Geometric mean of bimetric spacetimes

TL;DR

This paper introduces a novel parametrization of bimetric spacetimes using the geometric mean of two metrics, facilitating the formulation of the initial-value problem in ghost-free bimetric theory.

Contribution

It proposes a new metric parametrization based on the geometric mean, ensuring the reality of the interaction potential and deriving the initial-value formulation.

Findings

Geometric mean metric lies between the null cones of the two metrics.

The parametrization guarantees the reality of the square root in the interaction potential.

Derived the evolution equations and constraints for the initial-value problem.

Abstract

We use the geometric mean to parametrize metrics in the Hassan-Rosen ghost-free bimetric theory and pose the initial-value problem. The geometric mean of two positive definite symmetric matrices is a well-established mathematical notion which can be, under certain conditions, extended to quadratic forms having the Lorentzian signature, say metrics $g$ and $f$. In such a case, the null cone of the geometric mean metric $h$ is in the middle of the null cones of $g$ and $f$ appearing as a geometric average of a bimetric spacetime. The parametrization based on $h$ ensures the reality of the square root in the ghost-free bimetric interaction potential. Subsequently, we derive the standard $n+1$ decomposition in a frame adapted to the geometric mean and state the initial-value problem, that is, the evolution equations, the constraints, and the preservation of the constraints equation.