Towards a numerical derivation of maximal Kerr trumpetinitial data

This thesis is structured in two parts. In part 1 we summarise the necessary theoretical background and the numerical tools in order to conduct the research in part 2. In chapter 2 we summarise the 3 + 1 approach to numerical relativity, with a focus on the initial data formalism in the weighted transverse and conformal thin-sandwich decompositions. We then move on to discuss initial data for black hole simulations in chapter 3, where we focus on the moving puncture approach. We note that the standard gauge evolution equations force the initial numerical wormhole slices to evolve into trumpet slices, which suggests the construction of a priori trumpet initial data. We summarise the literature on trumpet research, with a focus on maximal Schwarzschild trumpet initial data in its analytical form. In chapter 4 we provide the necessary numerical tools and techniques, which we use in part 2 to solve the constraint equations numerically. We focus on finite difference methods in connection with the Thomas algorithm and successive over-relaxation to solve boundary value problems in one and two variables. Part 2 contains the novel research of this thesis. In chapter 5 we discuss Kerr in quasi-isotropic coordinates and point out that it represents a maximal trumpet foliation for extreme Kerr, however a wormhole foliation for slow Kerr. We lay out our approach to numerically derive maximal Kerr trumpet initial data. The approach is based on the proposition that two nontrivial impositions on the constraints suffice to construct the data numerically. We propose that these relations can be generalised from Schwarzschild and extreme Kerr to slow Kerr. The main motivation for this stems from the observation that the square of the extrinsic curvature for Bowen-York shows the same behaviour as for Kerr for small radii, and Bowen-York trumpets have been constructed successfully. Our main results are then presented in chapters 6 and 7 in which we test our approach for the special cases of zero and maximal spin, ie for Schwarzschild and extreme Kerr. We succeed with Schwarzschild and present the first ever purely numerical derivation of maximal Schwarzschild trumpet initial data in the weighted transverse decomposition – our first main result. Because of the complexity of the problem for extreme Kerr, we proceed in steps and start out by using more information of the analytical solution to treat the constraints separately. For instance, we provide the conformal metric to the Hamiltonian and momentum constraints, and solve them successfully for the trumpet solution – our second main result. Finally we elaborate on how to relax the assumptions. In particular, we introduce an additional equation which we can solve successfully for the function which describes the deviation of the conformal metric from being flat – our third main result.