Limit m 0 mp 1 t 1 5 t fixed mp 79 75 all

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Unformatted text preview: eraction terms. The term suppressed by the smallest scale is the cubic scalar term n ¼ 3, nA ¼ nh ¼ 0, which is suppressed by the scale Ã5 ¼ ðMP m4 Þ1=5 , ^ The free graviton coupled to the source via ð1=MP Þh0 T  also survives the limit, as does the free decoupled vector. We can now understand the origin of the Vainshtein radius at which the linear expansion breaks down around heavy point sources. The scalar couples to the source through the We also get a whole slew of interaction terms, third order and higher in the fields, suppressed by various scales. We always assume m < MP .  always appears with two derivatives, A always appears with one derivative, and h always appears with none, so a generic term, with nh powers of h , nA powers of A , and n powers of , reads 2 $ m2 MP hnh ð@AÞnA ð@2 Þn 4Ành À2nA À3n ^ nh $ à ^ ^ h ð@AÞnA ð@2 Þn ; (7.4) where the scale suppressing the term is à ¼ ðMP mÀ1 Þ1= ; ¼ 3n þ 2nA þ nh À 4 : n þ nA þ nh À 2 Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 Kurt Hinterbichler: Theoretical aspects of massive gravity 694 ^ trace, ð1=MP ÞT . To linear order around a central source of mass M, we have M1 ^ : $ MP r (7.11) The nonlinear term is suppressed relative to the linear term by the factor ^ @4  M1 $ : M P à 5 r5 Ã5 5 5 (7.12) Nonlinearities become important when this factor becomes of order 1, which happens at the radius rV $  M MP 1=5   1 GM 1=5 $ : Ã5 m4 (7.13) When r & rV , linear perturbation theory breaks down and nonlinear effects become important. This is exactly the Vainshtein radius found in Sec. V.B by directly calculating the second order correction to spherical solutions. In the decoupling limit, the gauge symmetries (7.7) reduce to their linear forms, ^ ^ h ¼ @  þ @  ; ^ ^ A ¼ @ Ã;  ¼ 0: (7.14) Even though  is gauge invariant in the decoupling limit, the fact that it always comes with two derivatives means that the global Galileon symmetry (7.8) is still present, as is the shift symmetry on A . B. Ghosts Note that the Lagrangian (7.10) is a higher derivative action, and its equations of motion are fourth order. This means that this Lagrangian actually propagates two Lagrangian degrees of freedom rather than one, since we need to specify twice as many initial conditions to uniquely solve the fourth order equations of motion (de Urries and Julve, 1998), and by Ostrogradski’s theorem (Ostrogradski, 1850; Woodard, 2007), one of these degrees of freedom is a ghost. The decoupling limit contains 6 degrees of freedom: two in the massless tensor, two in the free vector, and two in the scalar. This matches the number of degrees of freedom in the full theory as determined in Sec. V.C, so the decoupling limit we have taken is smooth. The extra ghostly scalar degree of freedom is the Boulware-Deser ghost. Note that at linear order, the higher derivative scalar terms for the scalar are not visible, so the linear theory has...
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