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

Elias Zafiris

doi:10.3390/universe2030017

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.

Highlights

One hundred years after Einstein’s initial conception and formulation of the general theory of relativity, it still remains a vibrant subject of intense research and formidable depth
Our rationale is based on the idea that in an intrinsically dynamically-variable theory, like general relativity, it should be the pertinent physical conditions or the sources of the field themselves that determine the type of these extensions as solutions to the field equations
The primary motivations emanate from three distinct sources: The first comes from Clarke’s assertion concerning the problem of singularities, according to which, the answers “involve detailed considerations of distributional solutions to Einstein’s equations, leading into an area that is only starting to be explored . . . ”

Summary

Introduction

One hundred years after Einstein’s initial conception and formulation of the general theory of relativity, it still remains a vibrant subject of intense research and formidable depth. These equations constitute the irreducible kernel of general relativity and the possibility of retaining the form of Einstein’s equations, while concurrently extending their domain of validity is promising for shedding new light on old problems and guiding toward their effective resolution These problems are primarily related to the following perennial issues: (a) the smooth manifold background of the theory; (b) the existence of singular loci in spacetime where the metric breaks down or the curvature blows up; and (c) the non-geometric nature of the second part of Einstein’s equations involving the energy-momentum tensor. The geometrodynamical formalism is very instructive in relation to the proposed extensions because it leads to the conclusion that active positive gravitational mass may emerge from purely topological considerations taking into account the constraints imposed by Einstein’s field equations in the vacuum In this manner, we may re-assessfruitfully Wheeler’s insights referring to “mass without mass” and “charge without charge”, as well as re-evaluate the notion of wormhole solutions from a cohomological point of view. The cohomological expression of the Borromean link points to its physical interpretation as a higher order wormhole solution of the field equations

General Relativity from the Perspective of Sheaf Theory

Coping with Spacetime Singularities

Topological Links in Geometrodynamics

The Borromean Rings as a Universal Nowhere Dense Singular Link

Conclusions