Fierz–Pauli theory reloaded: from a theory of a symmetric tensor field to linearized massive gravity

Abstract

Modifying gravity at large distances by means of a massive graviton may explain the observed acceleration of the Universe without Dark Energy. The standard paradigm for Massive Gravity is the Fierz–Pauli theory, which, nonetheless, displays well known flaws in its massless limit. The most serious one is represented by the vDVZ discontinuity, which consists in a disagreement between the massless limit of the Fierz–Pauli theory and General Relativity. Our approach is based on a field-theoretical treatment of Massive Gravity: General Relativity, in the weak field approximation, is treated as a gauge theory of a symmetric rank-2 tensor field. This leads us to propose an alternative theory of linearized Massive Gravity, describing five degrees of freedom of the graviton, with a good massless limit, without vDVZ discontinuity, and depending on one mass parameter only, in agreement with the Fierz–Pauli theory.

The theory of symmetric tensor field: from fractons to gravitons and back

“…A few considerations are in order. First of all, we observe that in the Landau gauge a separate mass term for the trace of A µν (x) is forbidden, in close analogy to what happens in massive LG [25]. Moreover, having introduced a mass term, we have the new propagator (4.6) with coefficients (4.9)-(4.11), which we would like not to have tachyonic poles [34].…”

“…We interpreted this mismatch with the fact that the symmetry of pure LG is governed by a vector gauge parameter (2.4) rather than a scalar one as in (2.1). As a consequence, the gauge fixing needs a vector Nakanishi-Lautrup multiplier as done in [25,26,34,36,37] and not a scalar one as in (2.10). A final, important comment concerns the relation between the theory studied in this paper and the fracton excitations.…”

“…This transition is not continuous as far as the gauge parameter is concerned, because it changes from being a scalar (for the fracton symmetry (2.1)) to become a vector field (for the infinitesimal diffeomorphism (2.4)). As a consequence, the "scalar" gauge fixing realised in S gf (2.10) is not a proper gauge fixing for the a = 0 case, and one should choose instead a "vectorial" gauge fixing, as done in [25,34] for LG. This is the physical explanation of the divergence of the propagator (2.31) for a → 0.…”

“…To complete the analysis, we consider the massive case, and the generic mass term that can be added to the invariant action is [25,34,36]…”

“…the choice of a scalar ghost. The massive case is considered in Section 4, where appear analogies and differences with respect to LG: as in the Fierz-Pauli theory of LG only one mass parameter is allowed instead of two [25,26], as a symmetric tensor field would in principle admit, and, differently to what happens in massive LG [27,28,29], the number of degrees of freedom does not depend on the presence of a mass term. Finally, in Section 5 we discuss the main feature of fractons, i.e.…”