Free evolution of the hyperboloidal initial value problem in spherical simmetry

Résumé

The present work deals with the application of conformal compactification methods to the numerical solution of the Einstein equations, the field equations of General Relativity. They form a complex system of non-linear partial differential equations that can only be solved analytically for highly symmetric spacetimes. The most general spacetimes have to be obtained with the help of numerical techniques. In General Relativity, central physical quantities such as the total energy or radiation flux can only be defined unambiguously in the asymptotic region of a spacetime, which calls for the numerical treatment of infinite domains. The traditional approach in Numerical Relativity codes is based on spacelike slices that are cut at an artificial timelike boundary and whose data are extrapolated to infinity. The goal of this thesis is to further develop an elegant alternative approach, which aims to efficiently solve the Einstein equations for spacetimes of isolated radiating systems and compute the radiation signal without any approximations. Following a framework by Penrose, we use a finite unphysical spacetime related to the physical one by a conformal rescaling. On this rescaled spacetime, taking limits towards infinity is replaced by local differential geometry and observable physical quantities can be directly evaluated. In order to compute radiation quantities, it is convenient to foliate spacetime by hyperboloidal slices. These are smooth spacelike slices that reach future null infinity, the “place” in spacetime where light rays arrive. Among the advantages of evolving on compactified hyperboloidal slices are that no boundary conditions are required, because future null infinity is an ingoing null surface and it does not allow any information to enter the domain from beyond. The price to pay is that the conformally rescaled Einstein equations are singular at infinity and need to be regularized. Besides, the nontrivial background geometry of the hyperboloidal slices makes the evolution equations prone to continuum instabilities. As a first step towards developing numerical algorithms for the hyperboloidal initial value problem for strong field dynamical spacetimes, the numerical work in this thesis is restricted to spherical symmetry. Given that the regularization of the radial direction is common to spherical symmetry and the full three-dimensional case, the results obtained are expected to apply, at least to some degree, to the full system. This work's approach uses standard unconstrained formulations of General Relativity, specifically the BSSN (Baumgarte-Shapiro-Shibata-Nakamura) equations and the Z4 equations. The derivation of their spherically symmetric component equations will be described, as well as the calculation of appropriate initial data on the hyperboloidal slice given by a constant-mean-curvature foliation. A critical point is the treatment of the gauge conditions: both the specific requirements for the hyperboloidal value problem, such as scri-fixing or the preferred conformal gauge, and the adaptation of currently common gauge choices will be explained. As expected, the numerical implementation was difficult to stabilize, but by means of a variable transformation on the trace of the extrinsic curvature and the addition of a constraint damping term to the evolution equation of the contracted connection, the implementation finally became well-behaved. Stable simulations of the Einstein equations coupled to a massless scalar field have been performed with regular and strong field initial data. Small perturbations of regular initial data give stationary data that are stable forever, while larger scalar field perturbations result in the formation of a black hole. Schwarzschild trumpet initial data have been found to slowly drift away from the expected stationary values, but the effect for small perturbations is slow enough to allow the observation of the power-law decay tails of the scalar field.