The prolate electrically magnetized Schwarzschild solution with applications to ultra-relativistic magnetars

Abstract

We start our study by writing down the Klein–Gordon equation for charged bosons evolving in magnetars described by spherically prolate metrics. For massless particles, the variables can be separated and the radial equation can be solved for physically interesting cases of metric functions. For the interior Schwarzschild solution immersed in a magnetic universe, the corresponding wave function is expressed in terms of the general Heun functions. For the outer region of the magnetar, one has to consider besides the magnetic field, an electric intensity. A general relation between the metric functions is derived. For $g_{00} g_{11} =1$, this is leading to the Schwarzschild exterior electro-magnetized metric and to confluent Heun functions, as solutions to the corresponding Klein–Gordon equation.