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Analytic extremal and multiple-horizon comparison contracts

Statement

Two independent source comparison results are retained with separate hypotheses.

(1) A four-dimensional stationary rotating asymptotically flat analytic vacuum black hole with one degenerate horizon is extremal Kerr under the source's global black-hole regularity conditions. The precise connected $I^{+}$ -regular comparison data are Connected rotating degenerate vacuum comparison data. No bifurcation sphere is assumed for a degenerate generator.

(2) The stationary axisymmetric vacuum balance theorem excludes a regular configuration with $N > 1$ horizons when its singular harmonic map belongs to the explicit admissible mixed-rod class Exact mixed-rod boundary class for the balance theorem and every degenerate component has nonzero angular momentum. In fact each finite inter-horizon axis interval has strictly negative logarithmic angle defect. This allows arbitrary mixtures of nondegenerate rods and degenerate punctures and puts no angular-momentum condition on the nondegenerate rods.

Neither contract by itself treats non-rotating degenerate vacuum horizons outside its stated class or proves smooth nonanalytic uniqueness. To apply (2) to a geometric spacetime one must verify its membership in that class; regularity and axisymmetry are not substituted for the class's energy, trace and puncture requirements.

Facts & Assumptions

[A1]

Work in ZFC, The Axiom of Choice. Vacuum general relativity means the source-free zero-cosmological-constant specialization of Adopted Einstein field equation in the direct-equation presentation, so the geometric field equation is $Ric (g) = 0$ .

[F1]

The connected rotating degenerate comparison hypotheses are Connected rotating degenerate vacuum comparison data. The finite rod labels, twist gauge, reconstruction and complete admissibility conditions are Ordered horizon rods and twist-jump labels, Harmonic-map and Weyl reconstruction comparison data and Exact mixed-rod boundary class for the balance theorem. A regular weight-one axis has zero logarithmic angle defect by A regular weight-one axis has zero logarithmic angle defect.

[F2]

The exact uniqueness and balance results, and the more limited global-representation and degenerate-local-control inputs, are imported below with their separate hypotheses. The latter inputs are recorded without claiming that a generic geometric spacetime has passed the full HKWX admissibility test.

[F3]

The variational premise (V) and conditions (A1)–(A3) in the mixed-rod definition are the hypotheses of HKWX §2.1. Proposition 2.1(i)–(iv) supplies the asymptotic estimates needed by Theorem 1.1; (H) records the regular horizon-interior behavior. This regularity result is an imported premise, not an inference from harmonicity alone.

External input

The only four-dimensional stationary, rotating, asymptotically flat analytic vacuum black hole with a single degenerate horizon, under the source black-hole regularity assumptions, is the extremal Kerr black hole.

For $N > 1$ , let a stationary vacuum singular harmonic map with ordered black-hole rod data locally minimize renormalized energy with fixed rod traces and singularity data, and satisfy the source's ordinary-axis, nondegenerate-pole, regular horizon-interior, degenerate-puncture and infinity conditions. Reconstruct the Weyl–Papapetrou metric with $w \to 0$ and $α \to 0$ at infinity. If every degenerate component has $J_{i} \neq 0$ , every bounded axis rod has $b_{j} < 0$ for $2 \leq j \leq N$ . There is no regular asymptotically flat stationary axisymmetric multi-black-hole configuration in this class. No angular-momentum condition is imposed on nondegenerate rods.

HKWX Proposition 2.1 applies to harmonic maps satisfying all the hypotheses of §2.1, including local minimization of renormalized energy. Its parts (i)–(iv) provide ordinary-axis regularity and the puncture, nondegenerate-pole/interior and infinity estimates, with (2.12)–(2.26). These are the estimates identified just before Theorem 1.1. The current mixed-rod definition explicitly supplies those hypotheses and regular horizon-interior behavior; no additional asymptotic assumption is inferred from harmonicity alone.

A stationary, $I^{+}$ -regular, analytic electrovacuum spacetime with connected nonempty rotating degenerate future event horizon has a domain of outer communications isometrically diffeomorphic to that of a Kerr–Newman spacetime. Independently, in the stationary axisymmetric asymptotically flat $I^{+}$ -regular vacuum setting, near every degenerate horizon component the Ernst map is at bounded hyperbolic-plane distance from that of extreme Kerr with the same positive horizon cross-section area; its two adjacent axial twist constants differ in absolute value by $4 r_{0}^{2}$ , where $r_{0}$ is the positive area radius.

A stationary, axisymmetric, asymptotically flat, $I^{+}$ -regular, electrovacuum four-dimensional spacetime has a domain of outer communications isometric to one of the Weinstein solutions. If the event horizon is connected, that domain is a Kerr–Newman domain of outer communications. The preceding reduction discussion states that global isothermal coordinates and controlled boundary conditions extend to configurations with degenerate components.

Source notes

The uniqueness and balance contracts retain separate source hypotheses. The current mixed-rod definition supplies the variational and boundary hypotheses used by HKWX Proposition 2.1, which supplies the asymptotics used by its balance theorem. A geometric consumer must still establish that its spacetime yields data in this class and must justify nonzero degenerate angular momenta. Chruściel–Nguyen's local twist jump and the Weinstein representation input are retained for their stated purposes; they do not by themselves establish full HKWX class membership for an arbitrary geometric spacetime.

Depends on

Used by · 1

Analytic connected-horizon uniqueness including the degenerate case

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

External dependencies

Amsel–Horowitz–Marolf–Roberts, Uniqueness of Extremal Kerr and Kerr-Newman Black Holes · §2, Theorem 1, printed p.2; opening proof and footnote 2, printed p.3

Hypotheses: Four dimensions, stationarity, rotation, asymptotic flatness, analyticity, vacuum, and one degenerate horizon. Rotation is essential: footnote 2 explicitly sets aside the stationary but nonstatic zero-angular-velocity possibility. The source uses global Weyl–Papapetrou coordinates and a global axial twist potential on the domain of outer communication; these are not consequences of the mere local Killing-field definition. The I-plus-regular connected setting is stated precisely by Chruściel–Nguyen Theorem 1.1, cited separately below.

Reading limits: Consulted the arXiv:0906.2367v3 PDF text via the web reader: abstract, introduction, Theorem 1 on printed p.2, and opening proof and footnote 2 on printed p.3. The remaining uniqueness proof and its cited global-reduction proofs were not reconstructed; no local proof is claimed.

Han–Khuri–Weinstein–Xiong, Kerr Black Hole Uniqueness · §1, equations (1.1)–(1.11), source-class paragraph and Theorem 1.1; §2.1 variational hypothesis and (A1)–(A3), (2.3)–(2.10); §2.3 Proposition 2.1(i)–(iv), (2.12)–(2.26).

For $N>1$, let a stationary vacuum singular harmonic map with ordered black-hole rod data locally minimize renormalized energy with fixed rod traces and singularity data, and satisfy the source's ordinary-axis, nondegenerate-pole, regular horizon-interior, degenerate-puncture and infinity conditions. Reconstruct the Weyl–Papapetrou metric with $w\to0$ and $\alpha\to0$ at infinity. If every degenerate component has $\mathcal J_i\ne0$, every bounded axis rod has $\mathbf b_j<0$ for $2\le j\le N$. There is no regular asymptotically flat stationary axisymmetric multi-black-hole configuration in this class. No angular-momentum condition is imposed on nondegenerate rods. Proposition 2.1 derives the requisite asymptotic estimates under all the hypotheses of §2.1.

Hypotheses: The supplied mixed-rod definition includes (V), the renormalized-energy local-minimization hypothesis of §2.1, as well as (A1)–(A3), continuous axis traces and (H). Its reconstruction datum supplies smooth axisymmetric harmonic maps, the finite ordered rods and normalized reconstruction. Proposition 2.1 applies under §2.1 and supplies the axis, puncture, pole and infinity asymptotics required before Theorem 1.1. The nonzero degenerate twist jumps are explicit hypotheses; nondegenerate angular momenta are unrestricted. No arbitrary geometric spacetime is asserted to satisfy these premises.

Reading limits: Consulted the source HTML via the web reader, retaining its displayed TeX: §1 through Theorem 1.1 and its following paragraph; §2.1 in full, §2.2, and the complete Proposition 2.1 statement and attribution proof in §2.3. Checked the local definition against §2.1 and (2.23). The underlying regularity papers and substantial balance proof in §§3–6 were not read in full or reconstructed. Proposition 2.1 and Theorem 1.1 are imported, not locally proved.

Chruściel and Nguyen, A uniqueness theorem for degenerate Kerr-Newman black holes · Theorem 1.1, printed p.1; introduction describing missing global-reduction inputs; §3.3 Proposition 3.1 and §3.4; §3.6, (3.35) and Theorem 3.4, printed pp.20,22

A stationary, $I^+$-regular, analytic electrovacuum spacetime with connected nonempty rotating degenerate future event horizon has a domain of outer communications isometrically diffeomorphic to that of a Kerr–Newman spacetime. Independently, in the stationary axisymmetric asymptotically flat $I^+$-regular vacuum setting, near every degenerate horizon component the Ernst map is at bounded hyperbolic-plane distance from that of extreme Kerr with the same positive horizon cross-section area; its two adjacent axial twist constants differ in absolute value by $4r_0^2$, where $r_0$ is the positive area radius.

Hypotheses: Theorem 1.1 requires analyticity and connected rotating degenerate horizon; the local vacuum Theorem 3.4 is stated for a degenerate component of a stationary axisymmetric asymptotically flat I-plus-regular vacuum black hole. The surrounding analysis uses a spherical cross-section and a spacelike axial field wherever nonzero. This input does not assert that the whole mixed-rod map satisfies every HKWX (A1)–(A3) energy/trace requirement.

Reading limits: Consulted arXiv:1002.1737v1 PDF text via the web reader: Theorem 1.1 and introduction, opening of §2, the area-radius definition (3.1), Proposition 3.1/Remark 3.2 and the opening mixed-component discussion in §3.4, and §3.6 passages (3.33)–(3.46), including the twist jump (3.35) and Theorem 3.4. Did not reconstruct the global coordinate proof or earlier near-horizon classification. The cited local control is not a proof of full HKWX class membership.

Chruściel–Costa–Heusler, Stationary Black Holes: Uniqueness and Beyond · §3.2.3, printed p.14; §3.2.7, Theorem 3.2, printed p.15

A stationary, axisymmetric, asymptotically flat, $I^+$-regular, electrovacuum four-dimensional spacetime has a domain of outer communications isometric to one of the Weinstein solutions. If the event horizon is connected, that domain is a Kerr–Newman domain of outer communications. The preceding reduction discussion states that global isothermal coordinates and controlled boundary conditions extend to configurations with degenerate components.

Hypotheses: Stationarity, axisymmetry, asymptotic flatness, I-plus-regularity, four dimensions and electrovacuum. Vacuum is the zero electromagnetic field specialization. The theorem is used here to record the available global representation input, not as a verified identification of its full boundary class with the later HKWX quantitative admissibility class.

Reading limits: Consulted the source PDF text via the web reader at §3.2.3 (printed p.14), §§3.2.4–3.2.6, and the complete Theorem 3.2 in §3.2.7 (printed p.15). The underlying Weinstein construction and cited global-reduction proofs were not reconstructed; no identification with the later HKWX variational class is inferred.

Sources

Amsel–Horowitz–Marolf–Roberts, Uniqueness of Extremal Kerr and Kerr-Newman Black Holes · §2, Theorem 1, printed p.2; opening proof and footnote 2, printed p.3
Han–Khuri–Weinstein–Xiong, Kerr Black Hole Uniqueness · §1, equations (1.1)–(1.11), source-class paragraph and Theorem 1.1; §2.1 variational hypothesis and (A1)–(A3), (2.3)–(2.10); §2.3 Proposition 2.1(i)–(iv), (2.12)–(2.26).
Chruściel and Nguyen, A uniqueness theorem for degenerate Kerr-Newman black holes · Theorem 1.1, printed p.1; introduction describing missing global-reduction inputs; §3.3 Proposition 3.1 and §3.4; §3.6, (3.35) and Theorem 3.4, printed pp.20,22
Chruściel–Costa–Heusler, Stationary Black Holes: Uniqueness and Beyond · §3.2.3, printed p.14; §3.2.7, Theorem 3.2, printed p.15