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 $T(M) = \mathcal{D} \oplus \mathbb{R} \nabla \mathcal{T}$ of $M$, as exhibited by A.N. Bernal & M. Sanchez. We give sufficient geometric conditions for the existence of exact solutions $- \beta d \mathcal{T} \otimes d \mathcal{T} + \overline{g}$ to mixed field equations in free space. We linearize and solve the mixed field equations $\mathrm{Ric}_\mathcal{D} (g)_{\mu\nu} - \rho_\mathcal{D} (g) \, g_{\mu\nu} = 0$ for empty space, where $\rho_\mathcal{D} (g)$ is the mixed scalar curvature of foliated spacetime $(M, \mathcal{D})$. If $g_\epsilon = g_0 + \epsilon \gamma$ is a solution to the linearized field equations then each leaf of $\mathcal D$ is totally geodesic in $(\mathbb{R}^4 \setminus \mathbb{R}, g_\epsilon)$ to order $O(\epsilon )$. We derive the equations of motion of a material particle in gravitational field $g_{\mu\nu}$ governed by the mixed field equations $\mathrm{Ric}_\mathcal{D} (g)_{\mu\nu} - \rho_\mathcal{D}(g) \, \omega_\mu \omega_\nu - \Lambda g_{\mu\nu} = 2\pi \kappa c^{-2} \{ T_{\mu\nu} -(1/2) T g_{\mu\nu}\}$. In the weak field ($\epsilon << 1$) and low velocity ($\| \mathbf{v} \| /c << 1$) limit the motion equations are $d^2 \mathb{ r}/dt^2 = \nabla \phi + \mathbf{F}$ where $\phi = (\epsilon /2 ) c^2 \gamma_{00}$.