Einstein equation at singularities

TL;DR

This paper reformulates Einstein's equation to remain valid at certain singularities, including black hole and Big Bang singularities, by expressing it in terms of the Ricci curvature component.

Contribution

It introduces an equivalent form of Einstein's equation that is applicable at singularities like Schwarzschild and Big Bang, expanding the mathematical understanding of these extreme conditions.

Findings

Equation remains valid at Schwarzschild and Big Bang singularities

Applicable to isotropic and warped product singularities

Provides a new mathematical framework for singularities

Abstract

Einstein's equation is rewritten in an equivalent form, which remains valid at the singularities in some major cases. These cases include the Schwarzschild singularity, the Friedmann-Lema\^itre-Robertson-Walker Big Bang singularity, isotropic singularities, and a class of warped product singularities. This equation is constructed in terms of the Ricci part of the Riemann curvature (as the Kulkarni-Nomizu product between Einstein's equation and the metric tensor).