Distributional metrics and the action principle of Einstein–Hilbert gravity

Abstract

In this work, a subclass of the generalized Kerr-Schild class of spacetimes is specified, with respect to which the Ricci tensor (regardless of the position of indices) proves to be linear in the so-called profile function of the geometry. Considering Colombeau's nonlinear theory of generalized functions, this result is extended to apply to an associated class of distributional Kerr-Schild geometries, and then used to formulate a variational principle for these singular spacetimes. More specifically, it is shown in this regard that a variation of a suitably regularized Einstein-Hilbert action can be performed even if the metric of one of the corresponding generalized Kerr-Schild representatives contains a generalized delta function that converges in a suitable limit to a delta distribution.