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**Unformatted text preview: **action more symmetries. Note that this
method of introducing gauge invariance can be carried out on any
Lorentz invariant action, even one that does not contain a dynamical
metric g . For example, a plain old scalar ﬁeld in ﬂat space can be
made diffeomorphism invariant in this way. This highlights the fact
that general coordinate invariance is not the critical ingredient that
leads one to a theory of gravity, since it can be made to hold in any
theory.
Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 Y ¼ x À A ;
and using g ¼ gð 0 Þ
(6.24)
þ h , we have H ¼ h þ gð0Þ @ A þ gð0Þ @ A À gð0Þ @ A @ A :
(6.25)
Note the difference in sign for the term quadratic in A
compared with Eq. (6.16).
Under inﬁnitesimal gauge transformations we have
A ¼ À þ @ A ; (6.26) h ¼ rð0Þ þ rð0Þ þ L h ;
(6.27) where the covariant derivatives are with respect to gð0Þ and
the indices on are lowered using gð0Þ . To linear order, the
transformations are
A ¼ À ; (6.28) h ¼ rð0Þ þ rð0Þ ;
(6.29) ¨
which reproduces the linear Stuckelberg expansion.
In the case of a ﬂat background gð0Þ ¼ , the replace
ment is or inﬁnitesimally,
Y ¼ @ Y : Note that we do not need to make a replacement on the g ’s
pﬃﬃﬃﬃﬃﬃﬃﬃ
used to contract the indices, nor on the Àg out front of the
potential in Eq. (5.9).
Expanding H ¼ h þ @ A þ @ A À @ A @ A ; (6.30) with indices on A lowered by . Notice that this is the
complete expression; there are no higher powers of h ,
unlike Eq. (6.16).
We often follow this with the replacement A ! A þ
@ to extract the helicity 0 mode. The full expansion thus
reads
H ¼ h þ @ A þ @ A þ 2@ @ þ @ A @ A þ @ A @ @ þ @ @ @ A þ @ @ @ @ :
(6.31)
Under inﬁnitesimal gauge transformations,
h ¼ @ þ @ þ L h ; (6.32) A ¼ @ Ã À þ @ A ; (6.33) ¼ ÀÃ: (6.34) ¨
Yet another way to introduce Stuckelberg ﬁelds is advocated by Alberte, Chamseddine, and Mukhanov (2010, 2011)
and Chamseddine and Mukhanov (2010), in which they make
the inverse metric g covariant through the introduction of
scalars g ! g @ Y @ Y . There have also been many
studies, initiated by ’t Hooft, of the so-called gravitational
¨
Higgs mechanism, which is also essentially a Stuckelberging
of different forms of massive gravity (Kirsch, 2005; Leclerc,
2006; ’t Hooft, 2007; Kakushadze, 2008a, 2008b; Demir and
Pak, 2009; Kluson, 2010; Oda, 2010a, 2010b). All of these Kurt Hinterbichler: Theoretical aspects of massive gravity are equivalent to the theories we study, as can be seen simply
by going to unitary gauge (Berezhiani and Mirbabayi, 2010).
At the end of the day, Eq. (5.3) is the most general Lorentz
invariant graviton potential, and any Lorentz invariant massive gravity theory will have a unitary gauge with a potential
which is equivalent to it for some choice of the...

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