A note on "symmetric" vielbeins in bimetric, massive, perturbative and non perturbative gravities

TL;DR

This paper investigates conditions for symmetric vielbeins in bimetric and massive gravity theories, clarifying and correcting previous misconceptions about their relations and gauge choices.

Contribution

It provides a detailed analysis of when symmetric vielbeins can be imposed in bimetric gravity, correcting prior misunderstandings and clarifying the gauge conditions.

Findings

Derived conditions for symmetric vielbeins in bimetric gravity.

Clarified the relation between vielbeins and metrics in different gravity frameworks.

Corrected previous statements regarding the Deser-van Nieuwenhuizen gauge.

Abstract

We consider a manifold endowed with two different vielbeins $E^{A}{}_{\mu}$ and $L^{A}{}_{\mu}$ corresponding to two different metrics $g_{\mu\nu}$ and $f_{\mu\nu}$. Such a situation arises generically in bimetric or massive gravity (including the recently discussed version of de Rham, Gabadadze and Tolley), as well as in perturbative quantum gravity where one vielbein parametrizes the background space-time and the other the dynamical degrees of freedom. We determine the conditions under which the relation $g^{\mu\nu} E^{A}{}_{\mu} L^{B}{}_{\nu} = g^{\mu\nu} E^{B}{}_{\mu} L^{A}{}_{\nu}$ can be imposed (or the "Deser-van Nieuwenhuizen" gauge chosen). We clarify and correct various statements which have been made about this issue.