A Mathematical Study of Hawking Radiation on Collapsing, Spherically Symmetric Spacetimes
In this thesis, we give a mathematical treatment of the late time Hawking radiation of massless bosons emitted by a family of collapsing, spherically symmetric, charged models of black hole formation, including both extremal and sub-extremal black holes. We further bound the rate at which the late time behaviour is approached. This treatment relies heavily on analysing the behaviour of the linear scattering map for massless bosons (solutions to the wave equation), which will be discussed further in this thesis. The thesis will be split into three chapters. The first chapter will be an introduction and derivation of the underlying spacetime models, known as Reissner–Nordström Oppenheimer–Snyder (RNOS) models. We will discuss the derivation of the Oppenheimer– Snyder model [39], before moving on to the more general charged case. We will then summarise the interesting and useful properties of these models. The second chapter will cover the analysis of the scattering map for the wave equation on RNOS backgrounds. The main results will be the forward boundedness and backwards non-boundedness of the scattering map on the original Oppenheimer–Snyder space-time [39], and then the subsequent generalisation of this to RNOS models. These results will be achieved primarily by using vector field methods: by considering different energy currents and how they interact with the collapsing dust cloud, we will show that solutions of the linear wave equation have bounded energy when going from past null infinity up until a spacelike hypersurface which intersects the point of collapse of the dust cloud. Previous works allow us to extend this result to one on the whole spacetime. The final chapter of this thesis will apply the above results to the calculation first considered by Stephen Hawking [27, 28], in order to obtain the rate of radiation emitted by collapsing black holes. This result will further make use of some high frequency approximations and also an r ∗p weighted energy estimate. In particular, we will prove that for late times, the radiation given off by any RNOS model approaches its predicted Hawking radiation limit, that of a black body of fixed temperature. We will also prove a bound on the rate at which this limit is approached.
Mathematics, Differential Geometry, Partial Differential Equations, General Relativity
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