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Analytic connected-horizon uniqueness including the degenerate case

PDF p. 5: §1.3, first paragraph

Statement

Adopt the four-dimensional asymptotically flat stationary analytic vacuum model with the full $I^{+}$ -regularity and connected nonempty future event-horizon assumptions of the black-hole comparison data. Use the stationary vacuum model, the rotation and signed surface-gravity convention, and the Kerr convention listed below. Then the entire domain of outer communications is isometric to that of a Kerr spacetime.

More specifically, a connected rotating degenerate horizon has global axial symmetry and its vacuum exterior is extremal Kerr, under the precise connected-degenerate comparison data and the first contract of the analytic/extremal supplier. Electrovacuum means the source-free geometric Einstein–Maxwell equations and conventions listed below. The corresponding stationary analytic $I^{+}$ -regular electrovacuum spacetime with connected nonempty rotating degenerate future horizon has Kerr–Newman domain of outer communications, under Chruściel–Nguyen's exact source hypotheses. The full Kerr–Newman metric exterior meant here is specified in Comparison exterior below.

For a single nonrotating vacuum horizon, normalize the stationary generator by $- g (T, T) \to 1$ at the chosen end. Let $M = M_{K}$ be its normalized exterior Komar mass, equivalently the ADM energy in the asymptotic stationary rest frame, and let $A$ be the area of a compact horizon section. Then $M = κ A / (4 π)$ . The global transport needed for this identity uses the smooth horizon-reaching slice supplied by $I^{+}$ -regularity; it is established separately from the local horizon evaluation. In the degenerate case $κ = 0$ , the independently sourced Komar/positive-energy exclusion rules out a regular $I^{+}$ -regular black hole. No bifurcation-based staticity argument is used for a degenerate horizon. A nonzero degenerate generator has no regular bifurcation sphere for that same generator. Work in ZFC, including choice.

Statement references

Staticity, axisymmetry and black-hole regularity hypotheses

Stationary vacuum model and asymptotic normalization

Rotation, surface gravity and horizon degeneracy

Kerr metric, parameters, exterior and extremality

Connected rotating degenerate vacuum comparison data

Analytic extremal and multiple-horizon comparison contracts

Geometric electrovacuum equations and charge convention

Normalized Komar data on a nonrotating horizon section

A nonzero degenerate generator cannot vanish at a regular bifurcation section

Bifurcate Killing horizons, rotation and normalization

The Axiom of Choice

Comparison exterior

For this item the Kerr–Newman comparison exterior is the following full metric domain, rather than just a sufficiently distant tail. Let $m > 0$ , $a, q \in R$ , $m^{2} \geq a^{2} + q^{2}$ , $r_{+} = m + \sqrt{m^{2} - a^{2} - q^{2}}$ , and $D_{K N} = R_{t} \times (r_{+}, \infty)_{r} \times S^{2}$ . On the unit sphere use the smooth round metric $γ$ , coordinate functions $(X, Y, Z)$ , $b = 1 - Z^{2}$ , and $λ = X d Y - Y d X$ . Put $R = r^{2} + a^{2}$ , $Σ = r^{2} + a^{2} Z^{2}$ and $Δ = r^{2} - 2 m r + a^{2} + q^{2}$ . The full-sphere tensor is $g_{K N} = - \frac{Δ - a^{2} b}{Σ} d t^{2} + \frac{Σ}{Δ} d r^{2} + Σ γ + \frac{a^{2} (R + Σ - Δ)}{Σ} λ^{2} - \frac{2 a (R - Δ)}{Σ} d t ⊙ λ .$ Here $u ⊙ v = (u \otimes v + v \otimes u) / 2$ and $λ^{2} = λ \otimes λ$ . Off the axes this is the Boyer–Lindquist tensor of Kerr-Newman metric and potential on the exact Boyer-Lindquist domain, with $x^{0} = t$ and $M = m$ ; its global sphere expression is the same recipe as Full-sphere exterior extension of the Kerr-Newman datum, now on every $r > r_{+}$ . The azimuthal period is $2 π$ and the time orientation agrees near infinity with $\partial_{t}$ , using the regular ingoing extension in Regular ingoing KN metric, field and sphere axes. The independent exterior identification in Import 5 identifies this domain with the domain of outer communications of the standard Kerr–Newman comparison spacetime relative to its selected asymptotically flat end. The rotating degenerate comparison has $a \neq 0$ and $m^{2} = a^{2} + q^{2}$ , so $r_{+} = m$ . Only the metric is being compared; the parameter $q^{2}$ can represent the sum of squared electric and magnetic charges, as in Import 1. In particular no global electromagnetic potential is required by this formulation.

Facts & Assumptions

[A1]

Adopt exactly the stationary vacuum model, analytic and asymptotically flat comparison class, full $I^{+}$ -regularity, connected nonempty future horizon and stated field normalizations. Retain the ZFC convention. Stationary vacuum model and asymptotic normalization Staticity, axisymmetry and black-hole regularity hypotheses Rotation, surface gravity and horizon degeneracy Bifurcate Killing horizons, rotation and normalization The Axiom of Choice

[F1]

The rotating degenerate vacuum comparison is the first contract of the analytic/extremal supplier, with its exact connected-degenerate formulation. No multiple-horizon balance contract is used. Analytic extremal and multiple-horizon comparison contracts Connected rotating degenerate vacuum comparison data

[F2]

The local Komar suppliers fix the normal binormal, its boost invariance, the Killing-horizon contraction, the constant-gravity integral and the conditional zero-mass implication. In particular $Q_{C} [T] = κ A / (4 π)$ when $κ$ is constant, and $M_{K} = Q_{C} [T]$ must be supplied separately before inferring zero exterior mass. Normalized Komar data on a nonrotating horizon section The normal binormal is invariant under oriented boosts Killing-horizon derivative contracted with the binormal Constant surface gravity gives the normalized horizon integral Degeneracy forces zero mass when global Komar transport applies

[F3]

A regular bifurcation branch is incompatible with a nonzero generator having zero acceleration coefficient. A nonzero degenerate generator cannot vanish at a regular bifurcation section

[F4]

The Kerr definition fixes the subextremal/extremal parameter convention. The electrovacuum definition fixes the geometric field equations. Kerr metric, parameters, exterior and extremality Geometric electrovacuum equations and charge convention

[F5]

Imports 1–3 supply the rotating-degenerate electrovacuum comparison and global symmetry, the nonrotating-degenerate exclusion and mean-nondegenerate vacuum comparison, and the vacuum zeroth law and mean-gravity convention.

[F6]

Import 2 additionally supplies the smooth horizon-reaching spacelike slice, separation of distinct asymptotic ends, the adapted asymptotic stationary coordinates, and a smooth time coordinate whose spacelike levels extend to the horizon section and whose stationary flow is translation. Its future-increasing sign is fixed by the future stationary field at the selected end. Section 2.2 identifies the future horizon with the boundary of the chronological past of the selected stationary exterior. Import 4 supplies the stationary vacuum Komar–Stokes identity and the ADM-energy identification in that asymptotic frame. Neither input requires a maximal or bifurcation slice.

[F7]

The exact Boyer–Lindquist tensor, sphere recipe, and regular ingoing extension are mathematical inputs; only their metric and coordinate conclusions are used. For $m > 0$ and $m^{2} \geq a^{2} + q^{2} > 0$ , the local horizon-data proposition gives $κ_{+} = (r_{+} - r_{-}) / (2 (r_{+}^{2} + a^{2}))$ , including the double root. Import 5 supplies the standard comparison-exterior identification. Kerr-Newman metric and potential on the exact Boyer-Lindquist domain Full-sphere exterior extension of the Kerr-Newman datum Regular ingoing KN metric, field and sphere axes KN Killing generators, surface gravity, area and potential

[F8]

A spacelike hypersurface has a unique smooth future unit normal, obtained locally from its timelike normal covector and hence also up to a smooth spacelike boundary. A smooth embedded horizon has a local defining submersion. Chronological pasts are open past sets, and concatenation of future timelike curves gives chronological transitivity. These are mathematical inputs; the causal suppliers' Countable Choice and Dependent Choice assumptions hold in A1's ZFC setting. The future unit normal, induced metric and second fundamental form of a spacelike hypersurface Embedded submanifolds admit local defining submersions Openness of the chronological future and past, openness of the chronological relation, and closure of future and past sets Chart independence, concatenation, reparametrization and transitivity for causal curves

External input

Import 1: Chruściel–Nguyen Theorem 1.1 supplies the entire rotating-degenerate electrovacuum comparison; §2 supplies global axial symmetry. The opening of §3.2 treats electric and magnetic charges by electromagnetic duality. These inputs do not treat nonrotating degenerate horizons. Chruściel–Nguyen

Import 2: Chruściel–Costa §8 remark 2 supplies the nonexistence of stationary vacuum $I^{+}$ -regular black holes when every horizon component is nonrotating and degenerate, including the global applicability of its positive-energy argument. Theorem 1.3 supplies connected mean-nondegenerate analytic vacuum uniqueness. Theorem 4.5, with $s = 1$ , supplies a smooth acausal spacelike hypersurface up to its compact future-horizon boundary, in the finite-end compact-core setting of Definition 1.1; its smoothing agrees with the chosen end at large distances. Theorem 2.3 separates distinct asymptotically flat ends in a globally hyperbolic spacetime satisfying the null energy condition. Section 2.1 gives the adapted stationary end with asymptotically vanishing shift. Theorem 4.5 also gives a smooth time coordinate with spacelike level hypersurfaces up to their horizon boundary, with the stationary flow acting as translation in its product description. The sign increasing to the future is fixed by the future stationary field at the selected end. Section 2.2, (2.9)–(2.10), identifies the future event horizon as the boundary of $I^{-} (M_{e x t})$ . Chruściel–Costa

Import 3: Chruściel–Costa–Heusler §3.3.1 supplies the Killing-horizon conclusion under the analytic comparison hypotheses; §2.5.3 supplies constant vacuum Killing-horizon surface gravity and its compatibility with nonzero mean surface gravity. Section 7.2 supplies the horizon mass-integral convention and its same-generator restriction. Its static electric illustration alone is not used to establish global transport for a general stationary horizon. Chruściel–Costa–Heusler

Import 4: Reall §8.2, (8.18)–(8.24), gives $d (⋆_{g} d T^{♭}) = 0$ in vacuum and its Stokes identity on a smooth compact spacelike piece, with Komar factor $- (8 π)^{- 1}$ . Section 8.4 following (8.43) identifies this mass with ADM energy for an asymptotic stationary slicing whose normals approach the stationary time direction. The existence and boundary control of the piece used here come from Import 2, rather than from the local horizon integral. Reall

Import 5: Adamo–Newman §3.1, (3.1)–(3.2), and §3.3, (3.18)–(3.21) and the extremality paragraph, identify the standard Kerr–Newman exterior as $r > r_{+}$ , with outer event horizon $r_{+}$ and extremality $m^{2} = a^{2} + q^{2}$ . Their opposite signature is converted to $(-, +, +, +)$ here. The full-sphere construction and coordinate identification are justified below using the mathematical inputs in F7. Adamo–Newman

Proof

Given: The full comparison hypotheses in the Statement, and the independent inputs specified above.

Goal: Establish the entire connected-horizon analytic vacuum classification, the electrovacuum refinement, and the normalized nonrotating mass identity.

1.1givenF7algebra

In the comparison exterior, $r > r_{+} \geq m > 0$ gives $Σ > 0$ and $Δ > 0$ , so the displayed global sphere tensor is smooth. On the pole-free region it equals the Boyer–Lindquist metric by the algebraic identity in the sphere supplier, which uses no large-radius bound; the exact coordinate supplier gives Lorentzian signature there. At a pole $λ = 0$ and $b = 0$ , so the tensor is $- Δ Σ^{- 1} d t^{2} + Σ Δ^{- 1} d r^{2} + Σ γ$ , also Lorentzian. The advanced-coordinate change $d v = d t + R Δ^{- 1} d r$ , $d ψ = d ϕ + a Δ^{- 1} d r$ has radial primitives on the whole interval $(r_{+}, \infty)$ , and its angular shift is a smooth rotation of the full sphere. It therefore embeds the whole exterior into the regular ingoing model, with the specified time orientation. Import 5 identifies precisely this full exterior with the source comparison domain. F7's horizon formula makes zero outer surface gravity equivalent to the double root in the rotating range; it includes extremality.

1.2givenA1F2F6

In the nonrotating vacuum case, apply Import 2's structure theorem with $s = 1$ to the domain $D$ of outer communications. Vacuum implies the null energy condition. End separation applied within globally hyperbolic $D$ excludes another asymptotically flat end in causal communication with the selected end: any such end would itself lie in $D$ , contrary to Theorem 2.3. Thus the finite-end compact-core slice can be truncated at a single large end sphere $S_{R}$ to give a compact smooth spacelike hypersurface $Σ_{R}$ with boundary $S_{R} - C$ , where $C$ is its compact horizon section. The slice is spacelike up to $C$ , even for a degenerate horizon; it is not an orthogonal cylindrical-end slice. In the orientations of F2, Import 4 gives $\int_{S_{R}} ⋆_{g} d T^{♭} = \int_{C} ⋆_{g} d T^{♭}$ . Consequently the exterior Komar limit exists and equals this horizon flux. End spheres in Import 2's adapted stationary coordinates are homologous to these spheres within the same stationary vacuum end, so the same closed-form Stokes identity gives their equal flux. That adapted end satisfies Import 4's asymptotic-normal condition for $E_{A D M} = M_{K}$ .

2.1givenF1F4F5F7step 1.1

For a rotating degenerate vacuum horizon, F1 gives an extremal Kerr exterior. Import 1 gives the electrovacuum comparison and global axial symmetry under its exact analytic $I^{+}$ -regular assumptions. Step 1.1 specifies the complete Kerr–Newman comparison object, including its axes and extremal parameter range. Global symmetry is supplied by the comparison input, rather than inferred from the existence of a merely local Killing field.

2.2A1F2F5F6F8step 1.2algebra

Let $τ$ be Import 2's stationary-flow time coordinate, with sign fixed at the selected end so that $d τ (T) = 1$ and $τ$ increases to the future. Its spacelike levels extend smoothly to $C$ . Let $n$ be their smooth future unit normal from F8. The covector $d τ$ is timelike and future-increasing in the exterior; its smooth nonzero timelike boundary value therefore has $d τ (n) > 0$ on $C$ . Writing $T = N n + X$ with $X$ tangent to the level slice gives $N = - g (T, n) = 1 / d τ (n) > 0$ , including on $C$ . This also proves nonvanishing there. To fix the outward sign, put $P = I^{-} (M_{e x t})$ . Import 3 supplies the smooth Killing-horizon structure. Near $p \in C$ , this future horizon is $\partial P$ , with exterior slice side in $P$ . Choose a local defining function $ρ$ from F8 and shrink its hypersurface chart so that its connected side $ρ > 0$ lies in $P$ and contains the exterior slice side, and let $s$ be the unit normal to $C$ within the slice pointing toward it; then $d ρ (s) > 0$ . The null covector $d ρ$ annihilates $T_{p} C$ , so in the orthonormal normal plane $d ρ (n) = \pm d ρ (s)$ . The positive sign would make a short coordinate segment starting at $p$ with initial velocity $n$ future timelike and contained in $P$ for positive parameter: its initial derivative of $ρ$ would be positive. Chronological transitivity would then put $p$ in the open past set $P$ , contradicting $p \in \partial P$ . Thus $d ρ (n) = - d ρ (s)$ , and the future null normal to the horizon is $n + s$ . Nonrotation puts the nonzero $T$ in this line, and its coefficient is precisely $N > 0$ . Therefore $T = N (n + s)$ satisfies the positive-coefficient hypothesis of F2, without an added normalization or a bifurcation assumption.

3.1A1F2F5F6step 1.2step 2.2algebra

Use the future and outward frame and the positive coefficient established in step 2.2. In a positively oriented orthonormal frame $(n, s, e_{1}, e_{2})$ , the defining Hodge convention gives $(⋆_{g} d T^{♭}) (e_{1}, e_{2}) = (\nabla^{a} T^{b}) b_{a b}$ for $b_{a b} = s_{a} n_{b} - n_{a} s_{b}$ ; this fixes the same sign and factor as F2. Its boost invariance permits the specified normalized horizon frame. Hence step 1.2 yields $M_{K} = Q_{C} [T]$ . The vacuum zeroth law in Import 3 makes $κ$ constant, so F2 now gives $M = E_{A D M} = M_{K} = κ A / (4 π)$ . At $κ = 0$ its conditional zero-mass implication applies with its global-transport hypothesis now established.

4.1givenA1F5step 1.2step 3.1

A connected nonempty nonrotating degenerate horizon satisfies the all-components condition of Import 2's exclusion. That complete source application therefore rules it out in the exact $I^{+}$ -regular vacuum class. This use retains the source's positive-energy applicability; it does not apply a bare zero-mass rigidity theorem to an unspecified completion. Steps 1.2–3.1 supply the separate mass identity and explain the vanishing mass, without using bifurcation-based staticity.

5.1givenA1F4F5step 2.1step 4.1

In the nondegenerate analytic vacuum case, the Killing-horizon and zeroth-law inputs in Import 3 give constant nonzero surface gravity and thus nonzero mean under the source convention. Import 2's connected mean-nondegenerate comparison gives a Kerr exterior. Together with the rotating degenerate case in step 2.1 and the exclusion in step 4.1, this exhausts the connected-horizon alternatives and proves the entire-domain vacuum conclusion.

6.1F3step 2.1step 3.1step 4.1step 5.1∎

The assertion about no regular bifurcation sphere for the same nonzero degenerate generator is F3. No such sphere entered the transport, exclusion or comparison arguments. The conclusions retain their stated vacuum and electrovacuum hypotheses and field normalizations.

Source notes

Original article attribution only: The uniqueness of smooth stationary black holes, arXiv:2611.16853v1, §1.3, PDF p.5. Its rotating and nonrotating degenerate discussion and connected-horizon analytic conclusion are retained. The independent imports supply the prerequisites and argument. Chruściel–Nguyen's maximal-surface caution concerns the separate staticity route; neither that route nor a mixed-horizon exclusion is asserted here.

Depends on

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External dependencies

P. T. Chruściel and L. Nguyen, A uniqueness theorem for degenerate Kerr-Newman black holes · Theorem 1.1, printed p.1; §2 opening paragraph, p.2; §3.2 opening paragraph, p.11

A stationary I-plus-regular analytic electrovacuum spacetime with connected nonempty rotating degenerate future event horizon has a domain of outer communications isometrically diffeomorphic to that of Kerr-Newman. Under these analytic regularity assumptions the rotating case has a global commuting axial U(1) symmetry. Section 3.2 uses electric and magnetic charge parameters and electromagnetic duality for the Kerr–Newman comparison.

Hypotheses: Four-dimensional asymptotically flat stationary analytic electrovacuum spacetime in the terminology inherited from Chruściel–Costa; connected nonempty rotating degenerate future event horizon. No nonrotating case is included.

Reading limits: Read the web-extracted version-1 PDF at Theorem 1.1, the introduction, opening of §2, and opening of §3.2. The global comparison and symmetry are imported; the later global-coordinate and harmonic-map proofs and electromagnetic-duality calculations were not reconstructed. The maximal-surface caveat concerns a different staticity route.

P. T. Chruściel and J. L. Costa, On uniqueness of stationary vacuum black holes · Definition 1.1, printed p.2; Theorem 1.3, pp.3–4; §2.1, (2.5)–(2.8), p.6; Theorem 2.3, p.10; §2.2, (2.9)–(2.10), p.7; Theorem 4.5 and (4.15), p.18, with end-matching conclusion of its proof, p.22; §8 remark 2, p.55

There are no stationary vacuum I-plus-regular black holes all of whose event-horizon components are nonrotating and degenerate: the source explicitly states the Komar/positive-energy nonexistence application. Its connected one-horizon case is a direct specialization. Separately, a stationary asymptotically flat I-plus-regular analytic four-dimensional vacuum spacetime with connected mean-nondegenerate future horizon has Kerr domain of outer communications. Theorem 4.5 for s=1 supplies a smooth acausal asymptotically flat slice spacelike up to its compact future-horizon boundary, in Definition 1.1’s finite-end compact-core class; its smoothing agrees with the chosen end at large distances. Theorem 2.3 prohibits causal communication between distinct asymptotically flat ends under global hyperbolicity and the null energy condition. Section 2.1 supplies adapted stationary coordinates with asymptotically vanishing shift. Theorem 4.5 also supplies a smooth time coordinate with spacelike levels up to the horizon boundary, in whose product description the stationary flow is translation. Section 2.2 identifies the future horizon as the boundary of the chronological past of the selected stationary exterior.

Hypotheses: For the first application: stationary, vacuum and I-plus-regular black-hole spacetime, with every event-horizon component nonrotating and degenerate; connected nonempty event horizon is the specialization used here. For Theorem1.3: the additional four-dimensional analytic and asymptotically flat conditions, connected horizon and nonzero mean surface gravity. The exact I-plus-regularity is Definition1.1, not an arbitrary incomplete exterior slice. The structure theorem needs a stationary I-plus-regular spacetime with the R action of its complete stationary field; s=1 imposes no axial symmetry. End separation is applied to the globally hyperbolic domain of outer communications itself, with its asymptotically flat ends and vacuum null energy condition. The stationary vacuum end uses the source’s derivative-decay convention. No maximal-slice or horizon-nondegeneracy hypothesis is added to Theorem 4.5.

Reading limits: Read the cached PDF text at Definition 1.1, Theorem 1.3, §2.1 pp.6–7, §2.3 p.8, Theorem 2.3 p.10, Theorem 4.5 p.18, and §8 remark 2 p.55; cross-checked the structure theorem and its proof’s end-matching sentence in web PDF text. These theorems and the complete nonexistence application are imported. Their global smoothing, topological-censorship and positive-energy proofs were not reconstructed. Compact truncation and flux matching are the local application in Proof 1.2 and 2.2–3.1; no exclusion of mixed-horizon configurations is inferred. For this repair, independently read the web-extracted version-2 PDF at section 2.2, equations (2.9)–(2.10), the statement of Theorem 4.5 on printed page 18, equation (4.18) on page 20, and the closing paragraph on page 22. Only the stated time-coordinate/product and chronological-boundary conclusions are additionally imported; the generator sign and its application to the Komar frame are derived locally in Proof 2.2. No reconstruction of the source smoothing proof is claimed.

P. T. Chruściel, J. L. Costa and M. Heusler, Stationary Black Holes: Uniqueness and Beyond · §7.2, printed pp.38–39, especially (7.8) and its horizon-flux explanation; §2.5.3, pp.8–10; §3.3.1–3.3.2, including Theorem 3.3, p.16

The stationary mass identities evaluate the horizon Komar integral as kappa A/(4 pi). Evaluation for a nonrotating horizon uses the same Killing field for the asymptotic stationary normalization and the horizon generator; rotation requires the additional angular-momentum term. The source stresses that the mass-formula framework is not restricted to static configurations. The zeroth law gives constant vacuum Killing-horizon surface gravity; nonzero constant surface gravity has nonzero mean. Its analytic connected-horizon classification includes the mean-nondegenerate or rotating alternatives. Section 3.3.1 states that strong rigidity makes the event horizon a Killing horizon under the analytic comparison hypotheses.

Hypotheses: Stationary asymptotically flat smooth four-dimensional vacuum specialization of the source equations with compact horizon cross-section; nonrotating means the asymptotic stationary field itself generates the horizon. Global mass transport must have the source admissible hypersurface and boundary controls; it is not inferred from a local tensor identity. Analytic classification additionally uses the exact I-plus-regular connected-horizon hypotheses.

Reading limits: Read cached and web PDF text at §7.2 pp.38–39, §2.5.3 pp.8–10 and §3.3.1–3.3.2 p.16. Imported the stated vacuum zeroth law, Killing-horizon and mean-gravity conventions and classification statement. The static electric example in (7.8) is not generalized to prove stationary global transport: that transport is separately established using Chruściel–Costa’s slice and Reall’s identity. The underlying rigidity and uniqueness proofs were not reconstructed.

H. S. Reall, Black Holes, Part III lecture notes, 15 March 2024 · §8.2, (8.18)–(8.24), printed p.108 (PDF p.109); §8.4, paragraph after (8.43), printed p.113 (PDF p.114)

For a stationary Killing field, (8.22) makes star d(T-flat) closed in vacuum; (8.23) gives the Stokes identity on a smooth compact hypersurface piece. Komar mass is its asymptotic flux with factor -1/(8 pi). In a stationary asymptotically flat spacetime it equals ADM energy when the slicing is asymptotically orthogonal to the stationary field.

Hypotheses: Smooth oriented time-oriented four-dimensional stationary spacetime, signature (-,+,+,+), G=c=1, normalized stationary field, and vacuum for closedness. Stokes is applied only to compact smooth truncations with their complete oriented boundary. The ADM identification uses a stationary asymptotically flat end with slice normals approaching the stationary time direction.

Reading limits: Read the cached 149-page PDF’s pages 109 and 114, including the displayed equations and the paragraph after (8.43). The browser retrieval timed out; no fresh download or full-notes reading is claimed. The source supplies the identity and asymptotic identification, while the current proof checks its hypersurface and boundary hypotheses using Chruściel–Costa. No local reconstruction of the ADM identification is claimed.

T. Adamo and E. T. Newman, The Kerr-Newman metric: A Review · §3.1, (3.1)–(3.2), printed p.13; §3.3, (3.18)–(3.21), pp.17–18, and extremality paragraph on p.18

The Boyer–Lindquist Kerr–Newman metric has outer event horizon r-plus = m + sqrt(m squared - a squared - q squared), exterior r > r-plus, and extremality m squared = a squared + q squared. Advanced coordinates remove the horizon coordinate singularity.

Hypotheses: Standard asymptotically flat Kerr–Newman comparison spacetime, m>0 and m squared >= a squared + q squared; azimuthal period 2 pi. The metric signature in this source is opposite to the item’s (-,+,+,+) convention.

Reading limits: Read the web-extracted PDF at the complete (3.1)–(3.2) formulas and §3.3 through the extremality paragraph. Imported the standard exterior/horizon identification, not a reconstruction of the maximal analytic extension. Full-sphere tensor regularity is checked locally with the existing mathematical suppliers. No electromagnetic-potential or field-uniqueness claim is drawn from this reference.

Sources

P. T. Chruściel and L. Nguyen, A uniqueness theorem for degenerate Kerr-Newman black holes · Theorem 1.1, printed p.1; §2 opening paragraph, p.2; §3.2 opening paragraph, p.11
P. T. Chruściel and J. L. Costa, On uniqueness of stationary vacuum black holes · Definition 1.1, printed p.2; Theorem 1.3, pp.3–4; §2.1, (2.5)–(2.8), p.6; Theorem 2.3, p.10; §2.2, (2.9)–(2.10), p.7; Theorem 4.5 and (4.15), p.18, with end-matching conclusion of its proof, p.22; §8 remark 2, p.55
P. T. Chruściel, J. L. Costa and M. Heusler, Stationary Black Holes: Uniqueness and Beyond · §7.2, printed pp.38–39, especially (7.8) and its horizon-flux explanation; §2.5.3, pp.8–10; §3.3.1–3.3.2, including Theorem 3.3, p.16
H. S. Reall, Black Holes, Part III lecture notes, 15 March 2024 · §8.2, (8.18)–(8.24), printed p.108 (PDF p.109); §8.4, paragraph after (8.43), printed p.113 (PDF p.114)
T. Adamo and E. T. Newman, The Kerr-Newman metric: A Review · §3.1, (3.1)–(3.2), printed p.13; §3.3, (3.18)–(3.21), pp.17–18, and extremality paragraph on p.18