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.

arXiv comment22 Pages, no figures

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Cited by in corpus (3)

• Junction Conditions and local Spacetimes in General Relativity
• The gravitational Field of a massless Particle on the Horizon of a stationary Black Hole
• General Shells and Generalized Functions