Expansions there have been several extensive

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Unformatted text preview: Comelli et al. (2011), Koyama, Niz, and Tasinato (2011a, 2011b), and Nieuwenhuizen (2011). We now study the Hamiltonian of the nonlinear massive gravity action (5.1) with flat absolute metric  , S¼  1 Z D pffiffiffiffiffiffiffiffi d x ð ÀgRÞ 2 2  1 À m2   ðh h À h h Þ : 4 (5.36) We saw in Sec. II.A that the free theory carries 5 degrees of freedom in D ¼ 4, due to the fact that the time components h00 appeared as a Lagrange multiplier in the action. We see that this no longer remains true once the nonlinearities of Eq. (5.36) are taken into account, so there is now an extra degree of freedom. A particularly nice way to study gravity Hamiltonians is through the Arnowitt-Deser-Misner (ADM) formalism (Arnowitt, Deser, and Misner, 1960, 1962). A spacelike slicing of spacetime by hypersurfaces Æt is chosen, and we change variables from components of the metric g to the spatial metric gij , the lapse Ni , and the shift N , according to g00 ¼ ÀN 2 þ gij Ni Nj ; (5.37) g0i ¼ Ni ; (5.38) gij ¼ gij : (5.39) Here i; j; . . . are spatial indices, and gij is the inverse of the spatial metric gij (not the ij components of inverse metric g ). The Einstein-Hilbert part of the action in these variables reads [see Poisson (2004) and Dyer and Hinterbichler (2009) for detailed derivations and formulas] 1 Z D pffiffiffi ðdÞ (5.40) d x gN ½ R À K 2 þ K ij Kij Š; 2 2 Kurt Hinterbichler: Theoretical aspects of massive gravity where ðdÞ R is the curvature of the spatial metric gij . The quantity Kij is the extrinsic curvature of the spatial hypersurfaces, defined as Kij ¼ 1 _ ðg À ri Nj À rj Ni Þ; 2N ij (5.41) where the dot means a time derivative, and the covariant derivatives are with respect to the spatial metric gij . We then Legendre transform the spatial variables gij , defining the canonical momenta pij ¼ L 1 pffiffiffi ¼ 2 gðK ij À Kgij Þ; _ gij 2 and writing the action in Hamiltonian form  Z 2L ¼ d xpij g _ ij À H; 2 d N¼ (5.43) where the Hamiltonian H is defined by Z  Z _ H¼ dd xpab gab À L ¼ dd xN C þ Ni Ci ; (5.44) and the quantities C and Ci are pffiffiffi C ¼ g½ðdÞ R þ K 2 À K ij Kij Š; pffiffiffi Ci ¼ 2 grj ðK ij À Kgij Þ; (5.45) and here Kij should be thought of as a function of pij and gij , _ obtained by inverting Eq. (5.42) for gij and plugging into Eq. (5.41),   2 2 1 pgij : Kij ¼ pffiffiffi pij À DÀ2 g (5.46) All traces and index manipulations are performed with gij and its inverse. For m ¼ 0, the action is pure constraint, and the Hamiltonian vanishes, a characteristic of diffeomorphism invariance. The shift N and lapse Ni appear as Lagrange multipliers, enforcing the Hamiltonian constraint C ¼ 0 and momentum constraints Ci ¼ 0. It can be checked that these are first class constraints, generating the D diffeomorphism symmetries of the action. In D ¼ 4, we have 12 phase space metric components, minus four constraints, minus four gauge symmetries, leaves 4 phase space...
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