Anomalies, effective action and Hawking temperatures of a Schwarzschild black hole in the isotropic coordinates

TL;DR

This paper investigates Hawking radiation of a Schwarzschild black hole in isotropic coordinates using anomaly cancellation, revealing effects of different effective metrics on the anomaly analysis near the horizon.

Contribution

It analyzes the impact of different two-dimensional effective metrics derived from the isotropic Schwarzschild metric on Hawking radiation and anomaly cancellation.

Findings

Effective metrics can be conformally equivalent or inequivalent.

The determinant of the effective metric may vanish at the horizon.

The anomaly analysis may be invalid for metrics with non-unity determinant.

Abstract

Motivated by the universality of Hawking radiation and that of the anomaly cancellation technique as well as that of the effective action method, we investigate the Hawking radiation of a Schwarzschild black hole in the isotropic coordinates via the cancellation of gravitational anomaly. After performing a dimensional reduction from the four-dimensional isotropic Schwarzschild metric, we show that this reduction procedure will, in general, result in two classes of two-dimensional effective metrics: the conformal equivalent and the inequivalent ones. For the physically equivalent class, the two-dimensional effective metric displays such a distinct feature that the determinant is not equal to the unity ($\sqrt{-g} \neq 1$), but also vanishes at the horizon, the latter of which possibly invalidates the anomaly analysis there. ... This is an updated version to replace our e-print arXiv:0709.0044 [hep-th]. Abstract is too long to exceed the limit of 24 lines by arXiv.