Numerical general relativity in exotic settings

• Main Author: Adam, Alexander
• Other Authors: Wiseman, Toby
• Published: Imperial College London 2013
• Subjects: 530
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.656378

In this thesis, we discuss applications of numerical relativity in a variety of complex settings. After introducing aspects of black hole physics, extra dimensions, holography, and Einstein-Aether theory we discuss how one can frame the problem of solving the static Einstein equations as an elliptic boundary value problem by inclusion of a DeTurck gauge fixing term. Having setup this background, we turn to our simplest application of numerical relativity, namely fractionalisation in holographic condensed matter. We explain how one may describe this phenomenon by studying particular classes of hairy black holes and analysing whether bulk flux is sourced by a horizon or charged matter. This problem is our simplest application of numerical relativity as the Einstein equations reduce to ODEs and the problem may be solved by shooting methods. We next turn to a discussion of stationary numerical relativity and explain how one can also view the problem of finding stationary black hole solutions as an elliptic problem, generalising the static results discussed earlier. Ergoregions and horizons are naively a threat to ellipticity, but by considering a class of spacetimes describing a fibration of the stationary and axial Killing directions over a Riemannian base space manifold, we show how the problem can nevertheless still be phrased in this manner. Finally we close with a discussion of black holes in Einstein-Aether theory. These unusual objects have multiple horizons as a consequence of broken Lorentz symmetry, and in order to construct such solutions we explain how to generalise the PDE methods of previous sections to construct solutions interior to a metric horizon where the Harmonic Einstein equations cease to be elliptic. Using this new machinery we rediscover the spherically symmetric static black holes that have been found in the literature and moreover present the first known rotating solutions of the theory.