Transverse Fierz-Pauli symmetry

TL;DR

This paper explores theories of spin 2 gravitons with reduced symmetry, showing how different invariance conditions affect stability, ghost absence, and the form of mass terms, highlighting the uniqueness of the Fierz-Pauli Lagrangian.

Contribution

It demonstrates that transverse diffeomorphism invariance constrains the form of gravity theories and reveals the rigidity of the Fierz-Pauli mass term against deformations.

Findings

Massless theories require transverse diffeomorphism invariance for stability.

Adding Weyl symmetry leads to unimodular gravity.

Massive theories are constrained to the Fierz-Pauli form to avoid ghosts.

Abstract

We consider some flat space theories for spin 2 gravitons, with less invariance than full diffeomorphisms. For the massless case, classical stability and absence of ghosts require invariance under transverse diffeomorphisms (TDiff). Generic TDiff invariant theories contain a propagating scalar, which disappears if the symmetry is enhanced in one of two ways. One possibility is to consider full diffeomorphisms (Diff). The other (which we denote WTDiff) adds a Weyl symmetry, by which the Lagrangian becomes independent of the trace. The first possibility corresponds to General Relativity, whereas the second corresponds to "unimodular" gravity (in a certain gauge). Phenomenologically, both options are equally acceptable. For massive gravitons, the situation is more restrictive. Up to field redefinitions, classical stability and absence of ghosts lead directly to the standard Fierz-Pauli Lagrangian. In this sense, the WTDiff theory is more rigid against deformations than linearized GR, since a mass term cannot be added without provoking the appearance of ghosts.