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**Unformatted text preview: **ðY ðxÞÞ: (6.8) We now expand Y about the identity,
Y ðxÞ ¼ x þ A ðxÞ: (6.9) The quantity G is expanded as
G @Y ðxÞ @Y ðxÞ
¼
g ðY ðxÞÞ
@x
@x @ðx þ A Þ @ðx þ A Þ
g ðx þ AÞ
¼
@x
@x
¼ ð þ @ A Þð þ @ A Þ g þ A @ g
1
þ A A @ @ g þ Á Á Á
2
¼ g þ A @ g þ @ A g þ @ A g
1
þ A A @ @ g þ @ A @ A g 2
þ @ A A @ g þ @ A A @ g þ Á Á Á :
(6.10) We now look at the inﬁnitesimal transformation properties
of g, Y , G, and Y , under inﬁnitesimal general coordinate
transformations generated by fðxÞ ¼ x þ ðxÞ. The metric
transforms in the usual way,
g ¼ @ g þ @ g þ @ g :
(6.11) The transformation law for the A’s comes from the transformation of Y ,
Y ð xÞ ! fÀ1 ðY ðxÞÞ % Y ð xÞ À A ¼ À ðx þ AÞ
¼ À À A @ À 1A A @ @ À Á Á Á :
2
(6.12)
The A are the Goldstone bosons that nonlinearly carry the
broken diffeomorphism invariance in massive gravity. The
combination G , as we noted before, is gauge invariant
(6.13) ¨
We now have a recipe for Stuckelberg-ing the general
massive gravity action of the form (5.3). We leave the
Einstein-Hilbert term alone. In the mass term, we write all
the h ’s with lowered indices to get rid of the dependence on
the absolute metric, and then we replace all occurrences of
h with Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 where indices on A are lowered with the background metric.
¨
This is exactly the Stuckelberg substitution we made in the
linear case.
In the case where the absolute metric is ﬂat, gð0Þ ¼ ,
we have from Eq. (6.10),
H ¼ h þ @ A þ @ A þ @ A @ A þ Á Á Á :
(6.16)
Here indices on A are lowered with and the ellipsis are
terms quadratic and higher in the ﬁelds and containing at least
one power of h. This takes into account the full nonlinear
gauge transformation.
As in the linear case, we usually want to do another
¨
scalar Stuckelberg replacement to introduce a Uð1Þ gauge
symmetry,
A ! A þ @ : (6.17) Then the expansion for the ﬂat absolute metric takes the form
H ¼ h þ @ A þ @ A þ 2@ @ þ @ A @ A þ @ A @ @ þ @ @ @ A þ @ @ @ @ þ Á Á Á ; (6.18) where again the ellipsis are terms quadratic and higher in the
ﬁelds and containing at least one power of h. The gauge
transformation laws are
h ¼ @ þ @ þ L h ;
¼ ÀÃ: Y ¼ À ðY Þ; H ðxÞ ¼ G ðxÞ À gð0Þ ðxÞ;
(6.15) A ¼ @ Ã À À A @ À 1A A @ @ À Á Á Á ;
2 ðY ðxÞÞ; G ¼ 0: 691 (6.14) (6.19) ¨
This method of Stuckelberging can be extended to any
number of gravitons and general coordinate invariances, as
done by Arkani-Hamed, Georgi, and Schwartz (2003) and
Arkani-Hamed and Schwartz (2004), in analogy with the
gauge theory little Higgs models and dimensional deconstruction (Arkani-Hamed, Cohen, and Georgi, 2001a,
2001b). When multiple gravitons are...

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