The effect of quantum fields on black-hole interiors

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 14