Asymptotically null slices in numerical relativity: mathematical analysis and spherical wave equation tests

TL;DR

This paper explores a family of asymptotically null slices for numerical relativity, demonstrating their effectiveness in accurately simulating outgoing waves and tail phenomena in flat and Schwarzschild spacetimes with reduced computational costs.

Contribution

It introduces a new class of coordinate slices characterized by a parameter n, improving wave simulation accuracy and efficiency in numerical relativity.

Findings

Higher accuracy in wave propagation at large radii with lower computational cost.

Correct representation of power-law tails in Schwarzschild spacetime.

Slices with 0<n≤2 intersect null infinity, enhancing simulation capabilities.

Abstract

We investigate the use of asymptotically null slices combined with stretching or compactification of the radial coordinate for the numerical simulation of asymptotically flat spacetimes. We consider a 1-parameter family of coordinates characterised by the asymptotic relation $r\sim R^{1-n}$ between the physical radius $R$ and coordinate radius $r$, and the asymptotic relation $K\sim R^{n/2-1}$ for the extrinsic curvature of the slices. These slices are asymptotically null in the sense that their Lorentz factor relative to stationary observers diverges as $\Gamma\sim R^{n/2}$. While $1<n\le 2$ slices intersect $\scri$, $0< n\le 1$ slices end at $i^0$. We carry out numerical tests with the spherical wave equation on Minkowski and Schwarzschild spacetime. Simulations using our coordinates with $0<n\le 2$ achieve higher accuracy at lower computational cost in following outgoing waves to very large radius than using standard $n=0$ slices without compactification. Power-law tails in Schwarzschild are also correctly represented.