Deformation of contour and Hawking temperature

TL;DR

This paper investigates how contour deformation in tunneling calculations affects the Hawking temperature of black holes, showing that proper deformation yields the correct temperature even in coordinate systems where previous methods failed.

Contribution

It demonstrates that the Hawking temperature can be accurately obtained through contour deformation corresponding to coordinate transformations, resolving discrepancies in tunneling approaches.

Findings

Correct Hawking temperature obtained via contour deformation

Contour deformation linked to radial coordinate transformations

Addresses previous factor-of-two discrepancy in tunneling calculations

Abstract

It was found that, in an isotropic coordinate system, the tunneling approach brings a factor of 1/2 for the Hawking temperature of a Schwarzschild black hole. In this paper, we address this kind of problem by studying the relation between the Hawking temperature and the deformation of integral contour for the scalar and Dirac particles tunneling. We find that correct Hawking temperature can be obtained exactly as long as the integral contour deformed corresponding to the radial coordinate transform if the transformation is a non-regular or zero function at the event horizon.