Note on Covariant St\"uckelberg Formalism and Absence of Boulware-Deser Ghost in Bi-gravity

TL;DR

This paper uses covariant St"uckelberg formalism to confirm the absence of the Boulware-Deser ghost in bi-gravity theories at all orders in the decoupling limit, ensuring theoretical consistency.

Contribution

It extends previous analyses by demonstrating the ghost-free nature of bi-gravity at arbitrary perturbation orders using covariant St"uckelberg methods.

Findings

No higher time derivatives in equations of motion at leading order.

Boulware-Deser ghost does not appear at any order in the decoupling limit.

The formalism confirms the theoretical consistency of bi-gravity models.

Abstract

The covariant St\"{u}ckelberg formalism is applied to bi-gravity in order to revisit the issue of absence of the Boulware-Deser (BD) ghost. We first confirm that the leading order action in the decoupling limit for helicity-2 modes of metrics and helicity-0 mode of St\"{u}ckelberg perturbations does not lead to higher time derivative in equations of motion, which suggests the absence of the BD ghost. By extending this method, we reconfirm that the BD ghost does not appear for arbitrary order of the perturbations at the decoupling limit in bi-gravity.