On the discrete version of the black hole solution

TL;DR

This paper explores a discrete version of the Schwarzschild solution within Regge calculus, deriving a nonsingular metric at the origin that parallels Newtonian potential behavior, and connects to quantum gravity measures.

Contribution

It introduces a discrete Schwarzschild solution in Regge calculus that naturally regularizes the singularity at the origin, linking quantum measure effects to classical gravitational solutions.

Findings

The discrete solution is nonsingular at the origin.

The metric obeys a difference Poisson equation with a point source.

The effective action approximates the Regge action for large parameters.

Abstract

A Schwarzschild type solution in Regge calculus is considered. Earlier, we considered a mechanism of loose fixing of edge lengths due to the functional integral measure arising from integration over connection in the functional integral for the connection representation of the Regge action. The length scale depends on a free dimensionless parameter that determines the final functional measure. For this parameter and the length scale large in Planck units, the resulting effective action is close to the Regge action. Earlier, we considered the Regge action in terms of affine connection matrices as functions of the metric inside the 4-simplices and found that it is a difference form of the Hilbert-Einstein action in the leading order over metric variations between the 4-simplices. Now we take the (continuum) Schwarzschild problem in the form where spherical symmetry is not set a priori and arises just in the solution, take the difference form of the corresponding equations and get the metric (in fact, in the Lemaitre or Painlev\'{e}-Gullstrand like frame), which is nonsingular at the origin, just as the Newtonian gravitational potential, obeying the difference Poisson equation with a point source, is cut off at the elementary length and is finite at the source.