Gravity action on the rapidly varying metrics

TL;DR

This paper investigates the behavior of the Einstein action on a simplicial complex with rapidly varying metrics, showing that the path integral measure enforces continuity of the induced metric across faces in the limit of vanishing layer thickness.

Contribution

It demonstrates how the Einstein action's divergence in the thin layer limit leads to a measure that enforces metric continuity across simplices in the path integral formulation.

Findings

The action diverges as the layer thickness approaches zero.

The path integral measure includes a factor enforcing metric continuity.

The results align with previous symmetry-based analyses.

Abstract

We consider a four-dimensional simplicial complex and the minisuperspace general relativity system described by the metric flat in the most part of the interior of every 4-simplex with exception of a thin layer of thickness $\propto \varepsilon$ along the every three-dimensional face where the metric undergoes jump between the two 4-simplices sharing this face. At $\varepsilon \to 0$ this jump would become discontinuity. Since, however, discontinuity of the (induced on the face) metric is not allowed in general relativity, the terms in the Einstein action tending to infinity at $\varepsilon \to 0$ arise. In the path integral approach, these terms lead to the pre-exponent factor with \dfuns requiring that the induced on the faces metric be continuous, i. e. the 4-simplices fit on their common faces. The other part of the path integral measure corresponds to the action being the sum of independent terms over the 4-simplices. Therefore this part of the path integral measure is the product of independent measures over the 4-simplices. The result obtained is in accordance with our previous one obtained from the symmetry considerations.