Static Charged-Dust Distributions in General Relativity

The field equations of general relativity with electromagnetic stress tensor and charged incoherent matter, along with Maxwell equations in curved space, are investigated. The metric coefficients and the electromagnetic field tensor are assumed to be independent of $x^0$. It is then shown that, if one assumes a functional relation between $g_{00}$ and the electrostatic potential (which is a consequence of the Bianchi identities), one can then find a solution of the coupled Einstein-Maxweel equations in terms of a solution of the Einstein equations with incoherent matter as source.