New explicit spike solution -- non-local component of the generalized Mixmaster attractor.pdf

a r X i v : 0 7 1 0 .0 6 2 8 v 2 [ g r - q c ] 6 F e b 2 0 0 8 New explicit spike solution – non-local component of the generalized Mixmaster attractor Woei Chet Lim Department of Physics, Princeton University, Princeton, NJ 08544, USA. Email: wlim@princeton.edu February 6, 2008 Abstract By applying a standard solution-generating transformation to an arbi- trary vacuum Bianchi type II solution, one generates a new solution with spikes commonly observed in numerical simulations. It is conjectured that the spike solutions are part of the generalized Mixmaster attractor. 1 Introduction Berger and Moncrief [1] studied Gowdy spacetimes and found small-scale spa- tial structures develop on approach to the initial singularity. Since then many efforts have been spent trying to understand these spiky structures in Gowdy spacetimes and in more general G2 spacetimes through numerical simulations and analytical approximations [2, 3, 4, 5, 6, 7]. My motivation in studying spikes is to understand its role on approach to generic singularities. Lifshitz, Khalatnikov and Belinskii [8, 9, 10] were the first to provide heuristic arguments that the approach to generic spacelike singular- ities are vacuum dominated, local, and oscillatory (known as the BKL conjec- ture). Further evidence came from the study of Bianchi type IX cosmologies by Misner [11, 12, 13], who coined the term “Mixmaster” to describe the oscillatory behaviour. Uggla et al [14] provided a detailed description of the local attractor for generic singularities (called the generalized Mixmaster attractor). See [15] for a more complete introduction and the latest work on the attractor (called the billiard attractor). The local part of the BKL conjecture is increasingly under challenge from numerical evidence of the presence of recurring transient spikes, which are non-local structures, in the approach to singularities [6, 7]. In- sufficiently resolved spiky structures in the singular regime have been observed in n