For every globally hyperbolic spacetime $M$ we derive new gravitational field equations embodying the smooth Geroch infinitesimal splitting $T(M) = {\mathcal D} \oplus \nabla {\mathcal T}$ of $M$, as exhibited by A.N. Bernal \& M. Sánchez, \cite{BeSa2}. We solve the linearized field equations ${\rm Ric}_{\mathcal D} (g)_{\mu\nu} - \rho_{\mathcal D}(g) \, g_{\mu\nu} = 0$ for the empty space. If $g_\epsilon = g_0 + \epsilon \gamma$ is a solution to the linearized ($\epsilon << 1$) field equations then each leaf of $\mathcal D$ is totally geodesic in $({\mathbb R}^4 \setminus {\mathbb R}, g_\epsilon )$ to order $O(\epsilon )$.