Evolving Einstein's Field Equations with Matter: The ``Hydro without Hydro'' Test

TL;DR

This paper demonstrates a stable 3+1 evolution scheme for Einstein's equations with matter, successfully simulating static stars and collapsing dust spheres, and proposes a new approximation method for gravitational wave modeling in strong-field scenarios.

Contribution

The authors introduce a stable evolution scheme for Einstein's equations with matter and propose a novel approximation method for gravitational wave extraction in strong-field sources.

Findings

Stable evolution of static, strong-field stars for long durations.

Accurate simulation of dust sphere collapse past black hole formation.

Potential for simplified gravitational wave modeling using quasi-equilibrium approximations.

Abstract

We include matter sources in Einstein's field equations and show that our recently proposed 3+1 evolution scheme can stably evolve strong-field solutions. We insert in our code known matter solutions, namely the Oppenheimer-Volkoff solution for a static star and the Oppenheimer-Snyder solution for homogeneous dust sphere collapse to a black hole, and evolve the gravitational field equations. We find that we can evolve stably static, strong-field stars for arbitrarily long times and can follow dust sphere collapse accurately well past black hole formation. These tests are useful diagnostics for fully self-consistent, stable hydrodynamical simulations in 3+1 general relativity. Moreover, they suggest a successive approximation scheme for determining gravitational waveforms from strong-field sources dominated by longitudinal fields, like binary neutron stars: approximate quasi-equilibrium models can serve as sources for the transverse field equations, which can be evolved without having to re-solve the hydrodynamical equations (``hydro without hydro'').