The Vainshtein conditions: The Vainshtein mechanism in terms of St\"uckelberg functions

TL;DR

This paper presents a straightforward method to determine the operation of the Vainshtein mechanism using St"uckelberg functions, linking the Vainshtein scale to the effective graviton mass and solution parameters.

Contribution

It introduces a simple, invariant-based approach to evaluate the Vainshtein mechanism's presence via extremal conditions of the dynamical metric and effective graviton mass behavior.

Findings

Vainshtein scale as an extremal condition of the metric

Effective graviton mass correlates with the mechanism's operation

Mechanism absent when Vainshtein scale vanishes or is very small

Abstract

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 St\"uckelberg 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.