Omit from now on without loss of generality to linear

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Unformatted text preview: G½hv0 ; p þ qjT  jv; piŠ; L  ¼   q2 À   q2 À  q q   À  q q þ  q q þ  q q ;  (B5) Xavier Bekaert, Nicolas Boulanger, and Per A. Sundell: How higher-spin gravity surpasses the spin- . . . which is nothing but the Fourier transform of the symmetric ~  differential operator G  acting on the spin-two field h in the linearized (in h ) Einstein equations ~  G  h ¼ T ð’s ; ’s Þ þ Oð2 Þ; (B6) where T ð’s ; ’s Þ is the tensor bilinear in the spin-s field ’s that gives the cubic 2-s-s vertex in the action principle S½h ; ’s Š ¼ SPF ½h Š þ SFr ½’s Š Z D þ d xh T  ð’s ; ’s Þ þ Oð2 Þ: 2 (B7) To this same order in the metric fluctuation, a necessary condition is given by Porrati (2008) for the consistency of the gravitational interactions of high-spin massless particles: hv; p þ qjT  jvs ; pi ¼ L  Á ðqÞ (B8) with Á ðqÞ analytic in a neighborhood of q ¼ 0. Equation (B4) provided Porrati the most general condition for the decoupling of the so-called spurious polarization vs (that we call sometimes ‘‘pure-gauge’’ states) from the S-matrix amplitudes. Decoupling occurs when one can reabsorb the change in the matrix element due to the substitution v ! v þ vs with a local field redefinition of the graviton field. In the Lagrangian language, this can be seen to originate from the requirement of gauge invariance of the cubic action R Sð1Þ :¼ 1 dD xh T  ð’s ; ’s Þ under linearized gauge trans2 formations ð0Þ h ¼ [email protected] Þ ; (B9) ð0Þ ’1 ÁÁÁs ¼ [email protected]1 2 ÁÁÁs Þ (B10) up to terms that vanish on the surface of the free field equations: ð0Þ Sð1Þ þ ð1Þ Sð0Þ ¼ 0; (B11) where Sð0Þ denotes the free part of the action and ð1Þ denotes the gauge transformations taken at linear order in the field fh; ’g. The above equation can be rewritten as Z  Sð1Þ Sð1Þ þ ð0Þ ’1 ÁÁÁs dD x ð0Þ h h ’1 ÁÁÁs  ~ þ ð1Þ h G  h þ ð1Þ ’1 ÁÁÁs  Sð0Þ ¼ 0: ’1 ÁÁÁs If, as assumed in the S-matrix approach, one takes the spin-s particle on shell, then one sets Sð0Þ =’1 ÁÁÁs to zero. If, in addition, one takes the Euler-Lagrange derivative of the result with respect to the gravitational field, noting that the only structure for ð1Þ h which can contribute to Eq. (B11) with R Sð1Þ ¼ 1 dD xh T  ð’s ; ’s Þ is ð1Þ h ¼ R ð’s ; s Þ, one 2 finds ~  T ð’s ; ð0Þ ’s Þ þ G R ð’s ; s Þ ¼ 0 Rev. Mod. Phys., Vol. 84, No. 3, July–September 2012 (B12) 1007 which is (up to a convention of sign in front of the Fierz-Pauli R ~  action SFP ¼ 1 h G h ) the translation of Eq. (B8) in 2 Lagrangian language. Together with the principle of equivalence (B2), Eq. (B8) was the main assumption of the work (Porrati, 2008). We see that this condition (B8) is derived from the main equation (B11) in the Lagr...
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This document was uploaded on 09/28/2013.

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