Discontinuity surface instead of singularity

TL;DR

This paper explores solutions to Einstein equations with conformal invariance, showing they can have discontinuities that regularize singularities, and discusses conditions leading to such solutions.

Contribution

It introduces conformally invariant geometrodynamics equations that admit discontinuous solutions, providing a new mechanism for regularizing singularities in Einstein's equations.

Findings

Discontinuous solutions can remove singularities in Einstein equations.

Discontinuities occur on space-like hypersurfaces similar to shock waves.

Certain physical conditions induce the transition from smooth to discontinuous solutions.

Abstract

Einstein equations are addressed with the energy-momentum tensor that appears if the equations under discussion are required to possess conformal invariance. It is proved that thus derived equations (equations of conformally invariant geometrodynamics) can have not only smooth solutions, but also solutions with discontinuities on space-like hypersurfaces. The solutions obtained are similar to the well-known discontinuous Einstein equation solutions like shock-wave solutions, extended-source solutions, etc. For the centrally symmetric stationary solution discussed in the paper, the discontinuity surface removes the singularity. The degree of generality of this solution regularization mechanism is discussed. The issue of the mechanism that forces any smooth solution in the conformally invariant geometrodynamics to be rearranged into the discontinuous one when certain conditions are met is also discussed. The conditions can be: (1) sound speed becoming to be higher than light speed; (2) the solution becoming intolerant to smaller and smaller-scale perturbation modes.