Non-rotating staticity and Schwarzschild uniqueness

In the non-rotating case, tangency to both transverse horizon sheets places $T$ in $TS$, while non-rotation places it in the null-generator line. The positive-definite metric on $TS$ therefore makes $T=0$ there. If $T{\mathrm{\mid }}_S=0$ is assumed directly, this conclusion is already given. The field is nontrivial because it is timelike in the asymptotic end, so [F1] identifies the given sheets as Killing horizons for $T$ near $S$.

Put $C={\mathrm{\Sigma }}_0\cap E$. Apply [F2] to the connected selected end and the given exterior Cauchy section. Then $E$ and $C$ are connected, every additional ambient closure point of $C$ lies in the double chronological boundary $S$, and $C$ is the component containing the original stationary end. Its closure $C\cup S$ is complete up to $S$, with a compact core and one asymptotically Euclidean end. The radial outer-component description supplied by [F2] uses an adapted identification, which may differ from the original section identification and from the physical end chart. By [F3], pass to the working spacetime $M^{\mathrm{\prime }}$ containing the same $E$ and a neighborhood of $S$. Choose the working-section parametrization supplied there and denote its radial parameter by $\rho$. It has an embedded acausal edgeless section ${\mathrm{\Sigma }}^{\mathrm{\prime }}\cong \{\rho >1\mathrm{/}2\}$, with $C=\{\rho >1\}$, unchanged near $S$ and on the physical end, for which ${\mathrm{\Sigma }}^{\mathrm{\prime }}\cap I_{M^{\mathrm{\prime }}}^{+}(M^{\mathrm{e}\mathrm{n}\mathrm{d}})={\mathrm{\Sigma }}^{\mathrm{\prime }}\cap I_{M^{\mathrm{\prime }}}^{-}(M^{\mathrm{e}\mathrm{n}\mathrm{d}})=C$. The parameter $\rho$ labels these topological section regions; the original end coordinates continue to supply the metric estimates. Its DOC is exactly $E$, its full double chronological boundary is $S$ by the working-collar result in [F3], and its local chronological horizon sheets are unchanged. Take the connected component containing $E$ and its collar if needed; all these data remain in it.

By [F4] and step 1.1, $M_0=M_{\mathrm{A}\mathrm{D}\mathrm{M}}>0$ on this nonmaximal compact-core section. Match the contract of [F5] in the working spacetime: the restricted geometry is smooth, oriented, time-oriented and vacuum with the same Killing field; the unchanged end has future $T={\mathrm{\partial }}_t$, the original $O_6$ expansions and $t=0$ section alignment, with its mass now verified positive. Its DOC is the same globally hyperbolic $E$; the complete exterior stationary orbits meet $C$, since they meet ${\mathrm{\Sigma }}_0$ within $E$. Step 1.2 supplies the required partial Cauchy section, radial parametrization and both exact chronological section equalities. By [F3] and step 1.2, the full double chronological boundary in $M^{\mathrm{\prime }}$ is exactly $S$, and the two local sheets remain smooth, transverse, null, nonexpanding and tangent to $T$. Thus GR and SBS, including the zero-field condition from step 1.1, hold with no additional original hypothesis.

Apply the full nonrotating conclusion of [F5] to the data verified in step 2.1. It gives staticity throughout $E$ and a global isometry of $E$ with the Schwarzschild exterior of mass $M_0>0$, understood with the metric, exterior domain and geometrized units of [F6]. This imports the source’s completion and maximal-slice staticity argument, including its final classification; no end graph or stronger differentiated bound for a selected auxiliary slice is inferred here. Since the working restriction and the source’s completion preserve exactly the original $E$ with its metric and stationary field, the conclusion is the required one for the original exterior.