RevModPhys.84.671

Traces and index manipulations are performed with gij

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Unformatted text preview: degrees of freedom, the same counting as in the linear theory. The nonlinear theory contains the same number of degrees of freedom as the linearized theory. Now looking at the mass term, in ADM variables we have   ðh h À h h Þ ¼ ik jl ðhij hkl À hik hjl Þ þ 2ij hij À 2N 2 ij hij þ 2Ni ðgij À ij ÞNi ; where hij  gij À ij . The action becomes Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 (5.47) (5.48) C ; m2 ij hij Ni ¼ 1 ij ðg À ij ÞÀ1 Cj : m2 (5.49) When these values are plugged back into Eq. (5.48), we have an action with no constraints or gauge symmetries at all, so all the phase space degrees of freedom are active. The resulting Hamiltonian is H¼ Æt 1 Z D ij _ d xp gij À N C À Ni Ci 2 2 m2 ik jl ½  ðhij hkl À hik hjl Þ þ 2ij hij À 4 À 2N 2 ij hij þ 2Ni ðgij À ij ÞNi Š: In the m Þ 0 case, the Fierz-Pauli term brings in contributions to the action that are quadratic in the lapse and shift (but still free of time derivatives). Thus the lapse and shift no longer serve as Lagrange multipliers, but rather as auxiliary fields, because their equations of motion can be algebraically solved to determine their values, (5.42) Æt Æt S¼ 689 1Zd 1 C2 1 i ij C ðg À ij ÞÀ1 Cj d x 2 ij þ 2 2 2m  hij 2m2 þ m2 ik jl ½  ðhij hkl À hik hjl Þ þ 2ij hij Š; 4 (5.50) which is nonvanishing, unlike in GR. In four dimensions, we thus have 12 phase space degrees of freedom, or 6 real degrees of freedom. The linearized theory had only 5 degrees of freedom, and we have here a case where the nonlinear theory contains more degrees of freedom than the linear theory. It should not necessarily be surprising that this can happen, because there is no reason nonlinearities cannot change the constraint structure of a theory or that kinetic terms cannot appear at higher order. As argued by Boulware and Deser (1972), the Hamiltonian (5.50) is not bounded, and since the system is nonlinear, it is not surprising that it has instabilities (Gabadadze and Gruzinov, 2005). The nature of the instability, i.e., whether it is a ghost of a tachyon, what backgrounds it appears around, and its severity, is hard to see in the Hamiltonian formalism. But in Sec. VII.B we see that this instability is a ghost, a scalar with a negative kinetic term, and that its mass around a given background can be determined. It turns out that around flat space, the ghost degree of freedom is not excited because its mass is infinite, but around nontrivial backgrounds its mass becomes finite. This ghostly extra degree of freedom is referred to as the Boulware-Deser ghost (Boulware and Deser, 1972). There is still the possibility that adding higher order interaction terms such as h3 terms and higher can remove the ghostly sixth degree of freedom. Boulware and Deser analyzed a large class of various mass terms, showing that the sixth degree of freedom remained (Boulware and Deser, 1972), but they did not consider the most general poss...
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