On the third law of black hole mechanics for supersymmetric black holes

Aidan M. McSharry; Harvey S. Reall

REVIEW 2 major objections 4 minor 1 cited by

On the third law of black hole mechanics for supersymmetric black holes

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Supersymmetric black holes cannot form in finite time from non-extremal ones, even with a negative cosmological constant.

desk verdict Genuine extension of Reall's supersymmetric third-law no-go to AdS and two-sided black holes, but the proof rests on an existence claim for the supercovariant Sen-Witten equation that Appendix B leaves heuristic. read the letter →

arxiv 2507.06870 v1 pith:CSOEVWRA submitted 2025-07-09 gr-qc hep-th

Aidan M. McSharry , Harvey S. Reall This is my paper

classification gr-qchep-th

keywords thirdlawofblackholemechanicssupersymmetricholesKerr-Newman-anti-deSitterlocalmass-chargeinequalityNester-WittenfunctionalSen-WittenequationmarginallyoutertrappedsurfaceEinstein-Maxwelltheory

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Classical general relativity was recently found to allow a violation of the third law of black hole mechanics: an exactly extremal Reissner-Nordström black hole can form in finite time from charged matter with a large charge-to-mass ratio. This paper asks whether the obstruction can be restored by restricting the matter, and answers in the affirmative for a strengthened energy condition: matter whose energy density dominates both its momentum density and its charge density. It proves that under this 'local mass-charge inequality', a supersymmetric black hole — including a supersymmetric Kerr-Newman-anti de Sitter black hole, or any supersymmetric AdS black hole with a Killing horizon — cannot be the end state of finite-time collapse, and more generally that an initially non-extremal black hole, identified by containing a trapped surface, cannot evolve into a supersymmetric black hole in finite time. The result is quasi-local, so it does not depend on boundary conditions at infinity in anti-de Sitter space. The theorem restores the third law in the supersymmetric sector while leaving open extremal but non-supersymmetric endpoints such as extremal Reissner-Nordström-anti de Sitter.

What carries the argument

The argument is carried by a spinorial energy functional and a rigidity bootstrap. On the outer boundary S one defines the supercovariant Nester-Witten functional $\hat I_S[\epsilon]$; a supersymmetric surface makes it vanish, while positivity under the local mass-charge inequality forces equality to imply that $\epsilon$ extends over Σ as a solution of the tangential supercovariant constancy condition $h^b{}_a \hat\nabla_b\epsilon=0$, that the matter satisfies the compatibility condition (24), and that a weakly outer trapped inner boundary is actually marginally outer trapped. Positivity requires solving the gauge-supercovariant Sen-Witten equation $\gamma^i \hat\nabla_i \tilde\epsilon=0$ with boundary data (25) on S and (39) on T; this solvability step is justified only heuristically in Appendix B. With $\epsilon$ in hand, the spinor bilinear $X^a$ constitutes Killing initial data, and the proof passes to the Killing development spacetime in which $X^a$ is genuinely Killing; a Komar-type identity, the fact that the scalar bilinear V vanishes on S and T, and the general form of the matter stress tensor in a supersymmetric spacetime (Appendix C) then produce an equation whose left-hand side is strictly positive for Λ < 0 — the contradiction. For Λ = 0, the same framework yields a dust-like matter form and forces the matter charge densities to vanish.

What would settle it

Produce a smooth four-dimensional Einstein-Maxwell spacetime with negative cosmological constant and matter satisfying the local mass-charge inequality, containing a compact spacelike surface whose outer boundary carries the induced metric, extrinsic curvature and Maxwell field of a supersymmetric Kerr-Newman-anti de Sitter horizon cross-section and whose inner boundary is weakly outer trapped; one such spacetime would refute the central theorem. Short of that, solve the Appendix B boundary-value problem on a concrete hypersurface and exhibit a nontrivial kernel or a missing solution for the supercovariant Sen-Witten operator.

Watch

Extended reading notes

Core claim

The paper's central claim is an extension of the third law: in smooth Einstein-Maxwell spacetimes with cosmological constant Λ ≤ 0 and charged matter obeying the local mass-charge inequality, a compact spacelike hypersurface Σ cannot have an outer boundary S that is at once a marginally outer trapped surface and a 'supersymmetric surface' — a surface carrying a nontrivial spinor annihilated by the tangential parts of the supercovariant derivative, exactly the data found on a horizon cross-section of a supersymmetric black hole such as supersymmetric Kerr-Newman-anti de Sitter. When Σ has no inner boundary (the gravitational-collapse case, Theorem 3.6 and Corollary 3.6.1), the existence of such an S already contradicts the positive structure of the Nester-Witten functional. When Σ has an inner boundary T that is weakly outer trapped (a pre-existing or two-sided black hole, Theorem 4.2 and Corollary 4.2.1), the same contradiction holds for Λ < 0; for Λ = 0 the contradiction is avoided only if the matter on Σ has vanishing electric and magnetic charge densities and T is itself marginally outer trapped and supersymmetric. The consequence is that a non-extremal black hole — one containing a trapped surface — cannot become supersymmetric in finite time, irrespective of whether the initial black hole formed by gravitational collapse.

Load-bearing premise

The proof stands on the assumption that the supercovariant Sen-Witten equation on the compact hypersurface always has a solution taking the prescribed values on the outer and inner boundaries, a step the paper justifies only heuristically.

Editorial extensions

If this is right

A supersymmetric Kerr-Newman-anti de Sitter black hole cannot be the end state of finite-time gravitational collapse of charged matter obeying the local mass-charge inequality.
An initially non-extremal black hole, signalled by a trapped surface on an initial hypersurface, cannot evolve in finite time to a supersymmetric black hole, even if the black hole was not formed by collapse and even for negative cosmological constant.
The obstruction is quasi-local: it holds regardless of the boundary conditions at infinity in anti-de Sitter space, such as reflecting or transparent conditions.
For vanishing cosmological constant, the same rigidity forces the matter on the hypersurface to have zero electric and magnetic charge densities and forces the inner boundary to become marginally outer trapped and supersymmetric before a supersymmetric outer horizon can appear.
Event-horizon cross-sections that share only the induced metric and gauge field with a supersymmetric horizon, but not the extrinsic curvature (in particular not the ingoing null convergence), are not covered by the theorem.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the local mass-charge inequality fails, the theorem's obstruction is expected to disappear; adapting charged scalar or Vlasov collapse to extremal Reissner-Nordström-anti de Sitter would settle whether the third law fails exactly outside the supersymmetric sector.
The rigidity mechanism suggests a quasi-local mass-charge inequality on surfaces with a trapped inner boundary, defined by the supercovariant Nester-Witten functional; proving its positivity directly would both extend the result and make the Appendix B solvability step testable.
The theorem is silent on infinite-time approaches, so solutions whose horizons asymptote to supersymmetric data at infinite time may exist even under the inequality; finding one would show the obstruction is specifically about finite time.
A numerical check of the boundary-value problem for the supercovariant Sen-Witten equation on a representative hypersurface could falsify the proof's keystone even before a full spacetime counterexample is found.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know. First, this is not a repackaging of Reall's earlier result: the Λ<0 extension needs a genuinely new P-term in the Nester-Witten identity (Appendix A), a Killing-development argument adapted to AdS, and the two-sided trapped-surface generalization is new for both Λ=0 and Λ<0. Second, the central no-go theorem is persuasive but not airtight: the non-negativity step that drives everything depends on existence of a solution to the supercovariant Sen-Witten equation with mixed boundary conditions on S and T, and Appendix B explicitly labels its proof heuristic. That is the soft spot to watch.

What is good. The paper is clearly written and the argument is structured honestly. The informal theorems are stated precisely later, and the authors flag limitations: they cover black holes whose horizons are Killing horizons, they don't handle magnetic monopoles for Λ<0, they don't exclude non-supersymmetric extremal formation. The proof that a supersymmetric surface cannot be trapped (Lemma 3.3) and the rigidity theorems (3.5, 4.1) are elegant. The new result for two-sided black holes extends Reall's theorem to initial black holes not formed by collapse, which is a conceptual advance. The reliance on Reall's own prior paper is heavy but that paper is cited properly and the machinery is legitimate.

Where it softens. The existence business in Appendix B. The Fredholm argument asserts (i) that the boundary conditions (59) define a self-adjoint Fredholm extension, (ii) that Ω is compact, and (iii) that indices of D̂ and D coincide. None of these is demonstrated; the boundary identity (63)-(65) is a symmetry check, not a proof of Fredholmness. The uniqueness part is fine. This is a load-bearing step because without it I_S[ε]≥0 in Theorems 3.5 and 4.1 doesn't follow. It is probably fixable: the operator is a Dirac-type operator with elliptic boundary conditions, and Bartnik–Chrusciel [24] likely supplies the needed abstract result, but the authors need to fill this in or cite a theorem that actually covers the mixed boundary value problem. I did not find a circularity elsewhere; the contradiction in Theorem 3.6 comes from a strictly positive Λ term and the sign of the matter terms, and the sign arguments look sound. I didn't independently check every gamma-matrix identity in Appendices A and C; the structure is standard for this literature.

Who this is for: people working on spinorial proofs in mathematical relativity and on the third law. I would bring it to a relativity reading group and I'd send it to peer review. The referee should ask for a complete existence proof or a precise citation, but the main line of argument is a real advance and deserves to be published after that gap is addressed.

Referee Report

2 major / 4 minor

Summary. The paper extends previous work by one of the authors on the third law of black hole mechanics for supersymmetric black holes. It proves that, under the local mass-charge inequality (1), a supersymmetric Kerr-Newman-AdS black hole cannot form in finite time in gravitational collapse (Theorem 3.6 and Corollary 3.6.1), and more generally that an initially non-extremal black hole cannot evolve into a supersymmetric black hole in finite time, allowing for two-sided black holes and cosmological constant Λ ≤ 0 (Theorem 4.2 and Corollary 4.2.1). The proof combines supersymmetric-surface rigidity with the Nester-Witten functional, a supercovariant Sen-Witten equation on a compact hypersurface with mixed boundary conditions, and a Killing-development argument. The central geometric and inequality structures are presented in detail, but a key analytic input—existence of a solution to the supercovariant Sen-Witten boundary value problem—is left heuristic in Appendix B.

Significance. If the analytic gap identified below is closed, the result is significant: it strengthens the third law of black hole mechanics in a setting with negative cosmological constant and removes the gravitational-collapse assumption for the Λ = 0 case. The paper is careful to state quasi-local results that are independent of boundary conditions at infinity, and it gives explicit rigidity statements (Theorems 3.5, 3.6, 4.1, 4.2) with a clear logical structure. The main technical novelty—the use of supercovariant Sen-Witten equations with mixed boundary conditions on an inner trapped surface and an outer supersymmetric surface—is well motivated and largely convincing. However, the paper is proof-based and the entire chain of theorems rests on an existence statement that is explicitly described as heuristic; this is the main obstacle to acceptance.

major comments (2)

[Appendix B, 'We now establish existence' (eqs. (59)–(66))] The existence of a solution to the supercovariant Sen-Witten equation (28) with boundary conditions (25) on S and (39) on T is not proved. The variational argument leading to (62)–(66) shows only that the boundary conditions are formally self-adjoint in the sense of (64)–(65); it does not establish that the boundary value problem defines a Fredholm operator of index zero. In particular, the assertions that Ω is a compact perturbation, that D is self-adjoint, and that the indices of D and Ω̂D coincide are stated without proof. This matters because Theorem 3.5 and Theorem 4.1 both use the existence of such a solution to conclude non-negativity of the Nester-Witten functional (29); without that non-negativity, Theorems 3.6 and 4.2 and their corollaries do not follow. Please replace the heuristic argument with a rigorous existence proof (for example via the methods of [24]) or state the existence as an explicit assumption and carefully delineate which results depend on it.
[Section 3.3, proof of Theorem 3.6, paragraph beginning 'We first use an argument from Appendix B of [22]'] The step from the rigidity equation h^a_b ∇_a ϵ = 0 on Σ to a supercovariantly constant spinor in the Killing development of (Σ, X) is asserted as a 'slight modification' of Appendix B of [22], but the details are not given. This extension is used to derive the Killing spinor identity (30), which is essential for the Komar-like identity (31), for the vanishing of ∫_S ⋆Ψ, and for the stress-tensor relation (68). Since the contradiction in Theorem 3.6 depends on these identities, the reader needs either a proof of the extension in the present conventions or a precise statement of the generalised result from [22] with the required hypotheses.

minor comments (4)

[Section 5] There are repeated typos in 'Reissner-Norstr¨om' (missing the 'd' in 'Nordström'); please correct them.
[Figure 2 caption] The label 'SUSY AadS' appears to be a typo; it should presumably read 'SUSY AdS'.
[Appendix B, around eq. (62)] The notation '^3∇_a' for the induced covariant derivative on Σ is introduced without definition; please define it explicitly.
[Appendix B, uniqueness paragraph] The statement that 'I_S[ϵ∆] = 0 ... so ϵ∆ = 0, as can be seen by inspection of (22)' is terse; a short explanation of why the vanishing of the integral forces both remaining spinor components to vanish would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the contradiction is driven by a positivity argument, and the cited prior results are independent published theorems.

full rationale

The paper's central claims (Theorems 3.6 and 4.2) are not assumed in their inputs; they follow from assuming a supersymmetric, marginally outer trapped boundary S and deriving a strictly positive integral (e.g., equation (35)) that must vanish. Non-negativity of the Nester-Witten functional is proven by explicit computation in Appendix A. The most load-bearing technical step, existence of a solution to the supercovariant Sen-Witten equation with boundary conditions (25) and (39), is admittedly heuristic in Appendix B: "Following [7], we now provide a heuristic argument for why, as stated in a footnote of [8], checking that the kernel of the adjoint problem is trivial is sufficient to show existence." This is a proof gap or correctness risk, not a circular reduction: the paper does not define the sought theorem to be equivalent to that existence statement, and no fitted data are renamed as predictions. The reliance on [7] and [22] is self-citation, but those are published, parameter-free results with stated assumptions that do not include the target conclusions of this paper; per the review rules such citations count as independent evidence and do not by themselves make the derivation circular. The supersymmetric-surface definition and the always-solvable subsidiary condition (19) are working definitions and algebraic choices, not conclusions smuggled in as premises. No step can be exhibited in which an equation reduces, by construction, to its own input, so the honest finding is no significant circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The proof is a theorem derivation with no fitted numbers. Its inputs are the Einstein-Maxwell equations, the local mass-charge inequality as a strengthened dominant energy condition, the existence of charged spinor boundary-value problems, and the Killing-development machinery imported from [21,22]. The least certain input is the existence and uniqueness of the supercovariant Sen-Witten solution, whose proof in Appendix B is explicitly heuristic.

assumptions (6)

domain assumption Matter satisfies the local mass-charge inequality (1)/(3) on Σ.
The entire theorem is conditional on this strengthened dominant energy condition, invoked in Section 2.2 and used in Theorems 3.5, 3.6, 4.1, and 4.2. The Kehle-Unger counterexamples violate it.
domain assumption Σ is a smooth, compact, connected spacelike 3-surface with boundary ∂Σ = S or ∂Σ = S ∪ T, with S connected and T a finite union of weakly outer trapped surfaces.
Finite-time collapse or non-extremal initial state is encoded as compactness and boundary trappedness; see Section 3 and Section 4.
domain assumption A nontrivial solution exists to the supercovariant Sen-Witten equation γ^i ∇_i ε̃ = 0 on Σ with boundary conditions (25) on S and (39) on T.
Justified only heuristically in Appendix B via Fredholm theory; non-negativity of the Nester-Witten functional and the rigidity equality conditions depend on it. This is the weakest technical link.
domain assumption The Killing development of the initial data (Σ, X^a) exists and the supercovariantly constant spinor extends to it.
Used in Theorems 3.6 and 4.2 to reach the Komar-like integral contradiction; justified by citing [21,22] and a slight modification of Appendix B of [22].
domain assumption For Λ<0 the Maxwell field admits a local U(1) potential, i.e. no magnetic currents are present.
Footnote 2 in the Introduction: for Λ<0, magnetic charge is excluded because the supercovariant derivative requires a gauge potential; this limits Theorem 1.1 to electrically charged collapse.
domain assumption In a spacetime with a supercovariantly constant spinor, the matter stress tensor takes the form (68) or equivalently T_ab = χ X_a X_b and J_a = α X_a.
Derived in Appendix C using the integrability condition from [20]; it is a new derivation in this paper confirming a result stated in [23], and it is needed for the strict positivity of the Komar integral.

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Pith. "Pith review of On the third law of black hole mechanics for supersymmetric black holes." pith.science (2026). https://pith.science/paper/CSOEVWRA

@misc{pith2026250706870,
  author       = {Pith},
  title        = {Pith review of: On the third law of black hole mechanics for supersymmetric black holes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CSOEVWRA}},
  note         = {Machine review of arXiv:2507.06870}
}

read the original abstract

Recently it has been shown that the third law of black hole mechanics can be violated: an exactly extremal Reissner-Nordstrom black hole can form in finite time in gravitational collapse of matter with a large charge to mass ratio. However, it has also been proved that this cannot happen if the matter satisfies a ``supersymmetric'' lower bound on its energy in terms of its charge. This paper proves an analogous result for black holes with a negative cosmological constant. The result states that a supersymmetric Kerr-Newman-anti de Sitter black hole cannot form in gravitational collapse of charged matter satisfying the supersymmetric bound. The results for zero or negative cosmological constant are extended to apply to two-sided black holes: it is proved that an initially non-extremal black hole cannot evolve to a supersymmetric black hole in finite time, irrespective of whether or not the initial black hole was formed in gravitational collapse.

Figures

Figures reproduced from arXiv: 2507.06870 by the authors.

**Figure 2.** (a) Penrose diagram showing a third law violating spacetime in which an initially non-extremal black [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Formation of extremal Reissner-Nordstr\"om black holes: insights from numerics
gr-qc 2025-12 conditional novelty 6.0 of 10

For several scalar-field profiles, extremal Reissner-Nordström black holes can be glued from collapse only above a profile-dependent threshold in eM, and only for scalar masses below about m/e≈0.28.

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Works this paper leans on

27 extracted references · 11 canonical work pages · cited by 1 Pith paper

[24]

Boundary value problems for Dirac--type equations, with applications

R. Bartnik and P. T. Chrusciel, “Boundary value problems for Dirac type equations, with applications,” [arXiv:math/0307278 [math.DG]]

work page Pith review arXiv
[22]

On Israel-Wilson-Perjes black holes,

P. T. Chrusciel, H. S. Reall and P. Tod, “On Israel-Wilson-Perjes black holes,” Class. Quant. Grav. 23, 2519- 2540 (2006) doi:10.1088/0264-9381/23/7/018 [arXiv:gr-qc/0512116 [gr-qc]]

arXiv 2006
[1]

The Four laws of black hole mechanics,

J. M. Bardeen, B. Carter and S. W. Hawking, “The Four laws of black hole mechanics,” Commun. Math. Phys. 31, 161-170 (1973) doi:10.1007/BF01645742

doi:10.1007/bf01645742 1973
[2]

Third Law of Black-Hole Dynamics: A Formulation and Proof,

W. Israel, “Third Law of Black-Hole Dynamics: A Formulation and Proof,” Phys. Rev. Lett. 57, no.4, 397 (1986) doi:10.1103/PhysRevLett.57.397

doi:10.1103/physrevlett.57.397 1986
[3]

Gravitational collapse to extremal black holes and the third law of black hole ther- modynamics,

C. Kehle and R. Unger, “Gravitational collapse to extremal black holes and the third law of black hole ther- modynamics,” [arXiv:2211.15742 [gr-qc]]

arXiv
[4]

Extremal black hole formation as a critical phenomenon,

C. Kehle and R. Unger, “Extremal black hole formation as a critical phenomenon,” [arXiv:2402.10190 [gr-qc]]

arXiv
[5]

A Bogomolny Bound for General Relativity and Solitons in N=2 Supergravity,

G. W. Gibbons and C. M. Hull, “A Bogomolny Bound for General Relativity and Solitons in N=2 Supergravity,” Phys. Lett. B 109, 190-194 (1982) doi:10.1016/0370-2693(82)90751-1

doi:10.1016/0370-2693(82)90751-1 1982
[6]

Positive Mass Theorems for Black Holes,

G. W. Gibbons, S. W. Hawking, G. T. Horowitz and M. J. Perry, “Positive Mass Theorems for Black Holes,” Commun. Math. Phys. 88, 295 (1983) doi:10.1007/BF01213209

work page doi:10.1007/bf01213209 1983

Show all 27 references

[7]

A third law of black hole mechanics for supersymmetric black holes and a quasi-local mass-charge inequality,

H. S. Reall, “A third law of black hole mechanics for supersymmetric black holes and a quasi-local mass-charge inequality,” Phys.Rev. D 110 124059 (2025) [arXiv:gr-qc/2410.11956 [gr-qc]]

2025 arXiv
[8]

Quasilocal mass constructions with positive energy,

A. J. Dougan and L. J. Mason, “Quasilocal mass constructions with positive energy,” Phys. Rev. Lett. 67, 2119-2122 (1991) doi:10.1103/PhysRevLett.67.2119

1991 doi
[9]

Spinorial quasilocal mass for spacetimes with negative cosmological constant,

V. Rallabhandi, “Spinorial quasilocal mass for spacetimes with negative cosmological constant,” [arXiv:gr- qc/2504.11971 [grqc]]

arXiv
[10]

A New gravitational energy expression with a simple positivity proof,

J. A. Nester, “A New gravitational energy expression with a simple positivity proof,” Phys. Lett. A 83, 241 (1981) doi:10.1016/0375-9601(81)90972-5

1981 doi
[11]

Solitonic black holes in gauged N = 2 supergravity,

A. V. Kosteleck´ y and M. J. Perry, “Solitonic black holes in gauged N = 2 supergravity,” Phys. Lett. B 371 (1996), 191-198 [arXiv:hep-th/9512222 [hep-th]]

1996 arXiv
[12]

Supersymmetry of anti-de Sitter black holes,

M. M. Caldarelli and D. Klemm, “Supersymmetry of anti-de Sitter black holes,” Nucl. Phys. B 545 (1999), 434-460 doi:10.1016/S0550-3213(98)00846-3 [arXiv:hep-th/9808097 [hep-th]]

1999 arXiv
[13]

Spinors and space-time,

R. Penrose and W. Rindler, “Spinors and space-time,” Vol 1, Cambridge Univ. Press, 1988, doi:10.1017/CBO9780511564048

1988 doi
[14]

A space-time calculus based on pairs of null directions,

R. P. Geroch, A. Held and R. Penrose, “A space-time calculus based on pairs of null directions,” J. Math. Phys. 14, 874-881 (1973) doi:10.1063/1.1666410

1973 doi
[15]

Gauge internal symmetry in extended supergravity,

D. Z. Freedman and A. Das, “Gauge internal symmetry in extended supergravity,” Nucl. Phys. B 120 (1977), 221-230 doi:10.1016/0550-3213(77)90041-4

1977 doi
[16]

Supersymmetric, cold and lukewarm black holes in cosmological Einstein-Maxwell theory,

L. J. Romans, “Supersymmetric, cold and lukewarm black holes in cosmological Einstein-Maxwell theory,” Nucl. Phys. B 383 (1992), 395-415 [arXiv:hep-th/9203018 [hep-th]]

1992 arXiv
[17]

All Metrics Admitting Supercovariantly Constant Spinors,

K. P. Tod, “All Metrics Admitting Supercovariantly Constant Spinors,” Phys. Lett. B 121, 241-244 (1983) doi:10.1016/0370-2693(83)90797-9

1983 doi
[18]

Positivity of energy in Einstein-Maxwell axion dilaton gravity,

M. Rogatko, “Positivity of energy in Einstein-Maxwell axion dilaton gravity,” Class. Quant. Grav. 19, 5063- 5072 (2002) doi:10.1088/0264-9381/19/20/303 [arXiv:hep-th/0209126 [hep-th]]

2002 arXiv
[19]

A Simple Proof of the Positive Energy Theorem,

E. Witten, “A Simple Proof of the Positive Energy Theorem,” Commun. Math. Phys. 80, 381 (1981) doi:10.1007/BF01208277

1981 doi
[20]

All supersymmetric solutions of N=2,D=4 gauged supergravity,

M. M. Caldarelli and D. Klemm, “All supersymmetric solutions of N=2,D=4 gauged supergravity,” JHEP 09 (2003), 019 doi:10.1088/1126-6708/2003/09/019 [arXiv:hep-th/0307022 [hep-th]]

2003 arXiv
[21]

Killing vectors in asymptotically flat space-times: I. Asymptotically trans- lational Killing vectors and the rigid positive energy theorem,

R. Beig and P. T. Chrusciel, “Killing vectors in asymptotically flat space-times: I. Asymptotically trans- lational Killing vectors and the rigid positive energy theorem,” J. Math. Phys. 37 (1996), 1939-1961 doi:10.1063/1.531497 [arXiv:gr-qc/9510015 [gr-qc]]

1996 arXiv
[23]

Supersymmetric G¨ odel-type universe in four dimensions,

M. M. Caldarelli and D. Klemm, “Supersymmetric G¨ odel-type universe in four dimensions,” Class. Quant. Grav. 21 (2004), L17-L20 doi:10.1088/0264-9381/21/4/L03 [arXiv:hep-th/0310081 [hep-th]]. 19

2004 arXiv
[25]

Event horizon gluing and black hole formation in vacuum: The very slowly rotating case,

C. Kehle and R. Unger, “Event horizon gluing and black hole formation in vacuum: The very slowly rotating case,” Adv. Math. 452, 109816 (2024) doi:10.1016/j.aim.2024.109816 [arXiv:2304.08455 [gr-qc]]

2024
[26]

What happens at the horizon(s) of an extreme black hole?,

K. Murata, H. S. Reall and N. Tanahashi, “What happens at the horizon(s) of an extreme black hole?,” Class. Quant. Grav. 30, 235007 (2013) doi:10.1088/0264-9381/30/23/235007 [arXiv:1307.6800 [gr-qc]]

2013 arXiv
[27]

Nonlinear stability of extremal Reissner-Nordstr¨ om black holes in spherical symmetry,

Y. Angelopoulos, C. Kehle and R. Unger, “Nonlinear stability of extremal Reissner-Nordstr¨ om black holes in spherical symmetry,” [arXiv:2410.16234 [gr-qc]]. 20

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [24]

Boundary value problems for Dirac--type equations, with applications

R. Bartnik and P. T. Chrusciel, “Boundary value problems for Dirac type equations, with applications,” [arXiv:math/0307278 [math.DG]]

work page Pith review arXiv

[2] [22]

On Israel-Wilson-Perjes black holes,

P. T. Chrusciel, H. S. Reall and P. Tod, “On Israel-Wilson-Perjes black holes,” Class. Quant. Grav. 23, 2519- 2540 (2006) doi:10.1088/0264-9381/23/7/018 [arXiv:gr-qc/0512116 [gr-qc]]

arXiv 2006

[3] [1]

The Four laws of black hole mechanics,

J. M. Bardeen, B. Carter and S. W. Hawking, “The Four laws of black hole mechanics,” Commun. Math. Phys. 31, 161-170 (1973) doi:10.1007/BF01645742

doi:10.1007/bf01645742 1973

[4] [2]

Third Law of Black-Hole Dynamics: A Formulation and Proof,

W. Israel, “Third Law of Black-Hole Dynamics: A Formulation and Proof,” Phys. Rev. Lett. 57, no.4, 397 (1986) doi:10.1103/PhysRevLett.57.397

doi:10.1103/physrevlett.57.397 1986

[5] [3]

Gravitational collapse to extremal black holes and the third law of black hole ther- modynamics,

C. Kehle and R. Unger, “Gravitational collapse to extremal black holes and the third law of black hole ther- modynamics,” [arXiv:2211.15742 [gr-qc]]

arXiv

[6] [4]

Extremal black hole formation as a critical phenomenon,

C. Kehle and R. Unger, “Extremal black hole formation as a critical phenomenon,” [arXiv:2402.10190 [gr-qc]]

arXiv

[7] [5]

A Bogomolny Bound for General Relativity and Solitons in N=2 Supergravity,

G. W. Gibbons and C. M. Hull, “A Bogomolny Bound for General Relativity and Solitons in N=2 Supergravity,” Phys. Lett. B 109, 190-194 (1982) doi:10.1016/0370-2693(82)90751-1

doi:10.1016/0370-2693(82)90751-1 1982

[8] [6]

Positive Mass Theorems for Black Holes,

G. W. Gibbons, S. W. Hawking, G. T. Horowitz and M. J. Perry, “Positive Mass Theorems for Black Holes,” Commun. Math. Phys. 88, 295 (1983) doi:10.1007/BF01213209

work page doi:10.1007/bf01213209 1983