What does this equation mean?

What must therefore vanish, identically, is not the slope but the correctly defined curvature-frame component built from it. For any static metric of the form d$s^2$ = -U($\mathbf x$)^2\,d$x_0^2$ + $\delta_{ij}$\,d$x^i$ d$x^j$ — flat spatial slices, all the position dependence carried by U — the only nonzero Christoffel symbols are $\Gamma^0_{0i}$ = $\partial_i$ $\ln$ U = $\phi_{,i}$ and $\Gamma^i_{00}$ = $U^2$$\phi_{,i}$ , both following directly from the metric’s definition. Substituting into the Riemann tensor’s defining combination for the mixed component $R^i${}_{0j0} , every term built from purely spatial Christoffels vanishes because the spatial slices are exactly flat, and the two surviving pieces are $\partial_j$$\Gamma^i_{00}$ and -$\Gamma^i_{0\lambda}$$\Gamma^\lambda_{j0}$ . Expanding $\partial_j(U^2\phi_{,i})$ = $U^2$$\phi_{,i}$$\phi_{,j}$ + $U^2$$\phi_{,ij}$ and $\Gamma^i_{00}$$\Gamma^0_{j0}$ = $U^2$$\phi_{,i}$$\phi_{,j}$ , the two copies of $\phi_{,i}$$\phi_{,j}$ cancel exactly, leaving $R^i{}_{0j0} = U^2\left(\phi_{,ij} + \phi_{,i}\,\phi_{,j}\right)$. The function U appearing throughout this construction is exactly what the initial-value formulation of general relativity calls the lapse: the local rate at which proper time elapses across a chosen family of spatial slices, the same static-spacetime object standard treatments of Killing symmetries already build on [ 14 ] . Nothing in the derivation above depends on that name; it is worth stating only because it confirms that U is not a quantity invented for this network — it is the same function relativists already use to slice a spacetime into space-plus-time, borrowed here rather than redefined.