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2.5 Rules for deriving the wall forms from the Lagrangian – Summary

We have recalled above that the generic behavior near a spacelike singularity of the system with action (2.1View Equation) can be described at each spatial point in terms of a billiard in hyperbolic space. The action for the billiard ball reads, in the gauge √ -- N = g,
[ ] ∫ 0 d βμ dβν μ S = dx G μν---0 --0-− V (β ) , (2.46 ) dx dx
where we recall that x0 → ∞ in the BKL-limit (proper time T → 0+), and G μν is the metric in the space of the scale factors,
( ) ( ) ∑d ∑d ∑d G μν dβ μdβ ν = dβidβi − dβi dβj + d φd φ (2.47 ) i=1 i=1 j=1
introduced in Equation (2.15View Equation) above. As stressed there, this metric is flat and of Lorentzian signature. Between two collisions, the motion is a free, geodesic motion. The collisions with the walls are controlled by the potential V(β μ), which is a sum of sharp wall potentials. The walls are hyperplanes and can be inferred from the Lagrangian. They are as follows:

  1. Gravity brings in the symmetry walls
    βi+1 − βi = 0, (2.48)
    with i = 1,2,⋅⋅⋅ ,d − 1, and the curvature wall
    2β1 + β2 + ⋅⋅⋅ + βd −2 = 0. (2.49)
  2. Each p-form brings in an electric wall
    1 p λ (p) β + ⋅⋅⋅ + β + --2-φ = 0, (2.50)
    and a magnetic wall
    λ(p) β1 + ⋅⋅⋅ + βd− p−1 −---φ = 0. (2.51) 2

We have written here only the (potentially) relevant walls. There are other walls present in the potential, but because these are behind the relevant walls, which are infinitely steep in the BKL-limit, they are irrelevant. They are relevant, however, when trying to exhibit the symmetry in a complete treatment where the BKL-limit is the zeroth order term in a gradient expansion yet to be understood [47Jump To The Next Citation Point].

The scalar product dual to the scalar product in the space of the scale factors is

( )( ) ∑ --1--- ∑ ∑ (F|G ) = FiGi − d − 1 Fi Gj + FφG φ (2.52 ) i i j
for two linear forms F = Fiβi + Fφφ, G = Gi βi + G φφ.

These recipes are all that we shall need for investigating the regularity properties of the billiards associated with the class of actions Equation (2.1View Equation).

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