RevModPhys.84.671

And the other must be sent to zero to prevent the

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Unformatted text preview: Oð2 Þ. Going through the same procedure, we find for the solution, when mr ( 1,   8 GM 1 GM 1À þ ÁÁÁ ; (5.31) BðrÞ À 1 ¼ À 3r 6 m 4 r5 CðrÞ À 1 ¼ À AðrÞ À 1 ¼  C. Nonlinear Hamiltonian and the Boulware-Deser mode  8 GM GM 1 À 14 4 5 þ Á Á Á ; 3 m 2 r3 mr   4 GM GM 1 À 4 4 5 þ ÁÁÁ : 3 4m2 r3 mr (5.32) (5.33) The dots represent higher powers in the nonlinearity parameter . We see that the nonlinearity expansion is an expansion in the parameter rV =r, where   GM 1=5 (5.34) rV  m4 is known as the Vainshtein radius. As the mass m approaches 0, rV grows, and hence the radius beyond which the solution can be trusted gets pushed out to infinity. As argued by Vainshtein (1972), this perturbation expansion breaks down and says nothing about the true nonlinear behavior of massive gravity in the massless limit. Thus there was reason to hope that the vDVZ discontinuity was merely an artifact of linear perturbation theory, and that the true nonlinear solutions showed a smooth limit (Vainshtein, 1972; Deffayet et al., 2002; Porrati, 2002; Gruzinov, 2005). One might hope that a smooth limit could be seen by setting up an alternative expansion in the mass m2 . We take a solution to the massless equations, the ordinary Schwarzschild solution, with metric coefficients B0 , C0 , and A0 , and then plug an expansion, BðrÞ ¼ B0 ðrÞ þ m2 B1 ðrÞ þ m4 B2 ðrÞ þ Á Á Á ; CðrÞ ¼ C0 ðrÞ þ m2 C1 ðrÞ þ m4 C2 ðrÞ þ Á Á Á ; (5.35) AðrÞ ¼ A0 ðrÞ þ m2 A1 ðrÞ þ m4 A2 ðrÞ þ Á Á Á ; 8 Naively, it is a second order equation in A1 and B1 , first order in C1 , and we think this requires five initial conditions, but, in fact, it is a degenerate system, and there are second class constraints bringing the required boundary data to 2. Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 into the equations of motion. Collecting powers of m yields a new perturbation equation at each order, but in this case the equations are generally nonlinear. Even the equation we obtain at Oðm2 Þ for the first correction to Schwarzschild is nonlinear, so working with this expansion is much more difficult than working with the linearized expansion. The linearity expansion is valid is the region r ) rV . If general relativity is restored at distances near the source, the mass expansion should be valid in the opposite regime r ( rV , and the full solutions should interpolate between the two expansions. There have been several extensive numerical studies of the full nonlinear solutions in Damour, Kogan, and Papazoglou (2003), in the decoupling limit in Babichev, Deffayet, and Ziour (2009b), and more extensively in the full theory in Deffayet (2008), Babichev, Deffayet, and Ziour (2009a, 2010), with the final result being that the nonlinearities can, in fact, work to restore continuity with GR. We see later the mechanism by which this occurs. Some analytic solutions in various cases are given by Berezhiani et al. (2008),...
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