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.

我们使用几何平均来参数化 Hassan-Rosen 无鬼双度量理论中的度量，并提出初始值问题。两个正定对称矩阵的几何平均值是一个完善的数学概念，它可以在某些条件下扩展到具有洛伦兹签名的二次形式，例如度量g和f。在这种情况下，几何平均度量h的零锥位于g和f的零锥的中间，表现为双度量时空的几何平均值。基于h的参数化确保了无重影双度量相互作用势中平方根的真实性。随后，我们推导出标准n + 1 分解在适应几何平均和状态的框架中的初始值问题，即演化方程、约束和约束方程的保存。