Analogue of the Theorem of Birkhoff in a Metric Nonsymmetric Theory of the Gravitational-Electromagnetic Field

Recently [1] we have shown that the solution of the field equations of a metric nonsymmetric theory of gravitation [2I,II] outside a spherically symmetric time-dependent matter distribution is necessarily static. This is the analogue of the theorem of Birkhoff of general relativity (GR). Here we extend the result to the situation in which electromagnetic field is also present, that is, when the time-dependent source is charged . This is the situation for instance for a radially pulsating charged sphere. The proof of the result, which we will call the static theorem, will run along the same lines of GR as presented, for instance, in [3]. The form of the field equations when the electromagnetic is included into the theory was studied in [4] and the solution for a point charged mass was obtained. In what follows we will be making use of the field equations of that paper. The paper is organized as follows. In Sec. II we present the vacuum field equations of the theory and in Sec. III, we prove the static theorem. In Sec. IV we draw our conclusions.