On the discrete version of the Reissner–Nordström solution

Abstract

This paper generalizes our previous work on the discrete Schwarzschild-type solution in Regge calculus to the case of a charge. The known in the literature simplicial electro-dynamics retaining like Regge calculus geometric features of the continuum counterpart is incorporated into the formalism. The functional integral provides a loose fixation of edge lengths around some scale and a perturbative expansion, for which we consider, in essence, finding the optimal starting (background) metric/field from the skeleton Regge and electro-dynamic equations. The simplest periodic simplicial structure and the expansion over metric/field variations between 4-simplices are considered. In the leading order of this expansion, the electromagnetic action, as we found earlier for the Regge action, is reducible to a finite-difference form of the continuum counterpart. Instead of infinite continuous metric/field variables at the center, we have finite discrete variables; the discrete metric in the Schwarzschild-type coordinates turns out to change the sign of its variation when approaching the center from the nearest vertices, so that [Formula: see text] is positive at the center (the continuum [Formula: see text] tends to [Formula: see text] at the center). The metric/field in the neighborhood of the center and the curvature and the Kretschmann scalar at the center are estimated.