100 Years of Relativity Space-Time Structure: Einstein and Beyond ...

Gravitational Billiards, Dualities and Hidden Symmetries 51For this frame field we adopt the so-called Iwasawa decompositionθ a = e −βa N a idx i (33)by splitting off the (logarithmic) scale factors β a from the off-diagonal frame(and metric) degrees of freedom N a i, which are represented by an uppertriangular matrix with 1’s on the diagonal. The spatial metric then becomesg ij =d∑e −2βa N a i N a j (34)a=1The main advantage of the Iwasawa decomposition is that it matches preciselywith the triangular decomposition (48) below, which is valid for anyKac Moody algebra. Furthermore, it turns out that, in the limit t → 0 allthe interesting action takes place in the scale factors β a , whereas the N aswell as the matter degrees of freedom asymptotically ‘come to rest’ in thislimit. Similarly, the metric and other degrees of freedom at different spatialpoints should decouple in this limit, as the spatial distance between themexceeds their horizon e . The basic hypothesis underlying the BKL analysisis therefore that spatial gradients should become less and less important incomparison with time derivatives as t → 0, such that the resulting theoryshould be effectively describable in terms of a one dimensional reduction,in which the complicated partial differential equations of Einstein’s theoryare effectively replaced by a continuous infinity of ordinary differentialequations.To spell out this idea in more detail, let us insert the above metric ansatzinto the Einstein-Hilbert action, and drop all spatial derivatives (gradients),so that this action is approximated by a continuous superposition of onedimensionalsystems. One then obtains (still in d spatial dimensions)S[g ij ] = 1 4∫∫d d xdx 0 Ñ −1 [( tr (g −1 ġ) 2 − (tr g −1 ġ) 2)] (35)in a matrix notation where g(t) ∈ GL(d, R) stands for the matrix (g ij )representing the spatial components of the metric at each spatial point,and Ñ ≡ N√ g is a rescaled lapse function. Neglecting the off-diagonale One might even view this decoupling as a direct consequence of the spacelike nature ofthe singularity.