Fully pseudospectral solution of the conformally invariant wave equation on a Kerr background

TL;DR

This paper develops a pseudospectral numerical method to analyze the behavior of solutions to the conformally invariant wave equation on a Kerr black hole background, focusing on regularity near spacelike infinity and the effects of rotation.

Contribution

It extends previous Schwarzschild studies to Kerr backgrounds, constructing a conformal compactification and implementing a full 2+1 pseudospectral evolution code for axisymmetric waves.

Findings

Logarithmic singularities develop at the cylinder at infinity.

Rotation influences the regularity and singularity structure of solutions.

The pseudospectral method achieves high accuracy for the wave evolution.

Abstract

We study axisymmetric solution to the conformally invariant wave equation on a Kerr background by means of numerical and analytical methods. Our main focus is on the behaviour of the solutions near spacelike infinity, which is appropriately represented as a cylinder. Earlier studies of the wave equation on a Schwarzschild background have revealed important details about the regularity of the corresponding solutions. It was found that, on the cylinder, the solutions generically develop logarithmic singularities at infinitely many orders. Moreover, these singularities also `spread' to future null infinity. However, by imposing certain regularity conditions on the initial data, the lowest-order singularities can be removed. Here we are interested in a generalisation of these results to a rotating black hole background and study the influence of the rotation rate on the properties of the solutions. To this aim, we first construct a conformal compactification of the Kerr solution which yields a suitable representation of the cylinder at spatial infinity. Besides analytical investigations on the cylinder, we numerically solve the wave equation with a fully pseudospectral method, which allows us to obtain highly accurate numerical solutions. This is crucial for a detailed analysis of the regularity of the solutions. In the Schwarzschild case, the numerical problem could effectively be reduced to solving $(1+1)$-dimensional equations. Here we present a code that can perform the full $2+1$ evolution as required for axisymmetric waves on a Kerr background.