Redshift of a photon emitted along the black hole horizon

TL;DR

This paper analyzes how the frequency of photons emitted along black hole horizons changes for observers crossing the horizon, revealing conditions for redshift, blueshift, or no shift, with explicit formulas in various coordinates.

Contribution

It generalizes previous results by deriving explicit formulas for photon frequency shifts along black hole horizons in different coordinate systems.

Findings

Redshift occurs at nonextremal outer horizons.

Blueshift occurs at inner horizons like Reissner-Nordström.

No frequency change at extremal horizons.

Abstract

In this work we derive some general features of the redshift measured by radially moving observers in the black hole background. Let observer 1 cross the black hole horizon emitting of a photon while observer 2 crossing the same horizon later receives it. We show that if (i) the horizon is the outer one (event horizon) and (ii) it is nonextremal, received frequency is redshifted. This generalizes previous recent results in literature. For the inner horizon (like in the Reissner-Nordstr\"{o}m metric) the frequency is blueshifted. If the horizon is extremal, the frequency does not change. We derive explicit formulas describing the frequency shift in generalized Kruskal- and Lemaitre-like coordinates.