On h with the absolute metric we see that if the

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Unformatted text preview: ackground metric, which is a solution to the full nonlinear equations, about which we may expand the action. Often the solution metric we are expanding around will be the same as the absolute metric, but if we were expanding around a different solution, say a black hole, there would be two distinct structures, the black hole solution metric and the absolute metric. If we add matter to the theory and agree to use only minimal coupling to the metric g , then the absolute metric does not directly influence the matter. It is the geodesics and lengths as measured by the full metric (i.e., the solution of the field equations) that we care about. In massive gravity, unlike in GR, if we have a solution, we cannot perform a diffeomorphism on it to obtain a second solution to the same theory. What we obtain instead is a solution to a different massive gravity theory, one in which the absolute metric is related to the original absolute metric by the same diffeomorphism. Going to more general interactions beyond Eq. (5.1), our main interest will be in adding interaction terms with no derivatives, since these are most important at low energies. The most general such potential which reduces to Fierz-Pauli at quadratic order involves adding terms cubic and higher in h in all possible ways   qffiffiffiffiffiffiffiffiffiffi 1 1Z pffiffiffiffiffiffiffiffi S ¼ 2 dD x ð ÀgRÞ À Àg0 m2 Uðgð0Þ ; hÞ ; 4 2 (5.3) where the interaction potential U is the most general one that reduces to Fierz-Pauli at linear order, Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 Uðgð0Þ ; hÞ ¼ U2 ðgð0Þ ; hÞ þ U3 ðgð0Þ ; hÞ þ U4 ðgð0Þ ; hÞ þ U5 ðgð0Þ ; hÞ þ Á Á Á ; (5.4) U2 ðgð0Þ ; hÞ ¼ ½h2 Š À ½hŠ2 ; (5.5) U3 ðgð0Þ ; hÞ ¼ þC1 ½h3 Š þ C2 ½h2 Š½hŠ þ C3 ½hŠ3 ; (5.6) U4 ðgð0Þ ; hÞ ¼ þD1 ½h4 Š þ D2 ½h3 Š½hŠ þ D3 ½h2 Š2 þ D4 ½h2 Š½hŠ2 þ D5 ½hŠ4 ; (5.7) U5 ðgð0Þ ; hÞ ¼ þF1 ½h5 Š þ F2 ½h4 Š½hŠ þ F3 ½h3 Š½hŠ2 þ F4 ½h3 Š½h2 Š þ F5 ½h2 Š2 ½hŠ þ F6 ½h2 Š½hŠ3 þ F7 ½hŠ5 ; . .: . (5.8) The square bracket indicates a trace, with indices raised with gð0Þ; , i.e., ½hŠ ¼ gð0Þ h , ½h2 Š ¼ gð0Þ h gð0Þ  h , etc. The coefficients C1 , C2 , etc. are generic coefficients. Note that the coefficients in Un ðgð0Þ ; hÞ for n > D are redundant by 1, because there is a combination of the various contractions, the characteristic polynomial LTD ðhÞ (see the Appendix), n which vanishes identically. Thus one of the coefficients in Un ðgð0Þ ; hÞ for n > D (or any one linear combination) can be set to zero. If we want, we can reorganize the terms in the potential by raising and lowering with the full metric g rather than the absolute metric gð0Þ ,   1Z pffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffi 1 S ¼ 2 dD x ð ÀgRÞ À Àg m2 V ðg; hÞ ; (5.9) 4 2 where V ðg; hÞ ¼ V2 ðg; hÞ þ V3 ðg; hÞ þ V4 ðg; hÞ þ V5 ðg; hÞ þ ÁÁÁ;...
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