The index of the cosmological horizon and the area-charge-inequality

Neilha Pinheiro

arxiv: 2508.19490 · v2 · submitted 2025-08-27 · 🧮 math.DG · gr-qc

The index of the cosmological horizon and the area-charge-inequality

Neilha Pinheiro This is my paper

Pith reviewed 2026-05-18 21:50 UTC · model grok-4.3

classification 🧮 math.DG gr-qc

keywords MOTScosmological horizonKerr-Newman-de SitterMorse indexarea-charge inequalitydominant energy conditiongeneral relativityspacetime geometry

0 comments

The pith

The cosmological horizon in Kerr-Newman-de Sitter spacetime has a MOTS of index one when mass is bounded from below.

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

The paper studies the index of the marginally outer trapped surface that forms a spatial cross-section of the cosmological horizon in the Kerr-Newman-de Sitter spacetime. It proves that this index is at least one in the symmetrized sense when the angular momentum parameter is small and positive. Under an additional lower bound on the mass, the index is shown to be exactly one. An upper bound on the mass instead implies the index is at least two in the symmetrized sense. The work further derives an inequality connecting the area and charge for any MOTS of index one in Cauchy data that obeys the dominant energy condition.

Core claim

In the Kerr-Newman-de Sitter spacetime the spatial cross section of the cosmological horizon is a MOTS whose symmetrized index is at least one for small positive angular momentum parameter a. Assuming a lower bound on the mass the index equals one, while an upper bound on the mass yields index at least two in the symmetrized sense. The paper also proves an area-charge estimate for index-one MOTS in dominant-energy-condition Cauchy data, thereby relating such surfaces to general relativity.

What carries the argument

The symmetrized index of the marginally outer trapped surface (MOTS) given by the spatial cross-section of the cosmological horizon.

If this is right

When the mass satisfies the lower bound the cosmological horizon MOTS has index exactly one.
When the mass satisfies the upper bound the index is at least two in the symmetrized sense.
Index one MOTS obey an area-charge inequality in dominant energy condition data.
The results connect the index to properties in general relativity via the area-charge estimate.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The area-charge inequality may constrain charge-to-area ratios for stable trapped surfaces in de Sitter spacetimes.
Similar index results could hold for other horizons in spacetimes with positive cosmological constant.
The index-one property might indicate a stability that affects the long-term behavior of these surfaces.
Numerical checks of the index for chosen mass values could test the bound thresholds.

Load-bearing premise

The mass must satisfy a lower bound or upper bound to fix the index as one or at least two.

What would settle it

A direct computation of the index for a Kerr-Newman-de Sitter solution whose mass lies outside the assumed bounds but yields a different index value would disprove the claim.

read the original abstract

In this article, we investigate the index of the MOTS given by a spatial cross section of the cosmological horizon in the Kerr-Newman-de Sitter spacetime. We show that its index is at least one in the symmetrized sense for a small positive parameter a, such parameter defines the angular momentum. Assuming a lower bound for the mass, we prove that this MOTS has index one. Also, considering an upper bound for the mass, we show that its index is at least two in the symmetrized sense. Moreover, we establish an estimate relating the area and the charge of a MOTS with index one in a Cauchy data satisfying the dominant energy condition, which give us a connection between MOTS with index one and General Relativity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper pins down symmetrized index bounds for the cosmological MOTS in Kerr-Newman-de Sitter under small a and mass windows, plus a standard area-charge estimate under DEC.

read the letter

The main takeaway is that the paper shows the cosmological horizon MOTS in Kerr-Newman-de Sitter has symmetrized index at least one for small positive angular momentum parameter a. A lower mass bound gives exactly index one, while an upper mass bound gives at least two. It also produces an area-charge estimate for index-one MOTS in Cauchy data satisfying the dominant energy condition. These are the concrete statements to take away. The work applies the usual spectral analysis of the MOTS stability operator to this rotating charged de Sitter background. The small-a expansion controls the lowest eigenvalues, and the mass thresholds then fix the index values. The area-charge relation follows from a standard integral identity once index one is established, which ties the result back to general relativity. The conditioned index claims look new for this spacetime. The main limitations are the restrictions themselves. Results hold only for small a and within the stated mass ranges, so they do not cover arbitrary parameters. That may be required for the eigenvalue estimates to close, but it narrows the reach. The symmetrized index needs a clear definition in the text, though the logical steps from the linearized operator to the index statements appear standard and free of gaps. This is for specialists in mathematical general relativity who work on marginally outer trapped surfaces, horizon stability, or inequalities in de Sitter spacetimes. A reader wanting explicit index calculations in exact solutions would find usable statements here. The paper is technically grounded enough and the claims specific enough to deserve a serious referee. I recommend sending it to peer review.

Referee Report

0 major / 3 minor

Summary. The paper investigates the index of the marginally outer trapped surface (MOTS) given by a spatial cross-section of the cosmological horizon in Kerr-Newman-de Sitter spacetime. It claims that for small positive values of the angular momentum parameter a the symmetrized index is at least one; that a lower bound on the mass implies the index equals one exactly; that an upper bound on the mass implies the symmetrized index is at least two; and that an area-charge estimate holds for any index-one MOTS in a Cauchy data set satisfying the dominant energy condition.

Significance. If the derivations hold, the work supplies explicit spectral control on the MOTS stability operator in the rotating charged de Sitter background via small-a expansions and mass restrictions, together with a direct link from index-one MOTS to an area-charge inequality under the dominant energy condition. These results sit within the existing literature on MOTS stability and could be useful for rigidity or inequality questions in cosmological spacetimes.

minor comments (3)

The abstract and introduction should cite the precise theorem numbers (or section headings) that contain the four main statements, so that the logical structure is immediately visible to readers.
Clarify at first use (likely §2 or §3) how the symmetrized index is defined in terms of the standard index of the stability operator; a short comparison with the usual Morse index would help.
Ensure uniform notation for the mass bounds (lower versus upper) and for the small-a regime when these assumptions are invoked in the statements of the main theorems.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our results on the index of the cosmological horizon MOTS in Kerr-Newman-de Sitter and the associated area-charge inequality, as well as for recommending minor revision. No specific major comments appear under the MAJOR COMMENTS heading in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper establishes index results for the cosmological horizon MOTS in Kerr-Newman-de Sitter by direct spectral analysis of the stability operator, using an explicit small-a expansion for angular momentum and separate lower/upper mass bounds as stated assumptions to control eigenvalues. The area-charge estimate for index-one MOTS under DEC is obtained via a standard integral identity on the linearized operator, which is independent of the specific background parameters and does not reduce to a fit or self-citation. No load-bearing step equates a claimed prediction to its input by construction, and the logical chain from the operator to the index statements remains non-tautological.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The claims rest on the standard dominant energy condition and the specific geometric setup of the Kerr-Newman-de Sitter cosmological horizon; no new entities are introduced.

free parameters (2)

small positive angular momentum parameter a
Used to obtain the lower bound of one on the symmetrized index.
mass lower and upper bounds
Invoked to sharpen the index to exactly one or at least two.

axioms (1)

domain assumption Dominant energy condition holds for the Cauchy data
Required to derive the area-charge inequality for index-one MOTS.

pith-pipeline@v0.9.0 · 5647 in / 1395 out tokens · 53056 ms · 2026-05-18T21:50:45.998701+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Lsψ = −Δhψ + bψ where b = (Sch/2 − G(l+,ξ) − |χ+|²/2); index = number of negative eigenvalues of Ls; λ1(Ls(a)) < 0 < λ2(Ls(a)) for small a>0 under mass bound (3)
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Λ|Σ| + 16π²Q(Σ)²/|Σ| ≤ 12π with equality iff χ+≡0, EΣ=cν, (Scg)Σ≡2Λ+2c²

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

[1]

Andersson, M

L. Andersson, M. Mars and W. Simon,Stability of marginally outer trapped surfaces and existence of marginally outer trapped tubes Adv. Theor. Math. Phys.12 (2008), no. 4, 853–888

work page 2008
[2]

Booth, H.K

I. Booth, H.K. Kunduri, A. O’Grady,Unstable marginally outer trapped surfaces in static spherically symmetric spacetimes. Phys. Rev. D96 (2017)

work page 2017
[3]

Bousso, Charged Nariai black holes with a dilaton

R. Bousso, Charged Nariai black holes with a dilaton. Phys. Rev. D. (3) 55 (1997), no. 12, 3614-3621. 14 NEILHA PINHEIRO

work page 1997
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P. T. Chruściel, G. J. Galloway and D. Solis,Topological censorship for Kaluza- Klein space-times, Ann. Henri Poincaré. 10 (2009), no. 5, 893–912

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T. Cruz, V. Lima and A. de Souza,Min-Max Minimal Surfaces, Horizons and Elet- rostatic Systems, J. Differential Geometry. 128 (2024), 583-637

work page 2024
[6]

Dain and M

S. Dain and M. E. Gabach-Clement,Geometrical inequalities bounding angular mo- mentum and charges in General Relativity, Living Reviews in Relativity21 (2018), no. 5, 74pp

work page 2018
[7]

G. J. Galloway,Stability and rigidity of extremal surfaces in Riemannian geometry and general relativity. In Surveys in geometric analysis and relativity. Dedicated to Richard Schoen in honor of his 60th birthday. (2011), 221–239. Somerville, MA: In- ternational Press; Beijing: Higher Education Press

work page 2011
[8]

G. J. Galloway and R. Schoen,A generalization of Hawking’s black hole topology theorem to higher dimensions. Commun. Math. Phys.. 266 (2006), no 2, 571–576

work page 2006
[9]

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E. Gourgoulhon,3+1 Formalism in General Relativity. Bases of numerical relativity. Springer, Heidelberg. (2012), xviii+294 pp. ISBN: 978-3-642-24524-4, MR2906202, Zbl 1254.83001

work page arXiv 2012
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S. W. Hawking and G. F. R. Ellis,The large scale structure of space-time, Cam- bridge University Press, London, Cambridge Monographs on Mathematical Physics. (1973), no. 1

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[11]

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T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag Berlin Heidel- berg. 132 (1995)

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[12]

Li and S

P. Li and S. T. Yau,A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces, Invent. Math.. 69 (1982), no.2, 269-291, MR0674407, Zbl 0503.53042

work page arXiv 1982
[13]

Mars,Stability of Marginally Outer Trapped Surfaces and Applications, Springer Proceedings in Mathematics and Statistics.26 (2013), 111-138

M. Mars,Stability of Marginally Outer Trapped Surfaces and Applications, Springer Proceedings in Mathematics and Statistics.26 (2013), 111-138

work page 2013
[14]

Yang and S.T

P.C. Yang and S.T. Yau,Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci.(4) 7 (1980), no. 1, 55-63, MR0577325, Zbl 0446.58017

work page arXiv 1980
[15]

Ritoré and A

M. Ritoré and A. Ros, Stable constant mean curvature tori and the isoperimetric problem in three space forms, Comment. Math. Helvet.67 (1992), no. 2, 293-305, MR1161286, Zbl 0760.53037. Universidade Federal do Amazonas, Instituto de Ciências Exatas, Manaus, AM 69080900, Brazil Email address: neilha@ufam.edu.br

work page arXiv 1992

[1] [1]

Andersson, M

L. Andersson, M. Mars and W. Simon,Stability of marginally outer trapped surfaces and existence of marginally outer trapped tubes Adv. Theor. Math. Phys.12 (2008), no. 4, 853–888

work page 2008

[2] [2]

Booth, H.K

I. Booth, H.K. Kunduri, A. O’Grady,Unstable marginally outer trapped surfaces in static spherically symmetric spacetimes. Phys. Rev. D96 (2017)

work page 2017

[3] [3]

Bousso, Charged Nariai black holes with a dilaton

R. Bousso, Charged Nariai black holes with a dilaton. Phys. Rev. D. (3) 55 (1997), no. 12, 3614-3621. 14 NEILHA PINHEIRO

work page 1997

[4] [4]

P. T. Chruściel, G. J. Galloway and D. Solis,Topological censorship for Kaluza- Klein space-times, Ann. Henri Poincaré. 10 (2009), no. 5, 893–912

work page 2009

[5] [5]

T. Cruz, V. Lima and A. de Souza,Min-Max Minimal Surfaces, Horizons and Elet- rostatic Systems, J. Differential Geometry. 128 (2024), 583-637

work page 2024

[6] [6]

Dain and M

S. Dain and M. E. Gabach-Clement,Geometrical inequalities bounding angular mo- mentum and charges in General Relativity, Living Reviews in Relativity21 (2018), no. 5, 74pp

work page 2018

[7] [7]

G. J. Galloway,Stability and rigidity of extremal surfaces in Riemannian geometry and general relativity. In Surveys in geometric analysis and relativity. Dedicated to Richard Schoen in honor of his 60th birthday. (2011), 221–239. Somerville, MA: In- ternational Press; Beijing: Higher Education Press

work page 2011

[8] [8]

G. J. Galloway and R. Schoen,A generalization of Hawking’s black hole topology theorem to higher dimensions. Commun. Math. Phys.. 266 (2006), no 2, 571–576

work page 2006

[9] [9]

Gourgoulhon,3+1 Formalism in General Relativity

E. Gourgoulhon,3+1 Formalism in General Relativity. Bases of numerical relativity. Springer, Heidelberg. (2012), xviii+294 pp. ISBN: 978-3-642-24524-4, MR2906202, Zbl 1254.83001

work page arXiv 2012

[10] [10]

S. W. Hawking and G. F. R. Ellis,The large scale structure of space-time, Cam- bridge University Press, London, Cambridge Monographs on Mathematical Physics. (1973), no. 1

work page 1973

[11] [11]

Kato, Perturbation Theory for Linear Operators, Springer-Verlag Berlin Heidel- berg

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag Berlin Heidel- berg. 132 (1995)

work page 1995

[12] [12]

Li and S

P. Li and S. T. Yau,A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces, Invent. Math.. 69 (1982), no.2, 269-291, MR0674407, Zbl 0503.53042

work page arXiv 1982

[13] [13]

Mars,Stability of Marginally Outer Trapped Surfaces and Applications, Springer Proceedings in Mathematics and Statistics.26 (2013), 111-138

M. Mars,Stability of Marginally Outer Trapped Surfaces and Applications, Springer Proceedings in Mathematics and Statistics.26 (2013), 111-138

work page 2013

[14] [14]

Yang and S.T

P.C. Yang and S.T. Yau,Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci.(4) 7 (1980), no. 1, 55-63, MR0577325, Zbl 0446.58017

work page arXiv 1980

[15] [15]

Ritoré and A

M. Ritoré and A. Ros, Stable constant mean curvature tori and the isoperimetric problem in three space forms, Comment. Math. Helvet.67 (1992), no. 2, 293-305, MR1161286, Zbl 0760.53037. Universidade Federal do Amazonas, Instituto de Ciências Exatas, Manaus, AM 69080900, Brazil Email address: neilha@ufam.edu.br

work page arXiv 1992