Quantities such as g0 g and other contractions will

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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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