On the Initial Boundary Value Problem in Numerical Relativity

Abstract

The principal goal of this thesis is to properly understand, characterize, and numerically implement initial boundary value problems in numerical relativity. Throughout the history of solving Einstein's field equations on computers, boundaries have been mostly dealt with in an approximate way. For example, boundaries might be placed far away from strongly gravitating sources, where approximations like linearized gravity are valid. It has become necessary however to place boundaries in the strong gravity regime of a dynamical spacetime to model complicated and interesting physics, which necessitates a complete understanding of the initial boundary value problem of Einstein's field equations.

One motivation for this comes from a need to simulate black hole echoes. In classical general relativity, black holes are perfectly absorbing objects, where the mass of radially incoming wavepackets of matter or gravitational waves is absorbed by the black hole. Thus conclusive evidence of modifications to general relativity, such as quantum gravity, could include partial reflections of radially incoming wavepackets, called black hole echoes. To properly understand the modifications this would bring to detectable gravitational wave signals, we require simulations where reflecting boundary conditions are imposed close to the horizon of a black hole. Another motivation comes from recent advances in Cauchy characteristic matching, which combines state of the art numerical techniques to obtain physically accurate gravitational waveforms from simulations. This can allow numerical relativists to dramatically save on the computational cost of black hole merger simulations, but only if boundaries can be placed in the strong gravity regime.

This thesis presents advances in simulating initial boundary value problems in numerical relativity. Starting with spherical symmetry, a framework for reflecting a scalar field in a fully dynamical spacetime is developed and implemented numerically using the Einstein-Christoffel formulation. The evolution of a wave packet and its numerical convergence, including when the location of a reflecting boundary is very close to the horizon of a black hole, is studied. Next, this approach is generalized to spacetimes with no symmetries and implemented numerically using the generalized harmonic formulation. The evolution equations are cast into a summation by parts scheme, which seats the numerical method closer to a class of provably numerically stable systems. State of the art numerical methods are demonstrated, including an embedded boundary numerical method that allows for arbitrarily shaped domains on a rectangular grid and even boundaries that evolve and move across the grid. As a demonstration of these frameworks, the evolution of gravitational wave scattering off of a boundary either inside or just outside the horizon of a black hole, is studied. Finally, a boundary condition framework designed to control quasi-local energy flux is proposed motivated by examples from electromagnetism.