Mixed gravitational field equations on globally hyperbolic spacetimes

For every globally hyperbolic spacetime M, we derive new mixed gravitational field equations embodying the smooth Geroch infinitesimal splitting of T(M) = D ⊕ ℝ T of M, as exhibited by Bernal and Sánchez (2005 Commun. Math. Phys. 257 43-50). We give sufficient geometric conditions (e.g. is isoparametric and D is totally umbilical) for the existence of exact solutions -β dT ⊕ +ḡ to mixed field equations in free space. We linearize and solve the mixed field equations Ric_D(g)μν - ρD(g)g_μν = 0 for empty space, where ρD(g) is the mixed scalar curvature of foliated spacetime (M,D)(due to Rovenski (2010 arXiv:1010.2986 v1[math.DG])). If g_ε = g ₀ + εγ is a solution to the linearized field equations, then each leaf of D is totally geodesic in (ℝ⁴\ℝ, g ε) to order O(ε). We derive the equations of motion of a material particle in the gravitational field g_μν governed by the mixed field equations Ric_D(g)μν - ρD(g)ω_μ ω_ν - λg_μν= 2πκc ^-2{T_μν -1/2T g_μν}. In the weak field (ε ≪ 1) and low velocity (∥v∥/c ≪ 1) limit, the motion equations are d²r/dt² = ∇φ + F, where φ = (ε/2)c²γ₀₀.