Summary Quantization of fields in a black hole background. Vacuum choice. Hawking radiation and black hole evaporation. Thermodynamics of black holes.

Hawking radiation

Black holes are massive objects which have such a strong gravitational field that even light cannot escape from them. According to the classical General Relativity, a black hole can only absorb matter and its size never decreases. In 1974 Hawking considered quantum fields in a classical black hole background and discovered that the black hole emits thermal particles and thus evaporates. This theoretical result came to a certain extent as a surprise. In fact, at that time one thought that particles can be produced only by a nonstatic gravitational field. For example, for a rotating black hole there exist negative-energy states outside its horizon, and therefore the gravitational field can convert a virtual particle-antiparticle pair into a pair of real particles with zero total energy. The positive-energy particle can then escape to infinity, while the negative-energy particle falls into the black hole. In this case the black hole can emit energy. This effect is known as superradiance. On the contrary, a nonrotating black hole has no negative energy states outside its horizon. Therefore, at first glance its mass cannot decrease and hence no particles can be produced from the quantum fluctuations outside the black hole horizon.

This book is an expanded and reorganized version of the lecture notes for a course taught at the Ludwig-Maximilians University, Munich, in the spring semester of 2003. The course is an elementary introduction to the basic concepts of quantum field theory in classical backgrounds. A certain level of familiarity with general relativity and quantum mechanics is required, although many of the necessary concepts are introduced in the text.

The audience consisted of advanced undergraduates and beginning graduate students. There were 11 three-hour lectures. Each lecture was accompanied by exercises that were an integral part of the exposition and encapsulated longer but straightforward calculations or illustrative numerical results. Detailed solutions were given for all the exercises. Exercises marked by an asterisk (*) are more difficult or cumbersome.

The book covers limited but essential material: quantization of free scalar fields; driven and time-dependent harmonic oscillators; mode expansions and Bogolyubov transformations; particle creation by classical backgrounds; quantum scalar fields in de Sitter spacetime and the growth of fluctuations; the Unruh effect; Hawking radiation; the Casimir effect; quantization by path integrals; the energy-momentum tensor for fields; effective action and backreaction; regularization of functional determinants using zeta functions and heat kernels. Topics such as quantization of higher-spin fields or interacting fields in curved spacetime, direct renormalization of the energy-momentum tensor, and the theory of cosmological perturbations are left out.