Hawking radiation inside a charged black hole

TL;DR

This paper investigates Hawking radiation within a charged black hole, revealing how the radiation spectrum and effective temperature vary near the inner and outer horizons, with implications for black hole thermodynamics.

Contribution

It introduces two methods to analyze Hawking radiation in Reissner-Nordström spacetime and details the behavior of the Hawking spectrum and temperature near horizons, including negative and divergent temperatures.

Findings

Effective temperature is finite and negative at the outer horizon when (Q/M)^2 > 8/9.

At the inner horizon, the temperature is always negative and diverges.

The spectrum becomes ultraviolet-divergent at the inner horizon.

Abstract

Here we analyze the Hawking radiation detected by an inertial observer in an arbitrary position in a Reissner-Nordstr\"om spacetime, with special emphasis on the asymptotic behavior of the Hawking spectrum as an observer approaches the inner or outer horizon. Two different methods are used to analyze the Hawking flux: first, we calculate an effective temperature quantifying the rate of exponential redshift experienced by an observer from an emitter's vacuum modes, which reproduces the Hawking effect provided the redshift is sufficiently adiabatic. Second, we compute the full Bogoliubov graybody spectrum observed in the three regimes where the wave equation can be solved analytically (at infinity and at the outer and inner horizons). We find that for an observer at the event horizon, the effective Hawking temperature is finite and becomes negative when $(Q/M)^2>8/9$, while at the inner horizon, the effective temperature is always negative and infinite in every direction the observer looks, coinciding with an ultraviolet-divergent spectrum.