Initial Value Problem of the Einstein-Maxwell Field

Abstract

On an initial hypersurface, ${x}^{0}=0$, in the presence of gravitation and source-free electromagnetism one can specify the metric tensor, ${g}_{\ensuremath{\mu}\ensuremath{\nu}}$, and its partial derivatives, ${g}_{\ensuremath{\mu}\ensuremath{\nu},0}$, as well as the electromagnetic tensor, ${f}_{\ensuremath{\mu}\ensuremath{\nu}}$. These quantities must be specified so that on the initial hypersurface two of Maxwell's equations are satisfied and so that the components ${{R}_{\ensuremath{\alpha}}}^{0}$ of the Ricci tensor are proportional to the components ${{T}_{\ensuremath{\alpha}}}^{0}$ of the electromagnetic energy-momentum tensor. It is sometimes possible to specify a different electromagnetic tensor on the initial hypersurface which together with the old metric and Ricci tensors will describe a properly set initial value problem such that the geometry in advance of the initial hypersurface is different for the different electromagnetic fields. Thus, the Ricci tensor on the initial hypersurface does not always uniquely describe the geometry off the hypersurface in the Einstein-Maxwell theory. The conditions when this nonuniqueness exists are explicitly derived. An initial value problem could be set by specifying ${g}_{\ensuremath{\mu}\ensuremath{\nu}}$ and ${g}_{\ensuremath{\mu}\ensuremath{\nu},0}$ on the initial hypersurface and deriving an appropriate ${f}_{\ensuremath{\mu}\ensuremath{\nu}}$; however, ${g}_{\ensuremath{\mu}\ensuremath{\nu}}$ and ${g}_{\ensuremath{\mu}\ensuremath{\nu},0}$ cannot be arbitrarily specified but are subject to one rather complicated constraint condition on the hypersurface.