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.
Highlights
The shifts of g and f are given by the separation p and the position relative to the shift q of h
In the rest of this section we review some technical properties of the field equations in General relativity (GR) and the Hassan & Rosen (HR) bimetric theory
We summarize some important properties of the elementary symmetric polynomials that are used throughout this work
Summary
We first consider a general class of bimetric actions consisting of two Einstein-Hilbert terms SEH[g] and SEH[f ] each coupled to separate matter fields, and the interaction term Lign,ft that depends on the scalar invariants of the operator gμρfρν (written g−1f in matrix notation).. By varying the action with respect to g and f , one obtains the field equations, Ggμν = κgVgμν + κgTgμν , Gf μν = κf Vf μν + κf Tf μν ,. The dependence of Lign,ft only on g−1f , combined with the definitions of Vg and Vf from (1.29), implies the following algebraic identities, μρ. It can be proved that the differential identity (1.30c) is a consequence of the definitions (1.29) provided that Lign,ft is a scalar function of g−1f. The equations (1.31) will be hereinafter referred to as the bimetric conservation law
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