# The mean gauges in bimetric relativity

## TL;DR

This paper explores the use of mean gauges based on the geometric mean metric in bimetric relativity, demonstrating how they can improve gauge choices and potentially lead to a well-posed evolution system.

## Contribution

It introduces the application of geometric mean gauges in bimetric relativity and shows how they can be used to make arbitrary choices dynamically, enhancing system well-posedness.

## Key findings

- Mean gauges can be computed with respect to the geometric mean metric.
- Using the geometric mean metric allows for dynamic, rather than arbitrary, gauge choices.
- Potential to recast the bimetric system in a well-posed form.

## Abstract

The choice of gauge in numerical relativity is crucial in avoiding coordinate and curvature singularities. In addition, the gauge can affect the well-posedness of the system. In this work, we consider the mean gauges, established with respect to the geometric mean metric $h = g \, (g^{-1}f)^{1/2}$ in bimetric relativity. We consider three gauge conditions widely used in numerical relativity and compute them with respect to the geometric mean: The 1+log gauge condition and the maximal slicing for the lapse function of $h$, and the $\Gamma$-driver gauge condition for the shift vector of $h$. In addition, in the bimetric covariant BSSN formalism, there are other arbitrary choices to be made before evolving the system. We show that it is possible to make them by using the geometric mean metric, which is determined dynamically by the system, rather than using an arbitrary external metric, as in general relativity. These choices represent opportunities to recast the system in a well-posed form.