Harmonic coordinate method for simulating generic singularities
David Garfinkle
TL;DR
The paper addresses numerically simulating generic spacetime singularities in the Einstein-scalar system with stiff matter to determine whether the approach to the singularity is locally Kasner-like. It introduces a generalized harmonic coordinate formulation, reduces the equations to first order via $P_{\alpha\beta}=\partial_t g_{\alpha\beta}$ and $P_{\phi}=\partial_t \phi$, and evolves the system on a topology $T^3\times \mathbb{R}$ using a three-step iterated Crank-Nicolson scheme. Validation includes Gowdy tests showing agreement with symmetry-reduced evolutions and a constraint-convergence study revealing second-order convergence; near-singularity results show $\log g_{ii}$ and $\phi$ becoming linear in time with directional contraction, consistent with Kasner-like dynamics. The work demonstrates the utility of the harmonic-coordinate approach for strong-field numerical relativity and lays groundwork for extending to vacuum spacetimes and asymptotically flat setups with boundary considerations.
Abstract
This paper presents both a numerical method for general relativity and an application of that method. The method involves the use of harmonic coordinates in a 3+1 code to evolve the Einstein equations with scalar field matter. In such coordinates, the terms in Einstein's equations with the highest number of derivatives take a form similar to that of the wave equation. The application is an exploration of the generic approach to the singularity for this type of matter. The preliminary results indicate that the dynamics as one approaches the singularity is locally the dynamics of the Kasner spacetimes.