Geometric mean of bimetric spacetimes

• Published in: arXiv.org
• Main Author: Kocic, Mikica
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 14.04.2019
• Subjects: Cones; Mathematical analysis; Matrix methods; Parameterization; Physics - General Relativity and Quantum Cosmology; Physics - High Energy Physics - Theory; Quadratic forms
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1803.09752

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.