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Universe 2016, 2(3), 17; doi:10.3390/universe2030017

What Is the Validity Domain of Einstein’s Equations? Distributional Solutions over Singularities and Topological Links in Geometrodynamics

Parmenides Foundation, Center for the Conceptual Foundations of Science, Kirchplatz 1, Pullach, 82049 Munich, Germany
Academic Editors: Stephon Alexander, Jean-Michel Alimi, Elias C. Vagenas and Lorenzo Iorio
Received: 13 July 2016 / Revised: 18 August 2016 / Accepted: 22 August 2016 / Published: 29 August 2016
View Full-Text   |   Download PDF [566 KB, uploaded 29 August 2016]

Abstract

The existence of singularities alerts that one of the highest priorities of a centennial perspective on general relativity should be a careful re-thinking of the validity domain of Einstein’s field equations. We address the problem of constructing distinguishable extensions of the smooth spacetime manifold model, which can incorporate singularities, while retaining the form of the field equations. The sheaf-theoretic formulation of this problem is tantamount to extending the algebra sheaf of smooth functions to a distribution-like algebra sheaf in which the former may be embedded, satisfying the pertinent cohomological conditions required for the coordinatization of all of the tensorial physical quantities, such that the form of the field equations is preserved. We present in detail the construction of these distribution-like algebra sheaves in terms of residue classes of sequences of smooth functions modulo the information of singular loci encoded in suitable ideals. Finally, we consider the application of these distribution-like solution sheaves in geometrodynamics by modeling topologically-circular boundaries of singular loci in three-dimensional space in terms of topological links. It turns out that the Borromean link represents higher order wormhole solutions. View Full-Text
Keywords: general relativity; sheaf cohomology; abstract differential geometry; singularities; geometrodynamics; distributions; generalized functions; nowhere dense algebras; algebra sheaves; topological links; wormholes; Borromean rings general relativity; sheaf cohomology; abstract differential geometry; singularities; geometrodynamics; distributions; generalized functions; nowhere dense algebras; algebra sheaves; topological links; wormholes; Borromean rings
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Zafiris, E. What Is the Validity Domain of Einstein’s Equations? Distributional Solutions over Singularities and Topological Links in Geometrodynamics. Universe 2016, 2, 17.

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