At space the ghost degree of freedom is not excited

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Unformatted text preview: ible potential. This was addressed by Creminelli et al. (2005), where the analysis was done perturbatively in powers of h. The lapse is expanded around its flat space values N ¼ 1 þ N . In this case, N plays the role of the Lagrange multiplier, and it is shown that at fourth order, interaction terms involving higher powers of N cannot be removed. It Kurt Hinterbichler: Theoretical aspects of massive gravity 690 was concluded by Creminelli et al. (2005) that the BoulwareDeser ghost is unavoidable, but this conclusion is too quick. It may be possible that there are field redefinitions under which the lapse is made to appear linearly. Alternatively, it may be possible that after one solves for the shift using its equation of motion, then replaces into the action, the resulting action is linear in the lapse, even though it contained higher powers of the lapse before integrating out the shift. It is also possible that the lapse appears linearly in the full nonlinear action, even though at any finite order the action contains higher powers of the lapse. [For discussions and examples of these points, see de Rham and Gabadadze (2010a) and de Rham, Gabadadze, and Tolley (2010)).] As it turns out, it is, in fact, possible to add appropriate interactions that eliminate the ghost (Hassan and Rosen, 2011a, 2011c). In D dimensions, there is a D À 2 parameter family of such interactions. We study these in Sec. VIII, where we see that they also have the effect of raising the maximum energy cutoff at which massive gravity is valid as an effective field theory.9 This class of theories solves the problem of the Boulware-Deser ghost.10. where fðxÞ is the arbitrary gauge function, which must be a diffeomorphism. In massive gravity this gauge invariance is broken only by the mass term. To restore it, we introduce a ¨ Stuckelberg field Y  ðxÞ, patterned after the gauge symmetry (6.1), and we apply it to the metric g , g ðxÞ ! G ¼ @Y @Y g ðY ðxÞÞ: @x @x pffiffiffiffiffiffiffiffi The Einstein-Hilbert term ÀgR will not change under this substitution, because it is gauge invariant, and the substitution looks similar to a gauge transformation with gauge parameter Y  ðxÞ, so no Y fields are introduced into the Einstein-Hilbert part of the action. The graviton mass term, however, will pick up dependence on Y ’s in such a way that it will now be invariant under the following gauge transformation: @f @f g ðfðxÞÞ; @x @x Y  ðxÞ ! fÀ1 ðY ðxÞÞ g ðxÞ ! ¨ VI. THE NONLINEAR STUCKELBERG FORMALISM ¨ In this section we extend the Stuckelberg trick to full nonlinear order. This is a powerful tool with which to elucidate the nonlinear dynamics of massive gravity. It allows us to trace the breakdown in the linear expansion to strong coupling of the longitudinal mode. It also tells us about quantum corrections, the scale of the effective field theory and where it breaks down, as well as the nature of the Boulware-Deser ghost and w...
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This document was uploaded on 09/28/2013.

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