RevModPhys.84.671

Directions the major outstanding question is whether

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Unformatted text preview: The major outstanding question is whether it is possible to UV extend the effective field theory of massive gravity to the Planck scale and what this UV extension may look like. This would provide a solution to the problem of making the small cosmological constant technically natural and is bound to be an interesting theory in its own right (the analogous question applied to massive vector bosons leads to the discovery of the Higgs mechanism and spontaneous symmetry breaking). In the case of massive gravity, there are indications that a UV completion may not have a local Lorentz invariant form, although the issue is not settled. Another long shot, if UV completion can be found, would be to take the m ! 0 limit of the completion and hope to obtain a UV completion to ordinary GR. As this review is focused on the theoretical aspects of Lorentz invariant massive gravity, we do not have much to say about the large literature on Lorentz-violating massive gravity. We also do not say much about the experimental search for a graviton mass, or what the most likely signals and search modes would be. There has been much work in these areas, and each could be the topic of a separate review. Conventions: Often we work in an arbitrary number of dimensions, just because it is easy to do so. In this case, D signifies the number of spacetime dimension and we stick to D ! 3. d signifies the number of space dimensions d ¼ D À 1. We use the mostly plus metric signature convention  ¼ ðÀ; þ; þ; þ; . . .Þ. Tensors are symmetrized and antisymmetrized with unit weight, i.e., TðÞ ¼ 1 ðT þ T Þ, 2 T½Š ¼ 1 ðT À T Þ. The reduced 4d Planck mass is MP ¼ 2 1=ð8GÞ1=2 % 2:43  1018 GeV. Conventions for the curvature tensors, covariant derivatives, and Lie derivatives are those of Carroll (2004). Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 675 II. THE FREE FIERZ-PAULI ACTION We start by displaying an action for a single massive spin 2 particle in flat space, carried by a symmetric tensor field h , Z 1 S ¼ dD x À @ h @ h þ @ h @ h À @ h @ h 2 1 1 þ @ h@ h À m2 ðh h À h2 Þ: (2.1) 2 2 This is known as the Fierz-Pauli action (Fierz and Pauli, 1939). Our point of view is to take this action as given and then show that it describes a massive spin 2. There are, however, some (less than thorough) ways of motivating this action. To start with, the action above contains all possible contractions of two powers of h, with up to two derivatives. The two derivative terms, those which survive when m ¼ 0, are chosen to exactly match those obtained by linearizing the Einstein-Hilbert action. The m ¼ 0 terms describe a massless helicity 2 graviton and have the gauge symmetry h ¼ @  þ @  ; (2.2) for a spacetime dependent gauge parameter  ðxÞ. This symmetry fixes all the coefficients of the two-derivative part of Eq. (2.1), up to an overall coefficient. The mass term, however, violates this gauge symmet...
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