Spacetimes with distributional semi-Riemannian metrics and their curvature

Nigsch, Eduard A.

doi:10.1016/j.geomphys.2020.103623

Cited by 4 publications

(22 citation statements)

References 26 publications

(23 reference statements)

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“…In contrast to [17], which made use of the local theory in a number of key places, in the current paper we give intrinsic definitions on the whole of the manifold M . As here we only outline the general theory, we refer for full proofs to [25].…”

Section: Smoothing Distributions and The Colombeau Algebra On Manifoldsmentioning

confidence: 99%

Nonlinear generalized functions on manifolds

Nigsch

Vickers

2020

Proc. R. Soc. A.

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In this work, we adopt a new approach to the construction of a global theory of algebras of generalized functions on manifolds based on the concept of smoothing operators. This produces a generalization of previous theories in a form which is suitable for applications to differential geometry. The generalized Lie derivative is introduced and shown to extend the Lie derivative of Schwartz distributions. A new feature of this theory is the ability to define a covariant derivative of generalized scalar fields which extends the covariant derivative of distributions at the level of association. We end by sketching some applications of the theory. This work also lays the foundations for a nonlinear theory of distributional geometry that is developed in a subsequent paper that is based on Colombeau algebras of tensor distributions on manifolds.

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Section: Smoothing Distributions and The Colombeau Algebra On Manifoldsmentioning

confidence: 99%

Nonlinear generalized functions on manifolds

Nigsch

Vickers

2020

Proc. R. Soc. A.

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“…The following definitions are the obvious generalizations of the scalar case (we refer to [31,37] for detailed proofs in a slightly extended setting). Note that the space of moderate tensors does not depend upon the choice of background metric used to define the above norm.…”

Section: Condition (I) Means That For Each Chart U On M and All Multimentioning

confidence: 99%

“…Note that for the sake of presentation we completely omit discussion of the sheaf property; to obtain it we actually would have to restrict the basic space to a somewhat smaller one. For details, we refer to [31,34].…”

Section: The Algebra Of Generalized Tensor Fieldsmentioning

confidence: 99%

“…The linear space corresponding to this affine space is introduced as Υ0false(Mfalse):=falsefalse{false(Ξεfalse)ε∈TOfalse(Mfalse)I | false(Υεfalse)ε∈normalΥfalse(Mfalse)false⇒false(Υεfalse)ε+false(Ξεfalse)ε∈normalΥfalse(Mfalse)falsefalse}. The following definitions are the obvious generalizations of the scalar case (we refer to [31,37] for detailed proofs in a slightly extended setting).…”

Section: The Quotient Construction and The Algebra Of Generalized Tensor Fieldsmentioning

confidence: 99%

“…This follows from the fact that if g ab is continuous then g~abfalse(Υε,ωεfalse) converges uniformly to g ab on compact subsets, which allows one to define the inverse metric via the cofactor formula along the lines of [31]. ▪…”

Section: Generalized Differential Geometry and Applications To General Relativitymentioning

confidence: 99%

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A nonlinear theory of distributional geometry

Nigsch

Vickers

2020

Proc. R. Soc. A.

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This paper builds on the theory of nonlinear generalized functions begun in Nigsch & Vickers (Nigsch, Vickers 2021 Proc. R. Soc. A 20200640 ( doi:10.1098/rspa.2020.0640 )) and extends this to a diffeomorphism-invariant nonlinear theory of generalized tensor fields with the sheaf property. The generalized Lie derivative is introduced and shown to commute with the embedding of distributional tensor fields and the generalized covariant derivative commutes with the embedding at the level of association. The concept of a generalized metric is introduced and used to develop a non-smooth theory of differential geometry. It is shown that the embedding of a continuous metric results in a generalized metric with well-defined connection and curvature and that for C 2 metrics the embedding preserves the curvature at the level of association. Finally, we consider an example of a conical metric outside the Geroch–Traschen class and show that the curvature is associated to a delta function.

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Nonlinear generalised functions on manifolds

Nigsch,

Vickers

2019

Preprint

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This paper lays the foundations for a nonlinear theory of differential geometry that is developed in a subsequent paper [1] which is based on Colombeau algebras of tensor distributions on manifolds. We adopt a new approach and construct a global theory of algebras of generalised functions on manifolds based on the concept of smoothing operators. This produces a generalisation of previous theories in a form which is suitable for applications to differential geometry. The generalised Lie derivative is introduced and shown to commute with the embedding of distributions. It is also shown that the covariant derivative of a generalised scalar field commutes with this embedding at the level of association.2010 Mathematics Subject Classification. 46F30, 46T30.

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Spacetimes with distributional semi-Riemannian metrics and their curvature

Abstract: We develop a comprehensive geometric framework for the rigorous treatment of metrics with low regularity by means of regularization methods. e resulting tensor calculus is used to calculate the curvature of the conical metric describing cosmic strings.

Cited by 4 publications

References 26 publications

Nonlinear generalized functions on manifolds

Nonlinear generalized functions on manifolds

A nonlinear theory of distributional geometry

Nonlinear generalised functions on manifolds

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