On the discrete version of the Kerr-Newman solution

TL;DR

This paper develops a discrete Kerr-Newman black hole model within Regge calculus, analyzing the electromagnetic and gravitational contributions near the singularity ring, and finds the electromagnetic effect dominates only for near-extremal black holes.

Contribution

It introduces a discrete version of the Kerr-Newman solution using Regge calculus, with a novel diagram technique and analysis of electromagnetic effects near the singularity.

Findings

Electromagnetic contribution is infinite as the edge length scale approaches zero.

The electromagnetic effect is significant mainly for non-rotating black holes.

The discrete model resolves the classical singularity in a novel way.

Abstract

This paper continues our work on black holes in the framework of the Regge calculus, where the discrete version (with a certain edge length scale $b$ proportional to the Planck scale) of the classical solution emerges as an optimal starting point for the perturbative expansion after functional integration over the connection, with the singularity resolved. An interest in the present discrete Kerr-Newman type solution (with the parameter $a \gg b$) may be to check the classical prediction that the electromagnetic contribution to the metric and curvature on the singularity ring is (infinitely) greater than the contribution of the $\delta$-function-like mass distribution, no matter how small the electric charge is. Here we encounter a kind of a discrete diagram technique, but with three-dimensional (static) diagrams and with only a few diagrams, although with modified (extended to complex coordinates) propagators. The metric (curvature) in the vicinity of the former singularity ring is considered. The electromagnetic contribution does indeed have a relative factor that is infinite at $b \to 0$, but, taking into account some existing estimates of the upper bound on the electric charge of known substances, it is not so large for habitual bodies and can only be significant for practically non-rotating black holes.