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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
T½ ¼ 1 ðT À T Þ. The reduced 4d Planck mass is MP ¼
2
1=ð8GÞ1=2 % 2:43 Â 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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