive elements, which we parametrize as
𝒜 = exp (− β ), β = diag (β1,β2,⋅⋅⋅ ,βd). (2.9 )
Both 𝒩 and 𝒜 depend on the spacetime coordinates. The spatial metric dσ2 becomes
2 i j ∑d (− 2βk) k 2 dσ = gij dx dx = e (ω ) (2.10 ) k=1
with
ωk = ∑ 𝒩 dxi. (2.11 ) ki i
The variables βi of the Iwasawa decomposition give the (logarithmic) scale factors in the new, orthogonal, basis. The variables 𝒩ij characterize the change of basis that diagonalizes the metric and hence they parametrize the off-diagonal components of the original g ij.

We extend the transformation Equation (2.8View Equation) in configuration space to a canonical transformation in phase space through the formula

ij i ∑ π dgij = π dβi + Pij d𝒩ij. (2.12 ) i<j

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),

⌊ ⌋ d ( d )2 K = 1⌈ ∑ π2− --1--- ∑ π + π2 ⌉ , (2.13 ) 4 i d − 1 i φ i=1 i=1
plus the rest, denoted by V, which will act as a potential for the scale factors. The Hamiltonian then becomes
ℋ = K + V, ∑ V = VS + VG + Vp + V φ, p 1-∑ −2(βj−βi)(∑ )2 VS = 2 e Pim𝒩jm , i<j m V = − Rg, G V(p) = V(epl) + V m(pa)gn, (2.14 ) p!e−λ(p)φ Ve(pl) = -------- πm(1p⋅)⋅⋅mpπ(p)m1⋅⋅⋅mp, 2 eλ(p)φ V m(pa)gn = --------- gF (mp)1⋅⋅⋅mp+1F (p)m1 ⋅⋅⋅mp+1, 2 (p + 1 )! ij Vφ = g g∂iφ ∂jφ.
The kinetic term K 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
( ) ∑ i 2 ∑ i 2 2 (dβ ) − dβ + (dφ) . (2.15 ) i
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 (i i β → β + ε) defines a timelike direction. It will be convenient in the following to collectively denote all the scale factors (the βi’s and the dilaton φ) as βμ, i.e., (β μ) = (βi,φ ).

The analysis is further simplified if we take for new p-form variables the components of the p-forms in the Iwasawa basis of the ωk’s,

(p) ∑ −1 − 1 𝒜 i1⋅⋅⋅ip = (𝒩 )m1i1 ⋅⋅⋅(𝒩 )mpipA (p)m1⋅⋅⋅mp, (2.16 ) m1,⋅⋅⋅,mp
and again extend this configuration space transformation to a point canonical transformation in phase space,
( ) ( ) 𝒩ij,Pij,A (p) ,πm1 ⋅⋅⋅mp → 𝒩ij,P′ ,𝒜(p) ,ℰi1⋅⋅⋅ip , (2.17 ) m1⋅⋅⋅mp (p) ij m1 ⋅⋅⋅mp (p)
using the formula ∑ pdq = ∑ p′dq′, which reads
∑ P ˙𝒩 + ∑ πm1 ⋅⋅⋅mpA˙(p) = ∑ P′ ˙𝒩 + ∑ ℰi1⋅⋅⋅ip𝒜˙(p) . (2.18 ) ij ij (p) m1⋅⋅⋅mp ij ij (p) m1 ⋅⋅⋅mp i<j p i<j p
Note that the scale factor variables are unaffected, while the momenta Pij conjugate to 𝒩ij get redefined by terms involving ℰ, 𝒩 and 𝒜 since the components (p) 𝒜m1 ⋅⋅⋅mp of the p-forms in the Iwasawa basis involve the 𝒩’s. On the other hand, the new p-form momenta, i.e., the components of the electric field m ⋅⋅⋅m π(p1) p in the basis {ωk} are simply given by
i1⋅⋅⋅ip ∑ m1⋅⋅⋅mp ℰ(p) = 𝒩i1m1𝒩i2m2 ⋅⋅⋅𝒩ipmpπ (p) . (2.19 ) m1,⋅⋅⋅,mp
In terms of the new variables, the electromagnetic potentials become
el p! ∑ −2ei1⋅⋅⋅ip(β) i1⋅⋅⋅ip 2 V(p) = 2 e (ℰ (p) ) , i1,i2,⋅⋅⋅,ip magn 1 ∑ −2mi ⋅⋅⋅i (β) 2 (2.20 ) V (p) = 2-(p +-1)! e 1 p+1 (ℱ (p)i1⋅⋅⋅ip+1) . i1,i2,⋅⋅⋅,ip+1
Here, ei1⋅⋅⋅ip(β ) are the electric linear forms
i1 ip λ(p)- ei1⋅⋅⋅ip(β ) = β + ⋅⋅⋅ + β + 2 φ (2.21 )
(the indices ij are all distinct because i1⋅⋅⋅ip ℰ(p) is completely antisymmetric) while ℱ (p)i1⋅⋅⋅ip+1 are the components of the magnetic field F (p)m1⋅⋅⋅mp+1 in the basis {ωk},
∑ ℱ(p)i1⋅⋅⋅ip+1 = (𝒩 −1)m1i1 ⋅⋅⋅(𝒩 −1)mp+1ip+1F(p)m1⋅⋅⋅mp+1, (2.22 ) m1,⋅⋅⋅,mp+1
and mi1⋅⋅⋅ip+1(β ) are the magnetic linear forms
(p) ∑ j λ--- mi1⋅⋅⋅ip+1(β) = β − 2 φ. (2.23 ) j∈∕{i1,i2,⋅⋅⋅ip+1}
One sometimes rewrites mi ⋅⋅⋅i (β) 1 p+1 as bi ⋅⋅⋅i (β) p+2 d, where {ip+2,ip+3, ⋅⋅⋅ ,id} is the set complementary to {i1,i2,⋅⋅ ⋅ip+1}, e.g.,
(p) b12⋅⋅⋅d− p−1(β) = β1 + ⋅⋅⋅ + βd −p−1 − λ--φ = md −p⋅⋅⋅d. (2.24 ) 2
The exterior derivative ℱ of 𝒜 in the non-holonomic frame {ωk} involves of course the structure coefficients Ci jk in that frame, i.e.,
ℱ (p)i1⋅⋅⋅ip+1 = ∂[i1𝒜i2⋅⋅⋅ip+1] + “C 𝒜 ”- terms, (2.25 )
where
∑ −1 m ∂i1 ≡ (𝒩 )m1i1(∂∕∂x 1) (2.26 ) m1
is here the frame derivative. Similarly, the potential Vφ reads
∑ −2¯mi(β) 2 Vφ = e (ℱi) , (2.27 ) i
where ℱi is
ℱi = (𝒩 −1)ji∂jφ (2.28 )
and
∑ j m¯i (β) = β . (2.29 ) j⁄=i

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