Gravity action on discontinuous metrics

TL;DR

This paper analyzes the gravitational action for discontinuous metrics in a minisuperspace model, showing how regularization leads to a measure enforcing metric continuity across tetrahedra, aligning with previous symmetry-based results.

Contribution

It demonstrates how regularization of the Einstein action for discontinuous metrics results in a path integral measure that enforces metric continuity, connecting Regge calculus with independent tetrahedra.

Findings

Path integral measure includes delta functions enforcing metric continuity.

Regularization leads to a measure consistent with previous symmetry-based results.

The approach relates discontinuous metrics to standard Regge calculus measures.

Abstract

We consider minisuperspace gravity system described by piecewise flat metric discontinuous on three-dimensional faces (tetrahedra). There are infinite terms in the Einstein action. However, starting from proper regularization, these terms in the exponential of path integral result in pre-exponent factor with $\delta$-functions requiring vanishing metric discontinuities. Thereby path integral measure in Regge calculus is related to path integral measure in Regge calculus where length of an edge is not constrained to be the same for all the 4-tetrahedra containing this edge, i.e. in Regge calculus with independent 4-tetrahedra. The result obtained is in accordance with our previous one obtained from symmetry considerations.