Borromean Link in Relativity Theory What Is the Validity Domain of Einstein’s Field Equations? Sheaf-Theoretic Distributional Solutions over Singularities and Topological Links in Geometrodynamics

Abstract

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. In this way, during all these years our understanding of gravitation in differential geometric terms is being continuously refined. We believe that one of the highest priorities of a centennial perspective on General Relativity should be a careful re-examination of the validity domain of Einstein’s field equations. 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 to old problems and guiding toward their effective resolution. These problems are primarily related with 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. It turns out that these problems are intrinsically related to each other and require a critical re-thinking of the initial assumptions referring to the domain of validity of Einstein’s equations.