Charged massless scalar fields in a charged $C$-metric black hole: Exact solutions, Hawking radiation, and scattering of scalar waves

TL;DR

The article develops a rigorous analytic framework for charged massless scalar fields in a charged $C$-metric black hole by transforming the radial and angular equations of the conformally invariant Klein–Gordon equation into the general Heun equation. It derives exact local Heun-function solutions and connection coefficients, enabling precise asymptotics, Hawking-radiation spectra, and boundary-condition analyses for quasinormal modes and superradiance. The work connects horizon-crossing behavior to explicit expressions for scattering coefficients and confirms consistency with Damour–Ruffini–Sannan results, while providing a path to high-precision QNM calculations and potential extensions to other fields. Overall, the paper offers a comprehensive analytic treatment of wave propagation, radiation, and stability phenomena in accelerating charged black-hole spacetimes, with broad applicability to Heun-function methods in gravitational backgrounds.

Abstract

We study Hawking radiation and wave scattering of charged scalar fields in a charged $C$-metric black hole background. The conformally invariant wave equation for charged scalar fields can be separated into radial and angular parts, each with five singularities. We first show that the radial and angular equations can be respectively transformed into the general Heun equation, and then we explore exact solutions of the radial Heun equation in terms of the local Heun functions and connection coefficients. Exact behaviors of the asymptotic wave functions are determined without approximations. We further apply the exact results to derive Hawking radiation, quasinormal modes and superradiance.