Connected rotating degenerate vacuum comparison data

Work in ZFC. A connected rotating degenerate vacuum comparison datum consists of the time-oriented stationary four-dimensional Lorentzian data $(M,g,T)$ of Stationary Lorentzian data and tensor conventions, an asymptotically flat selected end, and the $I^{+}$-regularity, analyticity and causal-exterior conventions of Staticity, axisymmetry and black-hole regularity hypotheses and Causal exterior, chronological boundaries and Cauchy data. Require $\mathrm{Ric}(g)=0$. Require that the future event horizon $H^{+}=I_M^{+}(M_{\mathrm{e}\mathrm{x}\mathrm{t}})\cap {\mathrm{\partial }}_{M}D$ be nonempty and connected, and that it be a smooth Killing horizon for a specified nonzero generator $K$ with $d(g(K,K))=-2\kappa K^{\mathrm{♭}}$ on its nonzero-generator part and $\kappa =0$ there. Require that $T$ fail to be tangent to the null normal line somewhere on $H^{+}$; this is rotation relative to $T$. No bifurcation surface is part of these data.

Here $D$ is the domain of outer communications and $M_{\mathrm{e}\mathrm{x}\mathrm{t}}$ the flow-saturated selected end. $I^{+}$-regularity requires completeness of $T$, global hyperbolicity of $D$, and the finite-end acausal section and once-only horizon-generator cross-section specified by the comparison definition. Connectedness is imposed on the horizon, not merely on that section. These are predicates on supplied data, not assertions that the data exist.

Given: The supplied geometric objects and the displayed requirements.

Goal: Verify that the comparison class is a well-defined predicate without assuming a bifurcation surface or deriving axial symmetry.

1.1 The causal and comparison interfaces determine $M_{\mathrm{e}\mathrm{x}\mathrm{t}}$, $D$ and $H^{+}$, so nonemptiness and connectedness are definite topological predicates. The stated completeness, regularity and analyticity requirements are likewise the explicit predicates in [F1]. [given, F1]

2.1 The horizon interface fixes the surface-gravity equation and rotation condition for the supplied fields. It therefore makes $\kappa =0$ and failure of stationary tangency unambiguous. Adding the tensor equation $\mathrm{Ric}(g)=0$ selects vacuum geometries. None of these operations produces an axial field or a bifurcation surface; those would be further results. [given, F1, F2] ∎