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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 0 x → ∞ (g → 0). Although the more recent “derivations” of the BKL-limit treat both elements at once [43Jump To The Next Citation Point44Jump To The Next Citation Point45Jump To The Next Citation Point48Jump To The Next Citation Point], it appears useful – especially for rigorous justifications – to separate two aspects of the BKL conjecture3.

The first aspect is that the spatial points decouple in the limit x0 → ∞, in the sense that one can replace the Hamiltonian by an effective “ultralocal” Hamiltonian UL H 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

ds2 = gij(x0 )ψiψj, (2.30 )
where the invariant forms fulfill
1 dψi = − --fijkψj ∧ ψk. 2
Here, the fijk are the structure constants of the spatial homogeneity group. Similarly, for a 1-form and a 2-form,
1 A (1) = Ai(x0)ψi, A(2) = -Aij(x0)ψi ∧ ψj, etc. (2.31 ) 2
The Hamiltonian constraint yielding the field equations in the spatially homogeneous context4 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 i f ik contain implicit spatial gradients. The Hamiltonian can now be decomposed as before and reads
ℋUL = K + V UL, ∑ ( ) V UL = VS + VGUL+ V(elp) + V(UpL),magn , (2.32 ) p
where K, VS and V el (p), which do not involve spatial gradients, are unchanged and where V φ disappears since ∂iφ = 0. The potential VG is given by [61Jump To The Next Citation Point]
1- ∑ − 2αijk(β) i 2 1∑ −2¯mj(β)( i k ) VG ≡ − gR = 4 e (C jk) + 2 e C jk C ji + “more” , (2.33 ) i⁄=j,i⁄=k,j⁄=k j
where the linear forms αijk(β) (with i,j,k distinct) read
∑ αijk(β) = 2βi + βm, (2.34 ) m :m ⁄=i,m ⁄=j,m⁄=k
and where “more” stands for the terms in the first sum that arise upon taking i = j or i = k. The structure constants in the Iwasawa frame (with respect to the coframe in Equation (2.30View Equation)) are related to the structure constants fijk through
Ci = ∑ f i′ ′′𝒩 −1𝒩 ′𝒩 ′ (2.35 ) jk ′ ′ ′ jk ii′ jj kk i ,j,k
and depend therefore on the dynamical variables. Similarly, the potential V magn (p) becomes
1 ∑ V m(pa)gn = --------- e−2mi1⋅⋅⋅ip+1(β)(ℱh(p)i1⋅⋅⋅ip+1)2, (2.36 ) 2 (p + 1)!i1,i2,⋅⋅⋅,ip+1
where the field strengths h ℱ (p)i1⋅⋅⋅ip+1 reduce to the “AC” terms in dA 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 {dx0, ψi} where the ψi’s form a “generic” non-holonomic frame in space,

i 1- i m j k dψ = − 2 f jk(x )ψ ∧ ψ . (2.37 )
Here the i f jk’s are in general space-dependent. In the non-holonomic frame, the exact Hamiltonian takes the form
UL gradient ℋ = ℋ + ℋ , (2.38 )
where the ultralocal part ℋUL is given by Equations (2.32View Equation) and (2.33View Equation) with the relevant fijk’s, and where ℋgradient involves the spatial gradients of f ijk, βm, φ and 𝒩ij.

The first part of the BKL conjecture states that one can drop gradient ℋ asymptotically; namely, the dynamics of a generic solution of the Einstein–p-form-dilaton equations (not necessarily spatially homogeneous) is asymptotically determined, as one goes to the spatial singularity, by the ultralocal Hamiltonian

∫ UL d UL H = d xℋ , (2.39 )
provided that the phase space constants i m i m f jk(x ) = − f kj(x ) are such that all exponentials in the above potentials do appear. In other words, the f’s must be chosen such that none of the coefficients of the exponentials, which involve f and the fields, identically vanishes – as would be the case, for example, if f i = 0 jk since then the potentials V G and Vmagn (p) are equal to zero. This is always possible because the i f jk, even though independent of the dynamical variables, may in fact depend on x and so are not required to fulfill relations “ff = 0” analogous to the Bianchi identity since one has instead “∂f + ff = 0”.


  1. As we shall see, the conditions on the f’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 fijk = 0 (and hence also Ci = 0 jk).
  2. The actual values of the i f jk (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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