Observer Dependent Horizon Temperatures: a Coordinate-Free Formulation of Hawking Radiation as Tunneling

TL;DR

This paper presents a coordinate-independent reformulation of the tunneling method for Hawking radiation, emphasizing observer-dependent horizon temperatures and clarifying the role of preferred frames in static, spherically symmetric spacetimes.

Contribution

It introduces a coordinate-free approach to tunneling calculations that explicitly incorporates preferred observer frames, enhancing understanding of horizon temperatures in various spacetimes.

Findings

Preferred static observers measure the lowest horizon temperatures.

The formalism reproduces known temperatures for Schwarzschild spacetime.

It clarifies the relation between constants and observer frames in temperature calculations.

Abstract

We reformulate the Hamilton-Jacobi tunneling method for calculating Hawking radiation in static, spherically-symmetric spacetimes by explicitly incorporating a preferred family of frames. These frames correspond to a family of observers tied to a locally static timelike Killing vector of the spacetime. This formulation separates the role of the coordinates from the choice of vacuum and thus provides a coordinate-independent formulation of the tunneling method. In addition, it clarifies the nature of certain constants and their relation to these preferred observers in the calculation of horizon temperatures. We first use this formalism to obtain the expected temperature for a static observer at finite radius in the Schwarzschild spacetime. We then apply this formalism to the Schwarzschild-de Sitter spacetime, where there is no static observer with 4-velocity equal to the static timelike Killing vector. It is shown that a preferred static observer, one whose trajectory is geodesic, measures the lowest temperature from each horizon. Furthermore, this observer measures horizon temperatures corresponding to the well-known Bousso-Hawking normalization.