Constraint-preserving boundary conditions in numerical relativity

TL;DR

This paper develops and tests boundary conditions that preserve constraints in 3D numerical relativity simulations of black holes, ensuring stable evolution even with matter fields and large boundary pulses.

Contribution

It introduces a method to implement constraint-preserving boundary conditions in the Einstein-Christoffel formulation for spherically symmetric black hole spacetimes, enabling stable long-term evolutions.

Findings

Successfully evolved black hole spacetimes with boundary conditions that preserve constraints.

Reproduced expected scalar field tail decay and trivial vacuum evolution.

Allowed large boundary pulses without compromising stability.

Abstract

This is the first paper in a series aimed to implement boundary conditions consistent with the constraints' propagation in 3D numerical relativity. Here we consider spherically symmetric black hole spacetimes in vacuum or with a minimally coupled scalar field, within the Einstein-Christoffel symmetric hyperbolic formulation of Einstein's equations. By exploiting the characteristic propagation of the main variables and constraints, we are able to single out the only free modes at the outer boundary for these problems. In the vacuum case a single free modes exists which corresponds to a gauge freedom, while in the matter case an extra mode exists which is associated with the scalar field. We make use of the fact that the EC formulation has no superluminal characteristic speeds to excise the singularity. We present a second-order, finite difference discretization to treat these scenarios, where we implement these constraint-preserving boundary conditions, and are able to evolve the system for essentially unlimited times. As a test of the robustness of our approach, we allow large pulses of gauge and scalar field enter the domain through the outer boundary. We reproduce expected results, such as trivial (in the physical sense) evolution in the vacuum case (even in gauge- dynamical simulations), and the tail decay for the scalar field.