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potential. This was addressed by Creminelli et al. (2005),
where the analysis was done perturbatively in powers of h.
The lapse is expanded around its ﬂat space values N ¼
1 þ N . In this case, N plays the role of the Lagrange
multiplier, and it is shown that at fourth order, interaction
terms involving higher powers of N cannot be removed. It Kurt Hinterbichler: Theoretical aspects of massive gravity 690 was concluded by Creminelli et al. (2005) that the BoulwareDeser ghost is unavoidable, but this conclusion is too quick. It
may be possible that there are ﬁeld redeﬁnitions under which
the lapse is made to appear linearly. Alternatively, it may be
possible that after one solves for the shift using its equation of
motion, then replaces into the action, the resulting action is
linear in the lapse, even though it contained higher powers of
the lapse before integrating out the shift. It is also possible
that the lapse appears linearly in the full nonlinear action,
even though at any ﬁnite order the action contains higher
powers of the lapse. [For discussions and examples of these
points, see de Rham and Gabadadze (2010a) and de Rham,
Gabadadze, and Tolley (2010)).]
As it turns out, it is, in fact, possible to add appropriate
interactions that eliminate the ghost (Hassan and Rosen,
2011a, 2011c). In D dimensions, there is a D À 2 parameter
family of such interactions. We study these in Sec. VIII,
where we see that they also have the effect of raising the
maximum energy cutoff at which massive gravity is valid as
an effective ﬁeld theory.9 This class of theories solves the
problem of the Boulware-Deser ghost.10. where fðxÞ is the arbitrary gauge function, which must be a
diffeomorphism. In massive gravity this gauge invariance is
broken only by the mass term. To restore it, we introduce a
¨
Stuckelberg ﬁeld Y ðxÞ, patterned after the gauge symmetry
(6.1), and we apply it to the metric g , g ðxÞ ! G ¼ @Y @Y g ðY ðxÞÞ:
@x @x pﬃﬃﬃﬃﬃﬃﬃﬃ
The Einstein-Hilbert term ÀgR will not change under this
substitution, because it is gauge invariant, and the substitution
looks similar to a gauge transformation with gauge parameter
Y ðxÞ, so no Y ﬁelds are introduced into the Einstein-Hilbert
part of the action.
The graviton mass term, however, will pick up dependence
on Y ’s in such a way that it will now be invariant under the
following gauge transformation:
@f @f g ðfðxÞÞ;
@x @x Y ðxÞ ! fÀ1 ðY ðxÞÞ g ðxÞ !
¨
VI. THE NONLINEAR STUCKELBERG FORMALISM ¨
In this section we extend the Stuckelberg trick to full
nonlinear order. This is a powerful tool with which to elucidate the nonlinear dynamics of massive gravity. It allows us to
trace the breakdown in the linear expansion to strong coupling of the longitudinal mode. It also tells us about quantum
corrections, the scale of the effective ﬁeld theory and where it
breaks down, as well as the nature of the Boulware-Deser
ghost and w...

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