# Bimetric interactions based on metric congruences

## TL;DR

This paper introduces a new approach to defining spin-2 interactions in massive gravity and bigravity using a congruence matrix, providing insights into the mathematical structure and equivalences of different formulations.

## Contribution

It presents a novel construction of bimetric interactions via a congruence matrix, establishing the uniqueness of the primary square root solution and linking metric and vielbein formulations.

## Key findings

- Primary square root matrix is the unique power series solution.
- Shift vector redefinition arises naturally from equations of motion.
- Bimetric congruence formulation is algebraically equivalent to the vielbein approach.

## Abstract

In massive gravity and bigravity, spin-2 interactions are defined in terms of a square root matrix that involves two metrics. In this work, the interactions are constructed using a congruence matrix between the metrics. It is established that the primary square root matrix function is the only power series solution to the equations of motion for the congruence. Moreover, the shift vector redefinition that is used in the bimetric ghost-free proofs follows from the $N+1$ form of the equations of motion. The analysis also gives an insight into the vielbein formulation of spin-2 interactions since the bimetric formulation in terms of a congruence is algebraically equivalent to the unconstrained vielbein formulation.