One-loop corrections for a Schwarzschild black hole via 2D dilaton gravity

Abstract

We study quantum corrections for the Schwarzshild black hole by considering it as a vacuum solution of a 2D dilaton gravity theory obtained by spherical reduction of 4D gravity coupled with matter. We find perturbatively the vacuum solution for the standard one-loop effective action in the case of null-dust matter and in the case of minimally coupled scalar field. The corresponding state is in both cases 2D Hartle-Hawking vacuum, and we evaluate the corresponding quantum corrections for the thermodynamic parameters of the black hole. We also find that the standard effective action does not allow boundary conditions corresponding to a 4D Hartle-Hawking vacuum state.

Hawking radiation by effective two-dimensional theories

“…So we have firm evidence that the 2D stress tensor T ab constructed from S aind in the |H state is in strong qualitative disagreement with the well established result of the 4D H|T µν |H which, we remind, describes a black hole in thermal equilibrium with a positive energy bath of radiation at the temperature T H . One can indeed find an equilibrium state |x ± which is regular on the horizons and unlike |H has a positive flux at infinity [27]. Mathematically this is done by fine tuning two constants.…”

Dilaton gravity in two dimensions

“…This suggests to perform the q i -integrals first, yielding (functional) δ-functions which, in turn, may be used to get rid of the p i -integrals afterwards 64 . The vanishing arguments of the three δ-functions for the respective (q 1 , q 2 , q 3 )-integrals yield three differential equations (h =ūu + v 2 from (7.36))…”

“…It was proposed in ref. [64] to convert (6.42) into a local action by introducing two auxiliary fields, f 1 = (1/2)R and f 2 = (1/2)(∇Φ) 2 . Various versions of this method were frequently used since (e.g.…”

Quantum corrections for a Bañados-Teitelboim-Zanelli black hole via the 2D reduced model

“…It is easier to use the local form of the action (12); it can be obtained by a suitable introduction of auxiliary fields [8,5]. The local form, however, differs in the cases we are going to discuss.…”

“…This problem is of great heuristic importance, as dimensional reduction is repeatedly done in different scenarios of string and brane theories, although the mechanism is fully understood only at the classical level. The study of dimensional reduction from four to two dimension in the case of Schwarzchild black hole was done previously [6,7,8,9]. There are also some new ideas in the literature, as dimensional-reduction anomaly [10,11].…”