Gravity action on rapidly varying metrics

Abstract

We consider a four-dimensional simplicial complex and the minisuperspace general relativity system on it. The metric is flat in most parts of the interior of every 4-simplex, with the exception of a thin layer of thickness ∝ ε along each three-dimensional face. In this layer the metric undergoes a jump between the two 4-simplices sharing this face. At ε → 0 this jump would become a discontinuity. However, a discontinuity of the metric induced on the face is not allowed in general relativity: terms arise in the Einstein action tending to infinity as ε → 0. In the path integral approach, these terms lead to the pre-exponent factor with δ-functions requiring that the metric induced on the faces be continuous. That is, the 4-simplices fit on their common faces. The other part of the path integral measure corresponds to the action, which is 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 symmetry considerations.