Diffeomorphism invariance of the effective gravitational action

TL;DR

This paper analyzes the diffeomorphism invariance of the effective gravitational action, demonstrating that the Fradkin-Vilkovisky measure is invariant while the Fujikawa measure is not, clarifying misconceptions in the literature.

Contribution

It provides detailed calculations confirming the invariance of the Fradkin-Vilkovisky measure and highlights the importance of $g^{00}$ factors for invariance, correcting previous claims.

Findings

Fradkin-Vilkovisky measure is diffeomorphism invariant

Fujikawa measure is not invariant

Presence of $g^{00}$ factors is crucial for invariance

Abstract

We investigate on the diffeomorphism invariance of the effective gravitational action, focusing in particular on the path integral measure. In the literature, two different measures are mainly considered, the Fradkin-Vilkovisky and the Fujikawa one. With the help of detailed calculations, we show that, despite claims to the contrary, the Fradkin-Vilkovisky measure is diffeomorphism invariant, while the Fujikawa measure is not. In particular, we see that, contrary to naive expectations, the presence of $g^{00}$ factors in the Fradkin-Vilkovisky measure is necessary to ensure the invariance of the effective gravitational action. We also comment on results recently appeared in the literature, and show that formal calculations can easily miss delicate points.