Static vacuum conformal spatial metric ($d{s}^2=-e^{2U}d{t}^2+e^{-2U}{\gamma }_{ij}d{x}^{i}d{x}^j$)

Rescaling the spatial metric by the squared static lapse function gives a useful three-dimensional quotient metric $\gamma$. In Vacuum Einstein equations, its Ricci tensor obeys $R_{ij}(\gamma )=2{\partial }_{i}U{\partial }_{j}U$, while $U$ is harmonic. The contracted Bianchi identity recovers the harmonic equation without dividing at critical points.

On a complete nonsingular spatial quotient with no inner boundary and standard asymptotic flatness, a bounded harmonic function tending to zero on all ends is zero. The static vacuum conformal spatial metric is then Ricci flat; three-dimensional curvature from the Ricci tensor makes it flat. Completeness and an ordinary Euclidean asymptotic end exclude nontrivial flat quotients, yielding Minkowski spacetime with the standard global time coordinate.