First-order action and Euclidean quantum gravity

TL;DR

This paper demonstrates that the on-shell path integral for asymptotically flat Euclidean spacetimes can be formulated in the first-order approach, accurately reproducing thermodynamic properties without boundary embedding assumptions or counter-terms.

Contribution

It introduces a first-order formulation of the on-shell path integral for Euclidean quantum gravity that avoids boundary embedding assumptions and counter-terms.

Findings

Successfully evaluated the first-order action for Schwarzschild and NUT spacetimes.

Derived correct thermodynamic quantities from the first-order path integral.

Validated the approach by matching known thermodynamic results.

Abstract

We show that the on-shell path integral for asymptotically flat Euclidean spacetimes can be given in the first-order formulation of general relativity, without assuming the boundary to be isometrically embedded in Euclidean space and without adding infinite counter-terms. For illustrative examples of our approach, we evaluate the first-order action for the four-dimensional Euclidean Schwarzschild and NUT-charged spacetimes to derive the corresponding on-shell partition functions, and show that the correct thermodynamic quantities for the solutions are reproduced.