Distributional metrics and the action principle of Einstein–Hilbert gravity

Abstract

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.