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The Vainshtein conditions: The Vainshtein mechanism in terms of Stückelberg functions

Here I develop the simplest method in order to evaluate whether or not the Vainshtein mechanism can operate for a given set of parameters in a given solution. The method is based on the formulation of the mechanism in terms of the Stückelberg functions given in Int.J.Mod.Phys. D24 (2015) 1550022 and arXiv:1305.0475 [gr-qc]. In such a case, the Vainshtein scale appears as an extremal condition of the dynamical metric. If we fix the graviton mass, we can define the effective Vainshtein scale. Then for parameters where the Vainshtein scale vanishes or becomes smaller than the gravitational radius, the mechanism should be absent. At the other extreme, if the Vainshtein scale is finite or infinite, then the mechanism can operate. For consistency, if we define the Vainshtein scale as an invariant, then we should expect the effective graviton mass to become very large when the Vainshtein mechanism operates. On the other hand, if the mass scale tends to zero, then the extra-degrees of freedom are free to propagate. For clarity, here I analyze the effective mass behavior for the different type of modes.