Exact Solution to the "Auxiliary Extra Dimension" Model of Massive Gravity

TL;DR

This paper resolves the $u$-equation of the Auxiliary Extra-Dimension (AuXD) model for massive gravity exactly, producing a purely 4D action with a non-linear mass term $F(g^{-1} f)$ that depends on a boundary metric $f_{\mu\nu}$. It shows that, under the standard interpretation with flat $f$, the theory develops a ghost at the fourth order and beyond, despite matching ghost-free cubic terms in some constructions. While certain non-standard boundary choices or higher-curvature additions could mitigate the ghost to finite orders, these options conflict with the AuXD variation principle or alter the geometric basis of the model. Consequently, AuXD remains not fully ghost-free within its original setup, clarifying its relation to other ghost-free massive gravity formulations and highlighting the need for a framework extension to achieve complete stability.

Abstract

The "Auxiliary Extra Dimension" model was proposed in order to provide a geometrical interpretation to modifications of general relativity, in particular to non-linear massive gravity. In this context, the theory was shown to be ghost free to third order in perturbations, in the decoupling limit. In this work, we exactly solve the equation of motion in the extra dimension, to obtain a purely 4-dimensional theory. Using this solution, it is shown that the ghost appears at the fourth order and beyond. We explore potential modifications to address the ghost issue and find that their consistent implementation requires going beyond the present framework.