RevModPhys.84.671

In addition the equations of motion remain second

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Unformatted text preview: e-Deser ghost, around any ¨ background. This can also been seen in the Stuckelberg language (de Rham, Gabadadze, and Tolley, 2011a). C. The Ã3 Vainshtein radius We now derive the scale at which the linear expansion breaks down around heavy point sources in the Ã3 theory. To linear order around a central source of mass M, the fields still have their usual Coulomb form M1 ^^ : ; h $ MP r (8.20) The nonlinear terms in Eqs. (8.10) or (8.18) are suppressed relative to the linear term by a different factor than in the Ã5 theory, ^ @2  M1 $ : (8.21) 3 M P Ã 3 r3 Ã3 3 Nonlinearities become important when this factor becomes of the order of 1, which happens at the radius    M1 GM 1=3 $ : (8.22) r ð3 Þ $ V MP Ã 3 m2 This is parametrically larger than the Vainshtein radius found in the Ã5 theory. It is important that the decoupling limit Lagrangian is ghost free. To see what could go wrong if there were a ghost, ^ expand around some spherical background  ¼ ÈðrÞ þ ’ 14 This is contrary to Creminelli et al. (2005) who claims that a ghost is still present at quartic order. As remarked, however, by de Rham and Gabadadze (2010a), they arrive at the incorrect decoupling limit Lagrangian, which can be traced to a minus sign mistake in their Eq. 5, which should be as in Eq. (8.4). Kurt Hinterbichler: Theoretical aspects of massive gravity 700 and similarly for h . The cubic coupling and quartic couplings could possibly give fourth order kinetic contributions of the schematic form, respectively, These would correspond to ghosts with r-dependent masses, We can see the Vainshtein mechanism at work already by calculating the ratio of the fifth force due to the scalar to the force from ordinary Newtonian gravity, 8 r 3=2 r ( r ð3 Þ ; ^ V F 0 ðrÞ=MP < ðrðV3Þ Þ ¼ $ (8.29) 2 : FNewton M=MP r2 1 r ) r ð3 Þ : V Ã3 Ã6 3 3 ; ; (8.24) È [email protected] È or, given that the background fields are similar to È $ ðM=MP Þð1=rÞ,  2 M MP Ã 6 r4 : (8.25) m2 ðrÞ $ P Ã3 r; ghost 3 3 M M There is a gravitational strength fifth force at distances much farther than the Vainshtein radius, but the force is suppressed at distances smaller than the Vainshtein radius. This suppression extends to all scalar interactions in the presence of the source. To see how this comes about, we study perturbations around a given background solution ÈðxÞ. Expanding 1 È ð @2 ’ Þ 2 ; Ã3 3 1 È @ 2 È ð @2 ’ Þ 2 : Ã6 3 (8.23) m2 ðrÞ $ ghost Thus the ghost mass sinks below the cutoff Ã3 at the radius   1=2 M1 M 1 rð3Þ $ ; : (8.26) ghost MP Ã 3 MP Ã3 As happened in the Ã5 theory, these radii are parametrically larger than the Vainshtein radius. This is a fatal instability which renders the whole nonlinear region inaccessible, unless we lower the cutoff of the effective theory so that the ghost stays above it, in which case unknown quantum corrections would also kick in at $rð3Þ , swamping the entire nonlinear ghost Vainshtein region. D. Th...
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