An Exploration of the Singularities in General Relativity

TL;DR

This paper investigates a class of smooth, degenerate metric singularities in General Relativity that allow well-defined invariants and equations at singular points, potentially resolving some physical and mathematical issues associated with traditional singularities.

Contribution

It introduces a large class of benign singularities with smooth metrics that include important cosmological and black hole singularities, and explores their implications for physics and quantum gravity.

Findings

Benign singularities allow well-defined invariants and equations at singularities.

Certain black hole singularities can be transformed into benign forms in suitable coordinates.

Benign singularities may relate to dimensional reduction phenomena in quantum theories.

Abstract

Seminar held at JINR, Dubna, May 15, 2012. In General Relativity, spacetime singularities raise a number of problems, both mathematical and physical. One can identify a class of singularities - with smooth but degenerate metric - which, under a set of conditions, allow us to define proper geometric invariants, and to write field equations, including equations which are equivalent to Einstein's at non-singular points, but remain well-defined and smooth at singularities. This class of singularities is large enough to contain isotropic singularities, warped-product singularities, including the Friedmann-Lemaitre-Robertson-Walker singularities, etc. Also a Big-Bang singularity of this type automatically satisfies Penrose's Weyl curvature hypothesis. The Schwarzschild, Reissner-Nordstrom, and Kerr-Newman singularities apparently are not of this benign type, but we can pass to coordinates in which they become benign. The charged black hole solutions Reissner-Nordstrom and Kerr-Newman can be used to model classical charged particles in General Relativity. Their electromagnetic potential and electromagnetic field are analytic in the new coordinates - they have finite values at r=0. There are hints from Quantum Field Theory and Quantum Gravity that a dimensional reduction is required at small scale. A possible explanation is provided by benign singularities, because some of their properties correspond to a reduction of dimensionality.