Geometric mean of bimetric spacetimes

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.