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A Rigorous Study of Hawking Radiation on Collapsing Charged Spherically Symmetric Spacetimes

Frederick Alford

Abstract

In this paper, we give a rigorous mathematical treatment of the late time Hawking radiation of massless bosons emitted by a class of collapsing, spherically symmetric, charged models of black hole formation, including both extremal and sub-extremal black holes. We will also prove a bound on the rate at which the radiation emitted approaches this late time limit. This includes an integrable decay rate of radiation emitted by extremal black holes, for which the late time limit vanishes. Thus, we show that the total expected quantity of any massless boson emitted by an extremal \RNS black hole is finite.

A Rigorous Study of Hawking Radiation on Collapsing Charged Spherically Symmetric Spacetimes

Abstract

In this paper, we give a rigorous mathematical treatment of the late time Hawking radiation of massless bosons emitted by a class of collapsing, spherically symmetric, charged models of black hole formation, including both extremal and sub-extremal black holes. We will also prove a bound on the rate at which the radiation emitted approaches this late time limit. This includes an integrable decay rate of radiation emitted by extremal black holes, for which the late time limit vanishes. Thus, we show that the total expected quantity of any massless boson emitted by an extremal \RNS black hole is finite.
Paper Structure (25 sections, 28 theorems, 206 equations, 7 figures)

This paper contains 25 sections, 28 theorems, 206 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Previous Work
  4. Mathematical Works
  5. Selected Works Within the Physics Literature
  6. Further Remarks
  7. RNOS Models
  8. Reissner--Nordström Spacetimes
  9. RNOS Definition
  10. Double Null Coordinates
  11. Killing Fields
  12. Notation
  13. Previous Results on Scalar Waves in Pure Reissner--Nordström Spacetimes
  14. Scattering Results on Pure Reissner--Nordström
  15. Further Properties of Reissner--Nordström
  16. ...and 10 more sections

Key Result

Theorem 1

Let $\alpha:[-1,1]\times\mathbb{R}\to\mathbb{R}$ be the smooth function given by: where $\kappa$ is the surface gravity of the Reissner--Nordström black hole, i.e. Let $\psi_+(u, \theta, \varphi)$ be a Schwartz function on the $\mathbb{R}\times S^2$, with $\hat{\psi}_+$ only supported on positive frequencies ($[0,\infty)\times S^2$). Fix $\mathcal{M}$ an RNOS spacetime (see Section Sec:RNOS) wit

Figures (7)

  • Figure 1: Penrose Diagram of RNOS Model, with null hyper surface $\Sigma_{v}$.
  • Figure 2: Penrose diagram of pure Reissner--Nordström spacetimes
  • Figure 3: Penrose Diagram of RNOS Model, with various hyper surfaces labelled.
  • Figure 4: The Domain of Dependence
  • Figure 5: The set-up for the Hawking radiation calculation
  • ...and 2 more figures

Theorems & Definitions (63)

  • Theorem 1: Late Time Emission of Hawking Radiation
  • Remark 1.0.1: The Reflective Boundary Condition
  • Remark 3.1.1: Inclusion of $\mathcal{H}^-$
  • Definition 3.1: RNOS Manifold
  • Theorem 5.1.1: Existence of Scattering Solutions in pure Reissner--Nordström
  • proof : Sketch of Proof
  • Proposition 5.1.2: Boundedness and Decay of Reflection and Transmission Coefficients
  • proof
  • Proposition 5.2.1: $T$-energy Conservation
  • proof
  • ...and 53 more