Analytical solutions of CPT-odd Maxwell equations in Schwarzschild spacetime

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.