Simulations of gravitational collapse in null coordinates: I. Formulation and weak-field tests in generalised Bondi gauges

TL;DR

This paper introduces a numerical code for simulating gravitational collapse to black holes using null coordinates, with a focus on generalised Bondi gauges, and demonstrates its effectiveness through convergence tests in weak-field scenarios.

Contribution

The paper presents a novel numerical scheme for null coordinate simulations of gravitational collapse that reduces angular resolution at small radii, improving computational efficiency.

Findings

Code successfully simulates black hole formation in axisymmetry with scalar fields.

Convergence tests match exact solutions in weak-field regimes.

Coordinate choice allows zooming on critical collapse features without mesh refinement.

Abstract

We present a code for numerical simulations of the collapse of regular initial data to a black hole in null coordinates. We restrict to twist-free axisymmetry with scalar field matter. Our coordinates are $(u,x,y,\varphi)$, where the retarded time $u$ labels outgoing null cones emerging from a regular central worldline, the angles $(\theta,\varphi)$ label the null generators of each null cone, and the radial coordinate $x$ labels points along these generators. We focus on a class of generalised Bondi radial coordinates $x$ with the twin properties that $x=0$ is the central world line and that the numerical domain $(u\ge0$, $0\le x\le x_\text{max})$ is a subset of the domain of dependence of the initial data on $(u=0$, $0\le x\le x_\text{max}$). In critical collapse, an appropriate choice of these coordinates can be made to zoom in on the accumulation point of scale echos of the critical solution, without the need for explicit mesh refinement. We introduce a novel numerical scheme that in effect reduces the angular resolution at small radius, such that the time step $\Delta u$ for an explicit numerical scheme is limited by the radial resolution $\Delta x$, rather than $\Delta x(\Delta\theta)^2$. We present convergence tests in the weak-field regime, where we have exact solutions to the linearised scalar and gravitational-wave equations.