Hawking effect in an extremal Kerr black hole spacetime

TL;DR

This paper investigates the Hawking effect in extremal Kerr black holes, demonstrating that the complete near-null coordinate relationship, including a logarithmic term, confirms the absence of Hawking radiation, with the inverse term being crucial.

Contribution

It shows that the full relationship between near-null coordinates, including logarithmic terms, supports the vanishing Hawking radiation in extremal Kerr black holes within the canonical formulation.

Findings

Complete coordinate relation confirms zero Hawking radiation.

Logarithmic term does not induce Hawking radiation in extremal case.

Inverse term is key to understanding the vanishing number density.

Abstract

It is well known that extremal black holes do not Hawking radiate, which is usually realized by taking an extremal limit from the nonextremal case. However, one cannot perceive the same phenomenon using the Bogoliubov transformation method starting from an extremal black hole itself, i.e., without the limiting case consideration. In that case, the Bogoliubov coefficients do not satisfy the required normalization condition. In canonical formulation, which closely mimics the Bogoliubov transformation method, one can consistently reproduce the vanishing number density of Hawking quanta for an extremal Kerr black hole. In this method, the relation between the spatial near-null coordinates, imperative in understanding the Hawking effect, was approximated into a sum of linear and inverse terms only. In the present work, we first show that one can reach the same conclusion in canonical formulation even with the complete relationship between the near-null coordinates, which contains an additional logarithmic term. It is worth mentioning that in the nonextremal case, a similar logarithmic term alone leads to the thermal Hawking radiation. Secondly, we study the case with only the inverse term in the relation (i.e., when the spatial near-null coordinates associated to the past and future observers are inversely related to each other) to understand whether it is the main contributing term in vanishing number density. Third, for a qualitative realization, we consider a simple thought experiment to understand the corresponding Hawking temperature and conclude that the inverse term indeed plays a crucial role in the vanishing number density.