Geometric mean of bimetric spacetimes

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.

Automated summary

The article introduces the use of geometric mean to parametrize metrics in the Hassan-Rosen ghost-free bimetric theory and solves the initial-value problem. The parametrization based on geometric mean metric ensures the reality of the square root in the ghost-free bimetric interaction potential. It also mentions the standard n + 1 decomposition derived in a frame adapted to the geometric mean.