Hawking Radiation

The following sections are included:

• Dedication
• Contents
• Introduction

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.

Here we summarize and discuss the famous paper of S. Hawking [Hawking (1975)]. See of course also [Hawking (1974)]. We will use his notations…

In this chapter we explore various aspects of thermality of Hawking radiation. We first take into account a masterpiece by Hartle and Hawking on black hole radiation [Hartle and Hawking (1976)]. Therein, thermality of black holes is shown to occur by using path integral techniques. A fundamental role is played by the fact that amplitudes for particle emission are linked with amplitudes for particle absorption. Both processes are actually involved in pair creation near the black hole horizon. On the grounds of analyticity properties of the propagator, which enters the probability calculations, and in which is implicit the selection of the quantum state for quantum fields on a black hole background, it is found that probability for particle absorption and for particle emission are related in such a way that particles perceived by a static observer at fixed radius r far from the horizon share a blackbody spectrum at the Hawking temperature. There appear both technical and physical relevant arguments, as we show below…

Since the original work by Hawking, the fundamental mechanism leading to Hawking radiation is represented by quantum mechanical tunneling of a particle through the black hole horizon [Hawking (1975)]. In particular, in the geometrical optics approximation, which is justified by the fact that near the horizon particle energies are severely boosted, because of an hard blueshift, a classically forbidden tunneling of a particle from inside the horizon to outside can occur, according to still different mechanisms. Together with other approaches, so-called tunneling methods appeared since the initial period in black hole evaporation calculations. Herein, we mean to delve into this very intuitive framework, in which the original suggestion on the pair creation mechanism is implemented. Of course, the present approach is strictly related to the Hartle-Hawking approach we have considered in Chapter 3. Still, we stress that the tunneling method is very important in its own, because it represents a simple tool to show that the Hawking effect is actually present if a non-degenerate horizon is at hand in a stationary black hole metric. All the main features can be reproduced, at a very lower cost, both from a conceptual point of view and also from a computational one. Furthermore, it also provides us an useful tool for generalizing the Hawking effect to situations where backreaction effects are taken into account, giving rise to non-strictly thermal features…

In this chapter, we describe the somewhat unexpected relation between Hawking radiation and quantum anomalies. The history of this subject begins at the early days of Hawking’s discovery, indeed in a seminal paper [Christensen and Fulling (1977)] it was shown that the conformal anomaly can be used in order to infer information about the renormalized quantum stress-energy tensor through its covariant conservation equation. In particular, in 2D, the quantum stress-energy tensor is completely determined by the anomaly, up to boundary conditions to be imposed in order to find out integration constants. This route inspired several works and developments, still it seems to be not at the roots of the phenomenon, because even in 2D we must impose suitable boundary conditions and it is specialized only to the case of conformally invariant couplings (which, of course, do not correspond to the general case)…

It is known that, for static black holes with bifurcate horizon, and temperature T_H, it is necessary to impose a periodicity β = β_H in the imaginary time in order to avoid distributional curvature singularities (β_H = ħc/(k_BT_H)). We reconsider this argument, and show that the Hawking temperature T_H can be found as a consequence of the requirement of extendability of the Euclidean section of the metric up to including the horizon. The extremal case is shown to be special…

With rigorous methods, we do not obviously intend to present the Hawking evaporation effect as a result of a general theorem stated in a rigorous mathematical setting, but only for very particular situations. If one considers the Schwarzschild black hole manifold as a solution of the vacuum Einstein equations, then a semi rigorous result can be obtained. With semi rigorous we mean that, first of all, the mathematical conditions allowing for the exact statements don’t take into account certain important physical conditions, like the Sachs-Bondi energy conservation. The final state of the black hole then appear as a stationary infinite flux of energy from the horizon. It is clear that such a result is unphysical, and a backreaction on the metric should be included. Moreover, the high level of symmetries required in such deductions are not realizable in a physical situation. However, this problems affects also the deductions performed in a less rigorous way. The main motivation for looking for rigorous deductions is due to the fact that the original computations by Hawking, and many other ones, make use of approximations which are uncontrollable, like, for example, the rapid variation of the refractive index during the last stage of the collapse, which is heuristically neglected by arguing that the very strong blueshift makes matter essentially transparent, but which is mathematically out of control, or how much the details of the collapse influence the final answer. The point is thus that a rigorous formulation, even though not yet including all physical conditions and regards quite specific situations, would show that the Hawking effect is essentially independent from the details of the collapse then showing the robustness of such effect…

We have seen that the Hawking phenomenon is quite robust from a theoretical point of view, in the sense that if does not exist a complete incontrovertible exact proof of the Hawking effect, able to take into account the back reaction and eventual quantum gravity effects, there are several ways for analyzing it, all leading to the same result (for a quasi stationary black hole): the black hole must radiate away field modes (particles) at a temperature determined by the surface gravity к. If this strongly motivates for thinking to Hawking radiation as a physical phenomenon, it is obviously not enough, since this only one half of the medal. To be really considered a physical phenomenon it has to be verified experimentally. However, the task of measuring the Hawking phenomenon via astrophysical observations is essentially hopeless: a black hole with mass M is expected to emit like a black body with a temperature

The aim of this chapter is to show how event horizons can appear in certain nonlinear optical systems and how this can be employed for measuring analogue Hawking radiation. The simplest way in order to understand such a phenomenon is to suppose that the light propagates in a very weakly dispersive medium in such a way that dispersion can be ignored at least at a first stage. However, in optics, dispersion is always relevant and almost non dispersive regimes can be eventually observed only at determinate frequency ranges for a given material, so that it is practically impossible to ignore dispersion effects. For this reason, despite the title, we will not ignore dispersion in the present chapter, but we will include it generically in a naive way, t.i. by putting it at hand in the equations previously deduced for an hypothetical dispersionless material, demanding a full treatment of the Hawking radiation in dispersive dielectrics to the next chapter. Surely, this procedure can be expected to be too much ingenuous, since one could expect that the presence of nontrivial dispersion relations may affect the physical processes at the basis of the investigated phenomenon, a problem that is common to all analogue gravity systems. For example, as we will see, the same concept of horizon will be not so clear in a dispersive dielectric…

In this chapter it will be presented a theoretical framework for studying Hawking radiation in homogeneous linearly dispersive media, where the dispersion is included mesoscopically from the beginning, from which the Sellmeier relations are deduced as a consequence…

This chapter is dedicated to experiments involved with the revelation of analogue Hawking radiation in labs. We start from the Como experiment, where a dielectric perturbation created by means of a intense laser pulse in fused silica plays the role of white hole in the comoving frame. Then we discuss the Vancouver experiment, in which analogue Hawking radiation is measured in water by means of induced waves which hit a suitably designed obstacle. Finally, a discussion will be devoted to the Technion experiment involving a system which is associated with a so-called black hole laser.

The following sections are included:

• Appendix A Algebraic methods in QFT
  • Araki-Haag-Kastler algebraic formulation of QFT
  • Haag-Hugenholz-Winnik formulation of quantum statistical systems
    • Structure of the statistical system
    • The Gibbs states
    • KMS condition and the infinite volume limit
    • The thermal representations
  • The Tomita-Takesaki theorem
    • Polar decomposition
    • Some simple facts about Von Neumann algebras
    • The Tomita-Takesaki theorem and the KMS condition
• Bibliography
• Index