Exact solution to the Einstein-Maxwell field equations for a charge and mass distribution with a quadrupole moment

Abstract

The method introduced by Erez and Rosen to obtain exact multipole-type solutions to the static axially symmetric Einstein field equations is generalized to the Einstein-Maxwell system for the case of a static axially symmetric distribution of mass and charge. This is accomplished by defining an auxiliary function which, when assumed to be related to the electrostatic potential through certain derivative conditions, reduces the Einstein-Maxwell equations for this case to the Weyl form of the pure gravitational field equations. A general expression is then given for the electrostatic potential in terms of the auxiliary function. We interpret the solutions, which contain the Reissner-Nordstr\"om solution as a special case, as fields produced by mass and charge distributions possessing multipole structure, and we give the explicit form of the metric for the monopole plus quadrupole case.