Nonzero constant surface gravity and global bifurcate extension

Let $(M,g,\mathrm{\Sigma },T,\varphi ,\xi ,\chi ,N)$ satisfy the complete Rácz–Wald collapse-class contract Precise Rácz–Wald global collapse class, including the exact single-end asymptotic conditions, global hyperbolicity, absence of a white hole and both specified complete actions. Suppose the signed surface gravity of the future horizon-normal generator $\xi$ is a nonzero constant on $N$, with the conventions of Bifurcate Killing horizons, rotation and normalization.

Then there is a smooth spacetime $(M^{\ast },g^{\ast })$ and an isometric embedding of the whole $(M,g)$ into it such that the image of $N$ is a proper subset of a bifurcate Killing horizon. The original group ${\chi }_u$ extends to $(M^{\ast },g^{\ast })$, and the extension can be chosen with a wedge-reflection isometry about its bifurcation surface. In the local horizon coordinates this reflection sends $(U,V,x^1,x^2)$ to $(-U,-V,x^1,x^2)$.

This is a global embedding conclusion, not a claim that $M^{\ast }$ is globally hyperbolic. Extension of additional tensor fields is a separate question, as stated at the start of §5 of the source. No Einstein equation or matter-field extension is asserted. The completeness and horizon-normal property of $\xi$ are hypotheses even if $\xi \mathrm{\ne }T$.

Apply Rácz–Wald Theorem 4.2 to [F1] and the nonzero-constant hypothesis [F2]. It supplies the isometric extension of the whole spacetime, the proper-subset bifurcate horizon, the extended group ${\chi }_u$, and wedge reflection. The source construction identifies the reflection as $(U,V)\mapsto (-U,-V)$. This is an external theorem input, not a locally reconstructed proof. Its §5 separates the further problem of extending invariant tensor fields; no tensor extension follows from this item alone.