Charged or rotating black holes possess an inner horizon beyond which determinism is
lost. However, the strong cosmic censorship conjecture claims that even small perturbations
will turn the horizon into a singularity beyond which the spacetime is inextendible,
preventing the loss of determinism. Motivated by this conjecture, this dissertation studies
free scalar quantum fields on various black-hole spacetimes to test whether quantum
effects can lead to the formation of a singularity at the inner horizon in cases where
classical perturbations cannot. The starting point is the investigation of the behaviour
of real-scalar-field observables near the inner horizon of Reissner-Nordström-de Sitter
spacetimes. Using semi-analytical methods, we find that quantum effects can indeed uphold
the censorship conjecture. Subsequently, we consider charged scalar fields on the
same spacetime and observe that a first-principle calculation is essential to accurately describe
the quantum effects at the inner horizon. As a first step towards an extension of
these results to rotating black holes, we rigorously construct the Unruh state for the real
scalar field on slowly rotating Kerr-de Sitter spacetimes. We show that it is a well-defined
Hadamard state and can therefore be used to compute expectation values of the stressenergy
tensor and other non-linear observables.:1 Introduction 7
2 An introduction to quantum fields and black holes 13
2.1 Notations and conventions . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 A brief introduction to AQFT . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 An introduction to microlocal analysis . . . . . . . . . . . . . . . . . . . 24
2.4 An introduction to black-hole spacetimes . . . . . . . . . . . . . . . . . 28
2.4.1 The Reissner-Nordström-de Sitter spacetime . . . . . . . . . . . 28
2.4.2 The Kerr-de Sitter spacetime . . . . . . . . . . . . . . . . . . . . 32
2.5 Free scalar fields in black-hole spacetimes . . . . . . . . . . . . . . . . . 37
3 Computing the energy flux of the real scalar field 43
3.1 Strong cosmic censorship on RNdS . . . . . . . . . . . . . . . . . . . . 43
3.2 The Klein-Gordon equation on RNdS . . . . . . . . . . . . . . . . . . . 45
3.3 Extension to the charged scalar field on RNdS . . . . . . . . . . . . . . . 52
3.4 The energy flux at the Cauchy horizon . . . . . . . . . . . . . . . . . . . 53
4 The charged scalar field in Reissner-Nordström-de Sitter 63
4.1 The Unruh state for the charged scalar field . . . . . . . . . . . . . . . . 65
4.2 The renormalized current . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.3 The current in the Unruh state - numerical results . . . . . . . . . . . . . 80
4.4 The charged scalar field at the inner horizon . . . . . . . . . . . . . . . . 86
5 The Unruh state on Kerr-de Sitter 97
5.1 Null geodesics in the Kerr-de Sitter spacetime . . . . . . . . . . . . . . . 98
5.2 The Unruh state on Kerr-de Sitter . . . . . . . . . . . . . . . . . . . . . . 107
5.3 The Hadamard property of the Unruh state . . . . . . . . . . . . . . . . . 120
5.3.1 The Hadamard condition in O . . . . . . . . . . . . . . . . . . . 123
5.3.2 The Hadamard condition on M\O . . . . . . . . . . . . . . . . . 128
6 Summary and discussion 139
A Bibliography 143
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:87395 |
| Date | 12 October 2023 |
| Creators | Klein, Christiane Katharina Maria |
| Contributors | Universität Leipzig |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | info:eu-repo/semantics/publishedVersion, doc-type:doctoralThesis, info:eu-repo/semantics/doctoralThesis, doc-type:Text |
| Rights | info:eu-repo/semantics/openAccess |
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