A note on Hawking radiation via complex path analysis

TL;DR

This paper demonstrates that, in semiclassical analysis, the Hawking radiation emission probability is independent of particle species and applies to arbitrary stationary spacetimes across various field types, confirming a universal feature of black hole radiation.

Contribution

It shows that equations of motion for different spin fields reduce to Klein-Gordon equations in the semiclassical limit, leading to a universal expression for emission probability across all particle species.

Findings

Emission probability is identical for scalar, spinor, vector, and higher spin fields.

The derived expression is coordinate independent and valid for arbitrary stationary spacetimes.

Explicit examples confirm the universality of Hawking radiation emission probability.

Abstract

As long as we neglect backreaction, the Hawking temperature of a given black hole would not depend upon the parameters of the particle species we are considering. In the semiclassical complex path analysis approach of Hawking radiation, this has been verified by taking scalar and Dirac spinors separately for different stationary spacetime metrics. Here we show, in a coordinate independent way that, for an arbitrary spacetime with any number of dimensions, the equations of motion for a Dirac spinor, a vector, spin-$2$ and spin-$\frac{3}{2}$ fields reduce to Klein-Gordon equations in the WKB semiclassical limit. We then obtain, under some suitable assumptions, the complex solutions of those resulting scalar equations across the Killing horizon of a stationary spacetime to get a coordinate independent expression for the emission probability identical for all particle species. Finally we consider some explicit examples to demonstrate the validity of that expression.