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A nonlinear theory of distributional geometry

A nonlinear theory of distributional geometry
A nonlinear theory of distributional geometry
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.
math.FA, gr-qc, math.DG, 46F30, 46T30
Nigsch, Eduard A.
ea9b17e3-8ab7-4222-b1b3-a9e0e428634e
Vickers, James A.
719cd73f-c462-417d-a341-0b042db88634
Nigsch, Eduard A.
ea9b17e3-8ab7-4222-b1b3-a9e0e428634e
Vickers, James A.
719cd73f-c462-417d-a341-0b042db88634

Nigsch, Eduard A. and Vickers, James A. (2019) A nonlinear theory of distributional geometry. arXiv. (In Press)

Record type: Article

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.

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1910.03426 - Accepted Manuscript
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More information

Accepted/In Press date: 8 October 2019
Keywords: math.FA, gr-qc, math.DG, 46F30, 46T30

Identifiers

Local EPrints ID: 435183
URI: http://eprints.soton.ac.uk/id/eprint/435183
PURE UUID: d23a18ef-c0df-4638-a30f-8d346377a2b5
ORCID for James A. Vickers: ORCID iD orcid.org/0000-0002-1531-6273

Catalogue record

Date deposited: 25 Oct 2019 16:30
Last modified: 15 May 2020 00:26

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