Test fields cannot destroy extremal de Sitter black holes

Jose Natario; Rodrigo Vicente

REVIEW 2 major objections 3 minor 2 cited by

Test fields cannot destroy extremal de Sitter black holes

T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proves that test fields satisfying the null energy condition cannot overcharge or overspin an extremal Kerr-Newman-de Sitter black hole.

desk verdict A solid and genuinely useful extension of the no-destruction theorem to Kerr-Newman–de Sitter, but the uniqueness proof for the energy Killing field is explicitly incomplete and the stated theorem overreaches as written. read the letter →

arxiv 1908.09854 v1 pith:YBQHSZA5 submitted 2019-08-26 gr-qc

Jose Natario , Rodrigo Vicente This is my paper · ORCID

classification gr-qc MSC 83C5783C75 PACS 04.70.Bw04.20.-q

keywords weakcosmiccensorshipextremalblackholesKerr-Newman-deSittertestfieldsKillingvectorfieldnullenergyconditiongedankenexperimentsdefinition

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

The paper establishes which timelike Killing vector field defines the energy of test matter in Kerr-Newman-de Sitter spacetime, and uses that choice to prove that test fields satisfying the null energy condition at the event horizon cannot push an extremal de Sitter black hole past extremality. The problem is that asymptotically de Sitter spacetime has no ADM mass and no timelike Killing vector at infinity, so the energy of test matter is ambiguous; earlier literature used different Killing fields without proof. The authors construct spacetimes that interpolate between two Kerr-Newman-de Sitter solutions with different mass and charge parameters, compute the energy of the interpolating field with respect to a candidate Killing field, and show it equals the physical mass difference exactly for one candidate, $K=\partial/\partial t + (a/l^2)\partial/\partial\varphi$. A linearized divergence-theorem argument then extends this energy identification to arbitrary test fields, and a power-series check on alternative Killing fields supports the uniqueness of this $K$, yielding the no-destruction result.

What carries the argument

The load-bearing object is the interpolation metric: take the Kerr-Newman-(A)dS metric and let $m=m(r)$, $q=q(r)$ pass smoothly from $(m_1,q_1)$ to $(m_2,q_2)$ between two radii. The Einstein equations then define the energy-momentum tensor of the interpolating (unphysical) field, and the radial integration identity $A(r)-B'(r)=1/\Xi^2$ turns the energy flux computed with $K$ into $\Delta M=(m_2-m_1)/\Xi^2$. The companion linearized argument uses the divergence theorem on a hollow cylinder to show that all test fields share the model field's energy, and the uniqueness argument constrains any other Killing candidate $K+\varepsilon Y$ by requiring $\gamma(1-\varepsilon a)=1$, then setting $\varepsilon_0=\varepsilon_1=0$ by a power series in $a$.

What would settle it

Compute the coefficients $\varepsilon_n$ for $n\ge 2$ in the uniqueness expansion; if any is nonzero, $K$ is not the unique energy-defining Killing field and the application of the general theorem needs re-examination. Alternatively, exhibit a test field satisfying the null energy condition at the horizon that takes an extremal Kerr-Newman-de Sitter black hole to a configuration beyond extremality, which would directly falsify the no-destruction claim.

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Extended reading notes

Core claim

The central claim is that in a Kerr-Newman-de Sitter background with rotation parameter $a$ and cosmological scale $l$, the timelike Killing vector field $K=\partial_t+(a/l^2)\partial_\varphi$ is the correct generator of energy for test fields, and that with this definition the horizon generator is $K+\Omega_H \partial_\varphi$. The proof uses an interpolation metric in which the mass parameter $m$ and charge parameter $q$ vary with the radial coordinate between two Kerr-Newman-(A)dS solutions; the Einstein equations fix the energy-momentum tensor of the interpolating field, and identities such as $A(r)-B'(r)=1/\Xi^2$ convert the energy integral into exactly the physical mass difference $M_2-M_1$. A linearized argument shows the same energy is obtained for any test-field configuration, and a power-series check on candidate Killing fields $\tilde K=\gamma(K+\varepsilon Y)$ yields $\varepsilon_0=\varepsilon_1=0$, so $K$ is unique at leading order. Applying the earlier general theorem, an extremal Kerr-Newman-de Sitter black hole that absorbs energy, angular momentum and charge from such test fields ends up either subextremal or extremal, never with a naked singularity.

Load-bearing premise

The argument assumes that $K$ is the unique timelike Killing field defining energy: the authors show the first two power-series coefficients of any alternative vanish, but they do not compute the higher-order coefficients, so a nonzero higher-order coefficient would undermine uniqueness.

Editorial extensions

If this is right

In asymptotically de Sitter black-hole spacetimes, test-field energies should be computed with $K=\partial_t+(a/l^2)\partial_\varphi$; using a different stationary Killing field generally gives the wrong energy and can lead to false cosmic-censorship-violation claims.
Extremal Kerr-Newman-de Sitter black holes are protected against test-field destruction: any absorption obeying the null energy condition at the horizon leaves the spacetime subextremal or extremal.
The same interpolation technique confirms the energy Killing field used earlier for the anti-de Sitter case, $K=\partial_t-(a/l^2)\partial_\varphi$.
A finite-energy condition at infinity is essential: the no-destruction theorem applies to test fields satisfying the stated boundary conditions, not to fields with unbounded energy.
The result extends the original gedanken-experiment no-go to backgrounds with a positive cosmological constant, removing the need for an ADM mass.

Reading between the lines

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

If the higher-order coefficients $\varepsilon_n$ for $n\ge 2$ are computed and found to vanish, $K$ is the unique energy-defining Killing field; if one is nonzero, the physical-mass interpretation would need revisiting.
The interpolation construction suggests a route to a quasilocal mass for asymptotically de Sitter spacetimes: integrate the same energy flux up to the cosmological horizon rather than to infinity.
The same technique could be applied to near-extremal de Sitter black holes using second-order variations, paralleling the quasi-extremal analysis, to see whether the subextremal fallback persists.
A numerical experiment with finite-energy wave packets in extremal Kerr-Newman-de Sitter could map the boundary of the allowed parameter changes, making the theorem's inequality explicit.

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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

The paper is worth reading: it extends Natário–Queimada–Vicente's no-destruction theorem to extremal Kerr-Newman–de Sitter, and it isolates the energy Killing field K = X + (a/l^2)Y for dS. The interpolation trick—letting m and q become functions of r—is genuinely useful, and the integral identities in Sections 2–3 are explicit and checkable. Section 4's divergence-theorem extension to arbitrary linearized perturbations (with the same spin parameter) is clean. Credit where due: this is careful work, and the dS case was missing.

The soft spot is Section 5. The authors want to rule out other Killing fields K̃ = γ(K + εY). They derive γ(1−εa)=1 and then, by varying the spin parameter a(r), show ε0=ε1=0 in the expansion ε(a)=Σε_n a^n. They then say "we have not computed the higher order coefficients... but we expect them to also vanish." That is an omitted proof, and it is exactly the point where the uniqueness claim needs support. Worse, the calculation is an expansion in the background spin a0; even the vanishing of ε0 and ε1 is established only in a small-spin expansion truncated at O(a0^3). A nonzero ε2 would produce another Killing field that matches ΔM on the constructed family to the order computed, and the claim that K is "the" correct definition of energy would not be established for arbitrary spin. Since Theorem 6.1 leans on that claim via Theorem 4.1 of [25], the central conclusion is conditional as written.

Is this a deal-breaker? I don't think so. The no-destruction result is very plausibly true, and the gap is explicit rather than hidden. A referee could reasonably ask the authors to complete the higher-order calculation or to re-state the theorem with the uniqueness claim softened to "there exists a choice of K for which...". It is also possible the uniqueness proof can be bypassed: for a generic test perturbation, ΔJ and ΔM are independent, and then any ε≠0 fails to make γ(ΔM−εΔJ)=ΔM for all perturbations. If that simpler argument works, Section 5 is overkill; if it does not, Section 5 needs finishing.

The references and citation pattern are fine; reliance on [25] is legitimate because it is an external published theorem. The paper is honestly written and clearly signals its own limitation.

Verdict: worth a serious referee. I would send it out, with a request to fix or explicitly soften the uniqueness claim before acceptance.

Referee Report

2 major / 3 minor

Summary. The paper addresses the choice of timelike Killing vector field for defining the energy of test fields in Kerr-Newman-de Sitter spacetime. In Sections 2 and 3 the authors interpolate between two Kerr-(A)dS or Kerr-Newman-(A)dS metrics by promoting the mass parameter m (and, in Section 3, the charge parameter q) to functions of the radial coordinate, compute the effective energy-momentum tensor from the Einstein equations, and show that for the Killing vector K = X + (a/l^2)Y in the dS case (and K = X - (a/l^2)Y in AdS) the integrated energy and angular momentum equal the differences of the physical mass and angular momentum. Section 4 uses a divergence-theorem argument to extend this result from the special interpolating linearized solution to arbitrary linearized solutions. Section 5 attempts to prove uniqueness of K among Killing fields of the form K̃ = γ(K + εY), deriving the condition γ(1 - εa) = 1 and using a power-series expansion in a to show that the first two coefficients ε0 and ε1 vanish; the authors explicitly state that higher coefficients have not been computed. Section 6 then invokes Theorem 4.1 of the authors' earlier paper [25] to conclude that test fields cannot destroy extremal Kerr-Newman-dS black holes.

Significance. If the uniqueness step were completed, the paper would close a significant gap in the weak cosmic censorship literature for asymptotically de Sitter black holes, where there is no ADM mass and the choice of energy-defining Killing field is otherwise ambiguous. The interpolation identities (17), (23), (37), and (48) are explicit and verifiable, and the divergence-theorem argument in Section 4 is an elegant method for passing from a special linearized solution to arbitrary test fields. The final no-destruction statement is not simply a restatement of [25], because identifying the physical mass variation with the energy computed using K is a necessary nontrivial input. However, the central claim is currently conditional on an omitted piece of the uniqueness proof.

major comments (2)

[Section 5, Eqs. (52)–(60)] The uniqueness proof establishes only that ε0 = ε1 = 0 in the expansion ε(a) = Σ ε_n a^n. The authors state in Section 5 that they 'have not computed the higher order coefficients ε_n with n ≥ 2, but we expect them to also vanish.' This is an explicit gap: a nonzero ε2 would yield another Killing field K̃ = (K + εY)/(1 - εa) that also reproduces E = ΔM on the constructed interpolating family, so K would not be established as 'the' energy-defining Killing field. Since the application of Theorem 4.1 of [25] in Section 6 depends on identifying the physical energy with K, the main theorem remains conditional. The authors should either compute the higher-order coefficients or provide a structural argument showing that only the first two coefficients can be nonzero.
[Section 5, restriction to q1 = q2 = 0] The uniqueness calculation is performed only in the uncharged Kerr-(A)dS case, whereas Theorem 6.1 concerns Kerr-Newman-dS black holes. The paper does not explain why the charged case reduces to the uncharged one, nor does it rule out an ε that depends on the charge parameter in addition to a. To support the claimed uniqueness of K in the charged setting, the authors should either extend the computation to nonzero fixed charge or give a clear argument that the charge terms cannot affect the relevant coefficients of ε.

minor comments (3)

[Section 2, Eq. (25)] The formula E + ωL = (1 + ωa)ΔM appears to conflict with the definition L = -∫ T_μν Y^μ N^ν dV_3 in Eq. (18), which gives ∫ T(K + ωY)N = E - ωL = (1 - ωa)ΔM. The conclusion ω = 0 is unchanged, but the sign should be corrected for consistency with Eq. (53).
[Section 4, Eqs. (50)–(51)] The hypotheses on the arbitrary linearized metric (vanishing for r ≤ r1 and matching the final KN solution for r ≥ r2) are stated at the beginning of the section but should be repeated in the divergence-theorem paragraph, since the independence of the lateral boundary integrals relies on these assumptions.
[General] The text contains several typographical artifacts, such as 'Kerr-Newm an' in the abstract and unusual spacing in some displayed equations; a careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Killing field is verified against the independent mass difference; the Section 5 gap and [25] citation are proof/citation issues, not circular.

full rationale

The derivation is self-contained in the relevant sense. Sections 2–3 construct an explicit interpolating spacetime with m(r) and q(r), compute T^{μν} from the Einstein equations, and verify by direct integration and identities (e.g., B''(r)=A'(r), A(r)-B'(r)=1/Ξ^2) that the integral of the energy flux with K equals the independently defined physical mass difference ΔM = M2 - M1 (Eqs. (17), (37)). The Killing vector K is not fitted to ΔM; it is checked against it. Section 4 extends this to arbitrary linearized perturbations by a divergence-theorem argument rather than assuming the conclusion. Section 5 does contain an acknowledged omitted proof: after deriving γ(1-εa)=1 for any alternative K̃=γ(K+εY), the authors say 'we have not computed the higher order coefficients ε_n with n ≥ 2, but we expect them to also vanish.' That is a real gap in the uniqueness claim, but it is a gap in the proof, not a circular reduction of the conclusion to its inputs. The final theorem relies on Theorem 4.1 of [25], a previously published theorem by two of the same authors; this is an external cited result with its own proof, not a self-referential restatement of the present construction. No fitted parameter is renamed as a prediction, and no ansatz is smuggled in merely by citation. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper's central claim rests on the interpretation of m/Ξ^2 as the physical mass in asymptotically de Sitter spacetimes, on the identification of the interpolating field energy with ΔM, and on the authors' previous theorem. No data fitting is involved.

assumptions (5)

domain assumption M = m/Ξ^2, J = ma/Ξ^2, Q = q/Ξ are the physical mass, angular momentum, and charge in Kerr-Newman-(A)dS spacetimes
Used throughout to identify ΔM, ΔJ, ΔQ with parameter differences. In asymptotically de Sitter spacetimes there is no generally accepted total mass, so this identification is an input assumption.
domain assumption The energy of the interpolating field is defined as M2 - M1
Used in Sections 2 and 3 as the benchmark to fix the energy Killing vector. This is a physical input, not derived from the metric alone.
domain assumption Theorem 4.1 of [25] applies to extremal Kerr-Newman-dS once the correct energy Killing field is chosen
The no-destruction result is inherited from the authors' previous published theorem; it is not re-derived here.
ad hoc to paper ε(a) is analytic and the power series expansion in a can be truncated for the uniqueness argument
Section 5 assumes analyticity and only computes the first two coefficients, leaving the rest unproven.
domain assumption Test fields satisfy the null energy condition at the event horizon
Standard physical condition assumed in [25] and required for Theorem 6.1.

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Pith. "Pith review of Test fields cannot destroy extremal de Sitter black holes." pith.science (2026). https://pith.science/paper/YBQHSZA5

@misc{pith2026190809854,
  author       = {Pith},
  title        = {Pith review of: Test fields cannot destroy extremal de Sitter black holes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YBQHSZA5}},
  note         = {Machine review of arXiv:1908.09854}
}

read the original abstract

We determine the timelike Killing vector field that gives the correct definition of energy for test fields propagating in a Kerr-Newman-de Sitter spacetime, and use this result to prove that test fields cannot destroy extremal Kerr-Newman-de Sitter black holes.

Figures

Figures reproduced from arXiv: 1908.09854 by the authors.

**Figure 2.** Penrose diagram illustrating the deformation of an unboun [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Domain for the application of the divergence theorem. [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

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

Weak Cosmic Censorship with spinning particles in Kerr-(A)dS spacetimes
gr-qc 2025-07 conditional novelty 6.0 of 10

Extremal Kerr-de Sitter black holes cannot be overspun by spinning test particles, and apparent Kerr-anti-de Sitter overspinning configurations are eliminated once the particle's finite size is respected.
Six-dimensional Myers-Perry rotating black hole cannot be overspun
gr-qc 2019-08 conditional novelty 5.0 of 10

A six-dimensional rotating black hole with two spins cannot be overspun by test particle accretion, so weak cosmic censorship holds for this case.

Reference graph

Works this paper leans on

38 extracted references · 36 canonical work pages · cited by 2 Pith papers

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [25]

Nat´ ario, L

J. Nat´ ario, L. Queimada and R. Vicente, Test ﬁelds cannot destroy extremal black holes , Class. Quant. Grav. 33 (2016), 175002

work page 2016

[2] [1]

Wald, Gedanken experiments to destroy a black hole , Ann

R. Wald, Gedanken experiments to destroy a black hole , Ann. of Phys. 83 (1974), 548–556

work page 1974

[3] [2]

Penrose, Gravitational collapse: the role of general relativity , Riv

R. Penrose, Gravitational collapse: the role of general relativity , Riv. Nuovo Cim. 1 (1969), 252–276

work page 1969

[4] [3]

Wald, Gravitational collapse and cosmic censorship , arXiv:gr-qc/9710068 (1997)

R. Wald, Gravitational collapse and cosmic censorship , arXiv:gr-qc/9710068 (1997)

arXiv 1997

[5] [4]

K. Tod, F. de Felice and M. Calvani, Spinning test particles in the ﬁeld of a black hole , Nuovo Cim. B 34 (1976), 365–379

work page 1976

[6] [5]

Needham, Cosmic censorship and test particles , Phys

T. Needham, Cosmic censorship and test particles , Phys. Rev. D 22 (1980), 791–796

work page 1980

[7] [6]

Semiz, Dyonic Kerr-Newman black holes, complex scalar ﬁeld and cos mic censorship , Gen

I. Semiz, Dyonic Kerr-Newman black holes, complex scalar ﬁeld and cos mic censorship , Gen. Rel. Grav. 43 (2011), 833–846

work page 2011

[8] [7]

Toth, Test of the weak cosmic censorship conjecture with a charged scalar ﬁeld and dyonic KerrNewman black holes , Gen

G. Toth, Test of the weak cosmic censorship conjecture with a charged scalar ﬁeld and dyonic KerrNewman black holes , Gen. Rel. Grav. 44 (2015), 2019–2035

work page 2015