Thus one of the coefcients in un g0 h for n d

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Unformatted text preview: (5.10) V2 ðg; hÞ ¼ hh2 i À hhi2 ; (5.11) V3 ðg; hÞ ¼ þc1 hh3 i þ c2 hh2 ihhi þ c3 hhi3 ; (5.12) V4 ðg; hÞ ¼ þd1 hh4 i þ d2 hh3 ihhi þ d3 hh2 i2 þ d4 hh2 ihhi2 þ d5 hhi4 ; (5.13) V5 ðg; hÞ ¼ þf1 hh5 i þ f2 hh4 ihhi þ f3 hh3 ihhi2 þ f4 hh3 ihh2 i þ f5 hh2 i2 hhi þ f6 hh2 ihhi3 þ f7 hhi5 ; . .; . (5.14) where the angled brackets are traces with the indices raised with respect to g . It does not matter whether we use potential (5.3) with indices raised by gð0Þ , or the potential (5.9) with indices raised by g . The two carry the same information and we can easily relate the coefficients of the two by expanding the inverse full metric and the full determinant in powers of h raised with the absolute metric, Kurt Hinterbichler: Theoretical aspects of massive gravity g ¼ gð0Þ À h þ h h  À h h  h  þ Á Á Á ; (5.15)    qffiffiffiffiffiffiffiffiffiffiffiffi 1 1 1 pffiffiffiffiffiffiffi ffi Àg ¼ Àgð0Þ 1 þ h À h h À h2 þ Á Á Á : 2 4 2 (5.16) The following is useful for this purpose:   lþnÀ1 ½hlþn Š: hhn i ¼ ðÀ1Þl l l¼ 0 1 X (5.17) While the zero derivative interaction terms we have written in Eq. (5.3) are general, the two derivative terms are not, since we have demanded they sum up to the Einstein-Hilbert form. The potential has broken the diffeomorphism invariance, so there is no symmetry reason for the two derivative interaction terms to take the Einstein-Hilbert form. We could deviate from it if we wanted, but we will see later that there are good reasons why it is better not to. We may also conceivably add general interactions with more than two derivatives, but we omit these for the same reasons we omit them in GR, because they are associated with higher order effective field theory effects which we hope will be small in suitable regimes. gð0Þ dx dx ¼ Àdt2 þ dr2 þ r2 d2 :  We consider a spherically symmetric static ansatz for the dynamical metric7 g ¼ þ CðrÞdr2 þ Plugging this ansatz into the equations of motion, we get the following from the tt equation, rr equation, and  equation (which is the same as the  equation by spherical symmetry), respectively, 4BC2 m2 r2 A3 þ ½2BðC À 3ÞC2 m2 r2 pffiffiffiffiffiffiffiffiffiffiffiffiffi À 4 A2 BCðC À rC0 ފA2 pffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2 A2 BC½2C2 À 2rð3A0 þ rA00 ÞC þ r2 A0 C0 ŠA pffiffiffiffiffiffiffiffiffiffiffiffiffi þ C A2 BCr2 ðA0 Þ2 ¼ 0; (5.19) In general, when there are two metrics staticity and spherical symmetry are not enough to put both in diagonal form. An r dependent off-diagonal drdt term can remain in one of them. We will not seek such off-diagonal metrics and will limit ourselves to the diagonal ansatz. Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 þ CrðB0 Þ2 gA2 pffiffiffiffiffiffiffiffiffiffiffiffiffi þ B A2 BC½CrA0 B0 þ Bð4CA0 À rC0 A0 þ 2CrA00 ފA pffiffiffiffiffiffiffiffiffiffiffiffiffi À B2 C A2 BCrðA0 Þ2 ¼ 0: (5.21)...
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