Hawking Flux

The aim of this project was to clarify open points in a particular approach for calculating Hawking radiation. The basis of this approach is a two-dimensional dilaton model, describing a four-dimensional Black Hole spactime, which is obtained by integrating out the superfluous angular variables. Beside the common scalar curvature it contains an additional scalar field, the dilaton field, describing the curvature of a sphere (surrounding the Black Hole) and its embedding into 4D spacetime. The Hawking radiation is calculated for a massless scalar field, being used in place of the more complicated photons, whereby quantum mechanical expectation values like energy and momentum are derived directly from the generator of Green functions (called effective action in this context), i.e. by a path integral over all scalar field states. In the preliminary and first phase of this project, the effective action for this particular dilaton model could be established by means of the Covariant Perturbation Theory, a method developped in the eighties by the Russian physicists A. O. Barvinsky and G. A. Vilkovisky. Although the results were very promising, reproducing the correct Hawking flux in the regular sector, the effective action was plagued by infrared divergences, caused by the masslessness of the involved scalar particles. The main part of the present project was to give a precise form to these infrared divergences by a mass-term regularisation and eventually find a method of renormalisation. The infrared divergent terms could be calculated exactly up to the second, and (at least partly) to third order of covariant perturbation theory. All terms contained increasing powers in the endomorphism, some curvature potential formed by the dilaton field, but no scalar curvature. In particular, we could show that to second order the dependence on the regularisation mass was given by a non-trivial function which could be determined exactly. In the limit of vanishing mass this function became a logarithm, whereby a logarithm of the wave-operator appeared as a by-product. The separation of the infrared divergence thus required an analysis of this ill-defined term which was realised by an expansion in terms of eigenfunctions. This step was non-trivial since the eigenfunctions of the wave-operator cannot be given in a closed form for a field on Black Hole spacetime. However, the regularity of the whole expression at the critical points could be shown, making a separation of the infrared divergence possible. Considering the divergences of the first three orders of perturbation theory, a summation scheme for the whole series could be conjectured, resulting in a single, non-local term including a logarithmic divergence in the mass. The coupling of a classical source-term of the scalar field to the effective action produced an ambiguous term of the same structure as the (conjectured) infrared divergence. Accordingly, a complete infrared renormalisation of the effective action could be achieved by simply adjusting this ambiguity.