**Unformatted text preview: **present, all but one
must become massive, since there are no nontrivial interactions between multiple massless gravitons (Boulanger et al.,
2001) [see Bachas and Petropoulos (1993) and Kiritsis (2006)
for string theory and holographic proofs of this], and these
gravitons mimic the Kaluza-Klein spectrum of a discrete
extra dimension. Other work in this area, including applications to bigravity and multigravity models, can be found in
Jejjala, Leigh, and Minic (2003), Kan and Shiraishi (2003),
Deffayet and Mourad (2004), Groot Nibbelink and Peloso
(2005), Nibbelink, Peloso, and Sexton (2007), and Deffayet
and Randjbar-Daemi (2011). Kurt Hinterbichler: Theoretical aspects of massive gravity 692 ¨
B. Another way to Stuckelberg In the last section, we introduced gauge invariance and the
¨
Stuckelberg ﬁelds by replacing the metric g with the gauge
invariant object G . This is well suited to the case where we
have a potential arranged in the form (5.3), because all the
background gð0Þ ’s appearing in the contractions and determinant of the mass term do not need replacing. The drawback
¨
is that the Stuckelberg expansion involves an inﬁnite number
of terms higher order in h . If we wish to keep track of the
h ’s, this is not very convenient.
Instead, we develop another method, which is to introduce
¨
the Stuckelberg ﬁelds through the background metric gð0Þ ,
and then allow g to transform covariantly. This method will
be better suited to a potential arranged in the form (5.9) and
¨
will have the advantage that the Stuckelberg expansion contains no higher powers of h .
We make the replacement
gð0Þ ! gð0Þ @ Y @ Y :
(6.20) The Y ðxÞ that are introduced are four ﬁelds, which despite
the index are to transform as scalars under diffeomorphisms
Y ðxÞ ! Y ðfðxÞÞ; (6.21) (6.22) This is to be contrasted with the transformation rule Y ¼
@ Y À ð@ ÞY which would hold if Y were a vector.
Given this scalar transformation rule for Y , the replaced gð0Þ
now transforms similar to a metric tensor. If we now assign
the usual diffeomorpshim transformation law to the metric
g (so that it is now covariant), quantities such as gð0Þ g
and other contractions will transform as diffeomorphism
scalars. We can take any action which is a scalar function
of gð0Þ and g , and introduce gauge invariance in this way.12.
This is convenient when we have a potential of the form
(5.9). First we lower all indices on the h ’s in the potential.
Now the background metric gð0Þ appears only through h ¼
g À gð0Þ , so we replace all occurrences of h with
H ¼ g À gð0Þ @ Y @ Y : 12 (6.23) This is essentially the technique of spurion analysis, where a
coupling constant is made to transform as a ﬁeld. A quantity which
is normally a background quantity, a coupling constant in the case of
spurions, or the background gð0Þ in this case, is made to transform in
some way that gives the...

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