Thermal Hawking radiation of black hole with supertranslation field

TL;DR

This paper analyzes how supertranslation fields affect Hawking radiation, showing that near the horizon, the radiation remains thermally identical to the Schwarzschild case despite metric modifications.

Contribution

It provides a detailed perturbative and non-perturbative analysis of eigenmodes and geodesics in supertranslated black hole metrics, demonstrating the invariance of Hawking radiation.

Findings

Eigenmodes differ from Schwarzschild by terms vanishing at the horizon.

Null geodesics near the horizon behave as in Schwarzschild.

Thermal radiation density matrices are unchanged by supertranslation fields.

Abstract

Using the analytical solution for the Schwarzschild metric containing supertranslation field, we consider two main ingredients of calculation of the thermal Hawking black hole radiation: solution for eigenmodes of the d'Alambertian and solution of the geodesic equations for null geodesics. For calculation of Hawking radiation it is essential to determine the behavior of both the eigenmodes and geodesics in the vicinity of horizon. The equation for the eigenmodes is solved, first, perturbatively in the ratio $O(C)/M$ of the supertranslation field to the mass of black hole, and, next, non-perturbatively in the near- horizon region. It is shown that in any order of perturbation theory solution for the eigenmodes in the metric containing supertranslation field differs from solution in the pure Schwarzschild metric by terms of order $L^{1/2}= (1-2M/r)^{1/2}$. In the non-perturbative approach, solution for the eigenmodes differs from solution in the Schwarzschild metric by terms of order $L^{1/2}$ which vanish on horizon. Using the simplified form of geodesic equations in vicinity of horizon, it is shown that in vicinity of horizon the null geodesics have the same behavior as in the Schwarzschild metric. As a result, the density matrices of thermal radiation in both cases are the same.