Wave Equations on Curved Spacetimes

There are still many important unsolved problems in general relativity, two of which are the stability problem and the strong cosmic censorship conjecture. Both of these are ex- tremely difficult, but we can gain some insight by studying linear versions of them. These simplifications give rise to wave equations on curved spacetimes. We investigate the classical stability of supersymmetric, asymptotically flat microstate geometries with 5 non-compact dimensions. These geometries possess an evanescent ergo-surface, where there are stably trapped null geodesics that have zero energy relative to an observer at infinity. We give a heuristic argument as to why this may lead to a non-linear instability, which can be seen at the linear level by studying the wave equation. We calculate the quasinormal mode frequencies and find that, due to the stable trapping, the rate of decay is extremely slow. This suggests that stability is very unlikely at the non-linear level. The behaviour of geodesics is crucial for this, so we also investigate the geodesics in these microstate geometries in some detail. There has recently been evidence to suggest that Christodoulou’s formulation of the strong cosmic censorship conjecture is actually false for Reissner-Nordstron-de Sitter black holes sufficiently close to extremality. We investigate this problem for the more physical rotating Kerr-de Sitter black holes. We look at the linear problem, and find that solutions of the wave equation decay sufficiently slowly to suggest that strong cosmic censorship is respected. The two problems mentioned so far are both related to predictability in general relativ- ity. We investigate predictability more generally for subluminal and superluminal Lorentz- invariant scalar wave equations. We study the Born-Infeld scalar in two dimensions, which has both a superluminal and subluminal formulation. Contrary to previous expectation, we find that, at least in some sense, the subluminal equation behaves worse the superluminal equation. It is possible to have multiple different maximal globally hyperbolic developments arising from the same initial data for the subluminal equation, but the solution is unique in the superluminal case.

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2019

Author dc:creator

• Eperon, Felicity Clare

Advisor dc:contributor.advisor

• Reall, Harvey

• General relativity
• Black Holes
• Wave equations

Licence

• https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/a4643402-120f-4e4c-ab57-e3d9d33f12cd/download
• https://www.rioxx.net/licenses/all-rights-reserved/

Language dc:language

en

DOI dc:identifier.doi

https://doi.org/10.17863/CAM.43848

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/296804