Geometric mean of bimetric spacetimes. (English) Zbl 1481.83084
Summary: 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.
MSC:
| 83D05 | Relativistic gravitational theories other than Einstein’s, including asymmetric field theories |
| 53Z05 | Applications of differential geometry to physics |
| 58J47 | Propagation of singularities; initial value problems on manifolds |
| 54B15 | Quotient spaces, decompositions in general topology |
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