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under transformations, so it ﬁxes none of this symmetry.
We add two corresponding gauge ﬁxing terms to the action,
resulting from either Fadeev-Popov gauge ﬁxing or classical
gauge ﬁxing,
SGF1 ¼ SGF2 ¼ Z 2
1
dD x À @ h0 À @ h0 þ mA ;
2 Z
2
1
DÀ1
:
dD x À @ A þ m h0 þ 2
2
DÀ2
(4.35) These have the effect of diagonalizing the action,
Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 (4.34) Up to this point, we have studied only the linear theory of
massive gravity, which is determined by the requirement that
it propagates only one massive spin 2 degree of freedom. We
now turn to the study of the possible interactions and nonlinearities for massive gravity.
A. Massive general relativity What we want in a full theory of massive gravity is some
nonlinear theory whose linear expansion around some background is the massive Fierz-Pauli theory (2.1). Unlike in GR,
where the gauge invariance constrains the full theory to be
Einstein gravity, the extension for massive gravity is not
unique. In fact, there is no obvious symmetry to preserve,
so any interaction terms whatsoever are allowed.
The ﬁrst extension we consider would be to deform GR by
simply adding the Fierz-Pauli term to the full nonlinear GR
action, that is, choosing the only nonlinear interactions to be
those of GR,
1Z
pﬃﬃﬃﬃﬃﬃﬃ
ﬃ
S ¼ 2 dD x ð ÀgRÞ
2
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1
0
2 ð0Þ ð0Þ À Àg m g
g
ðh h À h h Þ :
4
(5.1) Kurt Hinterbichler: Theoretical aspects of massive gravity 686 Here there are several subtleties. Unlike GR, the Lagrangian
now explicitly depends on a ﬁxed metric gð0Þ , which we call
the absolute metric, on which the linear massive graviton
propagates. We have h ¼ g À gð0Þ as before. The mass
term is unchanged from its linear version, so the indices on
h are raised and traced with the absolute metric. The
presence of this absolute metric in the mass term breaks the
diffeomorphism invariance of the Einstein-Hilbert term. Note
that there is no way to introduce a mass term using only the
full metric g , since tracing it with itself just gives a
constant, so the nondynamical absolute metric is required to
create the traces and contractions.
Varying with respect to g we obtain the equations of
motion
pﬃﬃﬃﬃﬃﬃﬃﬃ 1
Àg R À Rg
2
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2
m
þ Àgð0Þ ðgð0Þ gð0Þ h À gð0Þ h gð0Þ Þ ¼ 0:
2
(5.2)
Indices on R are raised with the full metric, and those on
h with the absolute metric. We see that if the absolute
metric gð0Þ satisﬁes the Einstein equations, then g ¼ gð0Þ ,
i.e., h ¼ 0, is a solution. When dealing with massive
gravity and more complicated nonlinear solutions thereof,
there can be at times two different background structures.
On the one hand, there is the absolute metric, the structure
which breaks explicitly the diffeomorphism invariance. On
the other hand, there is the b...

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