Covariant path integrals and black holes

The thermal nature of the propagator in a collapsed black-hole spacetime is shown to follow from the non-trivial topology of the configuration space in tortoise coordinates by using the path integral formalism. The path integral (PI) formalism is useful to calculate propagators in configuration spaces (CS) endowed with a non-trivial topology, such as in curved spacetimes. Moreover, even if the CS topology is trivial, this may not be the case for its Euclidian section, where the PI should always be computed. For example, in an eternal black-hole background or in Rindler spacetime, although the CS itself has a trivial topology in Kruskal or Rindler coordinates, the Euclidian CS has a periodic structure in the Euclidian time-like coordinate. In these spacetimes endowed with an event-horizon (EH), it can be shown that the thermal properties of the propagator follows from the periodic structure of the Euclidian CS [1–4]. In a collapsing black-hole spacetime, however, this periodic structure is missing. One has to find in this case another procedure to obtain the thermal properties of the propagator. It is the purpose of this paper to show that a similar periodic structure may be recovered in tortoise coordinates if one requires that they cover the entire spacetime by allowing them to take complex values. In the resulting complex CS, the EH has a cylinder-like topology. The propagator is then obtained from a PI by adding all the contributions of the classes of 1