# Algebraic Properties of Einstein Solutions in Ghost-Free Bimetric Theory

## TL;DR

This paper investigates the algebraic structure of Einstein solutions in ghost-free bimetric theory, revealing conditions under which solutions are proportional or block proportional with limited eigenvalues.

## Contribution

It generalizes previous results by analyzing arbitrary parameters and potential forms, establishing new algebraic constraints on solutions in bimetric theory.

## Key findings

- Einstein solutions imply proportional or block proportional solutions.
- Solutions have at most two distinct eigenvalues in the interaction square root.
- The results hold for general parameter choices and potential forms.

## Abstract

A known fact is that an Einstein solution in one sector in ghost-free bimetric theory implies an Einstein solution in the other sector. Earlier studies have also shown that some classes of bimetric models necessitate proportional solutions between the sectors. Here we consider a general setup of the parameters in the theory as well as the general algebraic form of the potential. We show that, if one sector has an Einstein solution, the solutions are either proportional or block proportional with at most two different eigenvalues in the square root governing metric interactions.