Hawking Radiation

From Astrophysical Black Holes to Analogous Systems in Lab

Francesco D Belgiorno, Sergio L Cacciatori, Daniele Faccio

• Publisher: WSPC
• Year: 2018
• eBook ISBN:
  9789814508551
• Topic: Physical Sciences
• Subtopic: Astronomy & Astrophysics

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The aim of this book is to provide the reader with a guide to Hawking radiation through a dual approach to the problem. After an introductory chapter containing some basic knowledge about black holes and quantum field theory in curved spacetime, the first part of the book consists in a survey of methods for deriving and studying Hawking radiation from astrophysical black holes, from the original calculation by S W Hawking to the most recent contributions involving tunneling and gravitational anomalies. In the second part, we introduce analogue gravity, and we focus our attention to dielectric black hole systems, to which the studies of the present authors are devoted. The mutual interchange of knowledge between the aforementioned parts is addressed to render a more comprehensive picture of this very fascinating quantum phenomenon associated with black holes.

--> Contents:

• First Part:
  • A Short Scrapbook on Classical Black Holes
  • The Seminal Paper
  • Thermality of Hawking Radiation: From Hartle-Hawking to Israel and Unruh
  • The Tunneling Approach
  • The Anomaly Route to Hawking Radiation
  • The Euclidean Section and Hawking Temperature
  • Rigorous Aspects of Hawking Radiation
• Second Part:
  • The Roots of Analogue Gravity
  • Hawking Radiation in a Non-Dispersive Nonlinear Kerr Dielectric
  • Hawking Radiation in a Dispersive Kerr Dielectric
  • Hawking Radiation in the Lab
• Appendix:
  • Algebraic Methods in QFT

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--> Readership: Graduate students and professionals studying astrophysics and theoretical physics. -->
Hawking Radiation;Black Holes;Quantum Field Theory;Analogue Gravity;Dielectric Black Hole Systems00

PART 1

First Part

Chapter 1

A short scrapbook on classical black holes

There is not a general definition of a black hole. The intuitive idea is that a black hole appears when there is a trapping region from which nothing can escape, bounded by an horizon, the event horizon. In this chapter we will recall the definition of black holes in a particular class of spacetimes, which are asymptotically flat and with a far future predictable from Cauchy surfaces, as introduced by Hawking and Ellis [Hawking and Ellis (2011)]. Next, we will illustrate the classical “physical” examples of Schwarzschild and Kerr black holes. One could expect that, given the present knowledge of cosmological data, the de Sitter versions of such solutions would be more physical. These can also be considered, but, then, one should include black holes in a eventually closed expanding universe. To our knowledge, no rigorous definition of such a black hole, generalizing the one of Hawking and Ellis, exists in the last class of spacetimes. Thus, in more general situations the intuitive definition of a black hole must be sufficient. In all the explicit known examples of black hole, the event horizon is also a Killing horizon. Since some properties of Hawking radiation (and more general phenomena) are related to this fact, we will end this chapter by defining Killing horizons and illustrating some of their properties.

1.1Mathematical black holes

Following [Hawking and Ellis (2011)], we now will introduce a rigorous mathematical definition of a black hole. The main purpose is to collect some of the mathematical tools which will allow to get some exact results as illustrated in chapter 7. This general definition may be easily extended to the cases in presence of electromagnetic fields end/or a cosmological constant, but not to all possible physically interesting spacetimes, so the reader not interested in mathematically exact results can skip this section.

A D-dimensional spacetime is a D-dimensional smooth manifold M, endowed with a Lorentzian metric g. We will indicate it with a pair (M, g). It is said to be time orientable if it admits a continuous vector field everywhere non-spacelike (i.e. never spacelike) and future directed. In an orientable spacetime, a non-spacelike curve is future (past) directed if its tangent vector is everywhere future (past) directed. Given two subsets A, B ⊂ M, one defines

•the chronological future of A relative to B as the set I⁺(A, B)of all points of B that can be reached from A along future directed timelike curves;

•the chronological past of A relative to B as the set I⁻(A, B) of all points of B that can be reached from A along past directed timelike curves;

•the causal future of A relative to B as the set J⁺(A, B) obtained from the union of A ⋂ B to all points of B that can be reached from A along future directed non-spacelike curves;

•the causal past of A relative to B as the set J⁻(A, B) obtained from the union of A ⋂ B to all points of B that can be reached from A along past directed non-spacelike curves.

In particular, I^±(A):= I^± (A, M) and J^±(A):= J^±(A, M).

A set A is said to be a future set if it properly contains its chronological future, I⁺(A) ⊂ A. A is said to be achronal if A ⋂ I⁺(A) =

. It happens that the boundary of a future set is an achronal (D − 1)-dimensional C¹ manifold. A spacetime is said to satisfy the chronological condition if it does not contain closed time-like curves. It satisfies the causality condition...

Table of contents

• Cover Page
• Title Page
• Copyright
• Contents
• Introduction
• First Part
• Second Part
• Appendix A Algebraic methods in QFT
• Bibliography
• Index