Nonlinear interactions h0 1 0 2 h a m

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Unformatted text preview: nt under  transformations, so it fixes none of this symmetry. We add two corresponding gauge fixing terms to the action, resulting from either Fadeev-Popov gauge fixing or classical gauge fixing, SGF1 ¼ SGF2 ¼ Z 2  1 dD x À @ h0 À @ h0 þ mA ;  2 Z  2  1 DÀ1 : dD x À @ A þ m h0 þ 2 2 DÀ2 (4.35) These have the effect of diagonalizing the action, Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 (4.34) Up to this point, we have studied only the linear theory of massive gravity, which is determined by the requirement that it propagates only one massive spin 2 degree of freedom. We now turn to the study of the possible interactions and nonlinearities for massive gravity. A. Massive general relativity What we want in a full theory of massive gravity is some nonlinear theory whose linear expansion around some background is the massive Fierz-Pauli theory (2.1). Unlike in GR, where the gauge invariance constrains the full theory to be Einstein gravity, the extension for massive gravity is not unique. In fact, there is no obvious symmetry to preserve, so any interaction terms whatsoever are allowed. The first extension we consider would be to deform GR by simply adding the Fierz-Pauli term to the full nonlinear GR action, that is, choosing the only nonlinear interactions to be those of GR,  1Z pffiffiffiffiffiffiffi ffi S ¼ 2 dD x ð ÀgRÞ 2  qffiffiffiffiffiffiffiffiffiffi 1 0 2 ð0Þ ð0Þ À Àg m g g ðh h À h h Þ : 4 (5.1) Kurt Hinterbichler: Theoretical aspects of massive gravity 686 Here there are several subtleties. Unlike GR, the Lagrangian now explicitly depends on a fixed metric gð0Þ , which we call  the absolute metric, on which the linear massive graviton propagates. We have h ¼ g À gð0Þ as before. The mass  term is unchanged from its linear version, so the indices on h are raised and traced with the absolute metric. The presence of this absolute metric in the mass term breaks the diffeomorphism invariance of the Einstein-Hilbert term. Note that there is no way to introduce a mass term using only the full metric g , since tracing it with itself just gives a constant, so the nondynamical absolute metric is required to create the traces and contractions. Varying with respect to g we obtain the equations of motion   pffiffiffiffiffiffiffiffi  1  Àg R À Rg 2 qffiffiffiffiffiffiffiffiffiffiffiffi 2 m þ Àgð0Þ ðgð0Þ gð0Þ h À gð0Þ h gð0Þ Þ ¼ 0: 2 (5.2) Indices on R are raised with the full metric, and those on h with the absolute metric. We see that if the absolute metric gð0Þ satisfies the Einstein equations, then g ¼ gð0Þ ,   i.e., h ¼ 0, is a solution. When dealing with massive gravity and more complicated nonlinear solutions thereof, there can be at times two different background structures. On the one hand, there is the absolute metric, the structure which breaks explicitly the diffeomorphism invariance. On the other hand, there is the b...
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