Analytical solutions of CPT-odd Maxwell equations in Schwarzschild spacetime

TL;DR

This paper derives semi-analytical solutions for CPT-violating Maxwell equations in Schwarzschild spacetime using the Newman-Penrose formalism, revealing how Lorentz violation modifies electromagnetic fields in curved spacetime.

Contribution

It provides the first semi-analytical perturbative solutions for CPT-odd Maxwell equations in Schwarzschild spacetime, incorporating spherical symmetry and hypergeometric functions.

Findings

Solutions expressed in hypergeometric functions

Angular parts described by spin-weighted spherical harmonics

Perturbative approach with linear CPT-odd terms

Abstract

In this work, we present the CPT-violating (CPTV) Maxwell equations in curved spacetime using the Newman-Penrose (NP) formalism. We obtain a semi-analytical solution to the Maxwell equations in Schwarzschild spacetime under the assumption that the CPT-odd $\left(k_{AF}\right)^\mu$ term exhibits spherical symmetry in the Schwarzschild background. Retaining only terms up to linear order in the $\left(k_{AF}\right)^\mu$ coefficient, we obtain perturbative solutions by treating the solutions of the Lorentz-invariant Maxwell equations as the zeroth-order approximation and incorporating the $\left(k_{AF}\right)^\mu$ terms as an additional source term alongside the external charge current. Each resulting NP scalar field can be factorized into two components: the radial component is expressed in terms of hypergeometric functions, while the angular component is described by spin-weighted spherical harmonics.