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

Cedric Deffayet, Jihad Mourad, George Zahariade

Abstract

We consider a manifold endowed with two different vielbeins $E^{A}{}_μ$ and $L^{A}{}_μ$ corresponding to two different metrics $g_{μν}$ and $f_{μν}$. 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 $g^{μν} E^{A}{}_μ L^{B}{}_ν = g^{μν} E^{B}{}_μ L^{A}{}_ν$ can be imposed (or the "Deser-van Nieuwenhuizen" gauge chosen). We clarify and correct various statements which have been made about this issue.

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

Abstract

We consider a manifold endowed with two different vielbeins and corresponding to two different metrics and . 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 can be imposed (or the "Deser-van Nieuwenhuizen" gauge chosen). We clarify and correct various statements which have been made about this issue.

Paper Structure

This paper contains 8 sections, 10 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Necessary and sufficient conditions
  3. Sufficient conditions
  4. Two dimensional case
  5. Three dimensional case
  6. Four dimensional case
  7. Time evolution and application to ghost-free massive gravity
  8. Conclusions

Key Result

Proposition 1

An arbitrary invertible matrix $M$ can be decomposed as $M = \lambda s$, $\lambda$ being the matrix of a Lorentz transformation and $s$ a symmetric matrix, if and only if (i) the real matrix $\eta M^{t} \eta M$ has a real square root, and (ii) at least one such square root can be written as the prod

Theorems & Definitions (12)

  • Proposition 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof : Direct proof
  • Theorem 1
  • Proposition 4
  • Proposition 5
  • Theorem 2
  • Theorem 3
  • ...and 2 more