Effective Action and Hawking Flux from Covariant Perturbation Theory
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The computation of the radiation flux related to the Hawking temperature of a Schwarzschild Black Hole or another geometric background is still well-known to be fraught with a number of delicate problems. In spherical reduction, as shown by one of the present authors (W. K.) with D.V. Vassilevich, the correct black body radiation follows when two ``basic components'' (conformal anomaly and a ``dilaton'' anomaly) are used as input in the integrated energy-momentum conservation equation. The main new element in the present work is the use of a quite different method, the covariant perturbation theory of Barvinsky and Vilkovisky, to establish directly the full effective action which determines these basic components. In the derivation of W. K. and D.V. Vassilevich the computation of the dilaton anomaly implied one potentially doubtful intermediate step which can be avoided here. Moreover, the present approach also is sensitive to IR (renormalisation) effects. We realize that the effective action naturally leads to expectation values in the Boulware vacuum which, making use of the conservation equation, suffice for the computation of the Hawking flux in other quantum states, in particular for the relevant Unruh state. Thus, a rather comprehensive discussion of the effects of (UV and IR) renormalisation upon radiation flux and energy density is possible.
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