X iv : g r - qc / 0 60 10 08 v 1 3 J an 2 00 6 Quantum Field Theory in Curved Spacetime ∗

Quantum Field Theory (QFT) in Curved Spacetime is a hybrid approximate theory in which quantum matter fields are assumed to propagate in a fixed classical background gravitational field. Its basic physical prediction is that strong gravitational fields can polarize the vacuum and, when timedependent, lead to pair creation just as a strong and/or time-dependent electromagnetic field can polarize the vacuum and/or give rise to pair-creation of charged particles. One expects it to be a good approximation to full quantum gravity provided the typical frequencies of the gravitational background are very much less than the Planck frequency (c5/G~)1/2 ∼ 1043s−1) and provided, with a suitable measure for energy, the energy of created particles is very much less than the energy of the background gravitational field or of its matter sources. Undoubtedly the most important prediction of the theory is the Hawking effect, according to which a, say spherically symmetric, classical black hole of mass M will emit thermal radiation at the Hawking temperature T = (8πM)−1 (here and from now on, we use Planck units where G, c, ~ and k [Boltzmann’s constant] are all taken to be 1.) On the mathematical side, the need to formulate the laws and derive the general properties of QFT on non-flat spacetimes forces one to state and prove results in local terms and, as a byproduct, thereby leads to an improved perspective on flat-space-time QFT too. It’s also interesting to formulate QFT on idealized spacetimes with particular global geometrical features. Thus, QFT on spacetimes with bifurcate Killing horizons is intimately related to the Hawking effect; QFT on spacetimes with closed timelike curves is intimately related to the question whether the laws of physics permit the manufacture of a time-machine. As is standard in General Relativity, a curved spacetime is modelled mathematically as a (paracompact, Hausdorff) manifold M equipped with