variances. (When the need arises to suspend the summation conventions for some non-tensorial indices, we shall explicitly mention it.) In expressions where there is a sum that the indices do not indicate, all sums in the expression are indicated explicitly by a summation symbol. Indices on the velocity-dominated metric and the velocity-dominated extrinsic curvature are raised and lowered with the velocity-dominated metric.

$d+1$ decomposition {#stepzero}
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Consider a solution to the Einstein’s equations following from (\[001\]), consisting of a Lorentz metric and matter fields on a $D$-dimensional manifold $M$ which is diffeomorphic to $(0,T) \times \Sigma$ for a $d$-dimensional manifold, $\Sigma$, such that the metric induced on each $t =$ constant hypersurface is Riemannian, for $t \in (0,T)$. Here $D$ is an integer strictly greater than two. Furthermore, consider a $d+1$ decomposition of the Einstein tensor, $G_{\alpha \beta}$, and the stress-energy tensor, $T_{\alpha \beta}$, with a Gaussian time coordinate, $t \in (0,T)$, and a local frame $\{e_a\}$ on $\Sigma$. Note that the frame $e_a = e_a^i(x) \partial_i$ is time-independent. The spacetime metric reads $ds^2 = - dt^2 + g_{a b}(t,x) e^a e^b$, where $e^a = e^a_i(x) d x^i$ (with $e^a_i e^i_b = \delta^a_b$) is the co-frame. Let $\rho = T_{00}$, 