On the Black Holes in alternative theories of gravity: The case of non-linear massive gravity

I derive general conditions in order to explain the origin of the Vainshtein radius inside dRGT. The set of equations, which I have called "Vainshtein" conditions are extremal conditions of the dynamical metric ($g_{\mu\nu}$) containing all the degrees of freedom of the theory. The Vainshtein conditions are able to explain the coincidence between the Vainshtein radius in dRGT and the scale $r_0=\left(\frac{3}{2}r_s r_\Lambda^2\right)^{1/3}$, obtained naturally from the Schwarzschild de-Sitter (S-dS) space inside General Relativity (GR). In GR, this scale was interpreted as the maximum distance in order to get bound orbits. The same scale corresponds to the static observer position if we want to define the black hole temperature in an asymptotically de-Sitter space. In dRGT, the scale marks a limit after which the extra degrees of freedom of the theory become relevant.