Static spherically symmetric electrovac Brans-Dicke spacetimes

We investigate the stability of static spherically symmetric electrovac BD spacetimes under an electrostatic perturbation by a point charge. The field equations are integrated directly, and we are able to give, for the first time, a solution describing a general static spherically symmetric charged Brans-Dicke (CBD) spacetimes that is reducible to all known BD spacetimes. We find there are nine classes of independent solutions. We are able to give the physical interpretation of the parameters contained within the metric and shed light on not only the CBD spacetimes but the BD spacetimes as well. We investigate the stability of the CBD spacetimes by electrostatically perturbing it with a point charge. By extending a method first introduced by Copson (1928) we are able to convert the partial differential equation on the electrostatic potential generated by the point charge into a solvable ordinary differential equation. In this way we are able to give an exact, closed-form solution for the electrostatic potential generated by a point charge in a CBD spacetime. We introduce a boundary condition, based on Gauss' divergence theorem, that enables us to determine the constants of integration such that the solution is representative of a single charge. Furthermore, we introduce a method by which the CBD metric can be converted from isotropic to Schwarzschild-type coordinates. In Schwarzschild-type coordinates we find that the CBD Class I solution (also referred to as the Brans-Dicke Reissner-Nordstrom or BDRN solution) exhibits an extra S2 singularity in addition to the generalized inner and outer ``horizons". This additional singularity and the behaviour of the electrostatic potential is investigated by graphically representing in isotropic and Schwarzschild-type coordinates the equipotential surfaces generated by the perturbing charge in the four backgrounds (BDRN, Brans-Dicke, Reissner-Nordstrom and Schwarzschild), alongside Copson's perturbed Schwarzschild solution of Copson (1928). We find that all nine classes of the formal, generalized CBD solution are stable under electrostatic perturbations.