Numerical evolution of general Robinson-Trautman spacetimes: code tests, wave forms and the efficiency of the gravitational wave extraction

TL;DR

This paper introduces an efficient spectral numerical code for evolving Robinson-Trautman spacetimes, demonstrating high accuracy and applying it to various black hole scenarios to analyze gravitational waveforms and extraction efficiency.

Contribution

The paper presents a new spectral method-based numerical code for Robinson-Trautman spacetimes, improving computational efficiency and accuracy in simulating black hole perturbations and gravitational waves.

Findings

High accuracy and convergence of the code.

Derived bounds on black hole velocity and wave extraction efficiency.

Numerical waveforms for different black hole collision scenarios.

Abstract

We present an efficient numerical code based on spectral methods to integrate the field equations of general Robinson-Trautmann spacetimes. The most natural basis functions for the spectral expansion of the metric functions are spherical harmonics. Using the values of appropriate combinations of the metric functions at the collocation points, we have managed to reduce expression swell when the number of spherical harmonics increases. Our numerical code runs with relatively little computational resources and the code tests have shown excellent accuracy and convergence. The code has been applied to situations of physical interest in the context of Robsinson-Trautmann geometries such as: perturbation of the exterior gravitational field of a spheroid of matter; perturbation of an initially boosted black hole; and the non-frontal collision of two Schwarzschild black holes. In dealing with these processes we have derived analytical lower and upper bounds on the velocity of the resulting black hole and the efficiency of the gravitational wave extraction, respectively. Numerical experiments were performed to determine the forms of the gravitational waves and the efficiency in each situation of interest.