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

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.

20 pages, 5 figures

References in corpus (6)

• Pseudo-spectral construction of non-uniform black string solutions in five and six spacetime dimensions
• The second order spin-2 system in flat space near space-like and null-infinity
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• Fully pseudospectral solution of the conformally invariant wave equation near the cylinder at spacelike infinity
• Fully pseudospectral solution of the conformally invariant wave equation near the cylinder at spacelike infinity. II: Schwarzschild background

Cited by in corpus (4)

• Spectral methods for the spin-2 equation near the cylinder at spatial infinity
• Fully pseudospectral solution of the conformally invariant wave equation on a Kerr background
• Explorations of the infinite regions of space-time
• The characteristic initial value problem for the conformally invariant wave equation on a Schwarzschild background