A nonlinear theory of distributional geometry

TL;DR

This paper develops a nonlinear, diffeomorphism-invariant theory of generalized tensor fields on manifolds, extending distributional geometry to include nonsmooth metrics and curvature, with applications to Einstein's equations.

Contribution

It introduces a nonlinear, diffeomorphism-invariant framework for generalized tensor fields, extending distributional geometry to nonsmooth metrics and curvature in general relativity.

Findings

Generalized Lie derivative commutes with distribution embedding.

Embedding of continuous metrics yields well-defined generalized metrics.

Solutions to Einstein's equations can be embedded with preserved properties.

Abstract

This paper builds on the theory of generalised functions begun in [1]. The Colombeau theory of generalised scalar fields on manifolds is extended to a nonlinear theory of generalised tensor fields which is diffeomorphism invariant and has the sheaf property. The generalised Lie derivative for generalised tensor fields is introduced and it is shown that this commutes with the embedding of distributional tensor fields. It is also shown that the covariant derivative of generalised tensor fields commutes with the embedding at the level of association. The concept of generalised metric is introduced and used to develop a nonsmooth theory of differential geometry. It is shown that the embedding of a continuous metric results in a generalised metric with well defined connection and curvature. It is also shown that a twice continuously differentiable metric which is a solution of the vacuum Einstein equations may be embedded into the algebra of generalised tensor fields and has generalised Ricci curvature associated to zero. Thus, the embedding preserves the Einstein equations at the level of association. Finally, we consider an example of a metric which lies outside the Geroch-Traschen class and show that in our diffeomorphism invariant theory the curvature of a cone is associated to a delta function.