Fully pseudospectral solution of the conformally invariant wave equation near the cylinder at spacelike infinity. III: Nonspherical Schwarzschild waves and singularities at null infinity

TL;DR

This paper extends the analysis of the conformally invariant wave equation on Schwarzschild backgrounds to nonspherical solutions, revealing how singularities develop at null infinity and demonstrating a highly accurate pseudospectral numerical scheme.

Contribution

It introduces a method to handle nonspherical solutions with singularities at null infinity using a pseudospectral approach and regularity conditions on initial data.

Findings

Logarithmic singularities occur at the cylinder and spread to null infinity.

Regularity conditions can prevent the first singularities.

Coordinate transformations enable high-accuracy solutions despite singularities.

Abstract

We extend earlier numerical and analytical considerations of the conformally invariant wave equation on a Schwarzschild background from the case of spherically symmetric solutions, discussed in Class. Quantum Grav. 34, 045005 (2017), to the case of general, nonsymmetric solutions. A key element of our approach is the modern standard representation of spacelike infinity as a cylinder. With a decomposition into spherical harmonics, we reduce the four-dimensional wave equation to a family of two-dimensional equations. These equations can be used to study the behaviour at the cylinder, where the solutions turn out to have logarithmic singularities at infinitely many orders. We derive regularity conditions that may be imposed on the initial data, in order to avoid the first singular terms. We then demonstrate that the fully pseudospectral time evolution scheme can be applied to this problem leading to a highly accurate numerical reconstruction of the nonsymmetric solutions. We are particularly interested in the behaviour of the solutions at future null infinity, and we numerically show that the singularities spread from the cylinder to null infinity. The observed numerical behaviour is consistent with similar logarithmic singularities found analytically on the cylinder. Finally, we demonstrate that even solutions with singularities at low orders can be obtained with high accuracy by virtue of a coordinate transformation that converts functions with logarithmic singularities into smooth solutions.