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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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