Pith. sign in

REVIEW 2 major objections 3 minor 29 references

Conservation Laws and Boundedness for Linearised Einstein--Maxwell Equations on the Reissner--Nordstr\"om Black Hole

T0 review · 2 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read On charged Reissner–Nordström black holes with $|Q|/M < \sqrt{15}/4$, the gauge-invariant Teukolsky-variable energy fluxes are uniformly bounded, and a conservation law alone supplies the control.

desk verdict A genuinely new conservation-law proof of boundedness for linearised Einstein–Maxwell on Reissner–Nordström, with a real charge restriction and a theorem statement that omits load-bearing decay hypotheses. read the letter →

arxiv 2506.04812 v1 pith:RPPS5TKR submitted 2025-06-05 gr-qc math.AP

Marios A. Apetroaie , Sam C. Collingbourne , Elena Giorgi This is my paper
classification gr-qcmath.AP MSC 83C2283C5783C0535Q75
keywords linearisedEinstein–MaxwellequationsReissner–NordströmcanonicalenergyconservationlawTeukolskyvariablesdoublenullgaugeblackholestabilitycharge-to-massratio
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

This paper proves a boundedness statement for linear perturbations of charged, non-rotating black holes: for the Reissner–Nordström exterior with charge-to-mass ratio $|Q|/M < \sqrt{15}/4$, the energy fluxes of the gauge-invariant Teukolsky variables ${}^{(1)}b$, ${}^{(1)}f$, ${}^{(1)}\alpha$ and their negative-spin counterparts are uniformly bounded along every outgoing null cone. The route is deliberately different from the standard Teukolsky-wave-equation analysis: the authors derive a canonical energy conservation law for the full linearised Einstein–Maxwell system in double null gauge and prove the conserved fluxes are coercive, so they actually bound a positive energy. From that coercive estimate plus a hierarchy of transport equations, they reach the Teukolsky variables without ever invoking the hyperbolic nature of their wave equations. The result yields uniform $L^2$ control on spheres and pointwise weighted bounds, offering an alternative path to linear stability of Reissner–Nordström in this charge range.

What carries the argument

The load-bearing object is the canonical energy conservation law in double null gauge, expressed as the flux balance $E_{u_1}[S](v_0,v_1)+E_{v_1}[S](u_0,u_1)=E_{u_0}[S](v_0,v_1)+E_{v_0}[S](u_0,u_1)$ for any solution $S$ of the linearised Einstein–Maxwell system. The fluxes are not obviously positive because they contain mixed curvature–connection terms, so the proof rewrites them as a sum of manifestly positive, gauge-invariant terms plus two boundary terms on the initial and final spheres. The final-sphere boundary term is absorbed through a transport estimate for the combination $\xi = \bigl(X + (1 - r_c/r)\,6M\lambda\bigr)/\sqrt{24M}$, where $X$ and $\lambda$ are mass-aspect-type quantities built from the perturbed connection and curvature; absorption succeeds precisely when the coefficient $1 - 32Q^2/(24M r_+)$ is non-negative, i.e., $|Q| \le \sqrt{15}\,M/4$. Once the resulting master energy controls the connection and electromagnetic variables, a hierarchy of transport estimates derived from the linearised Bianchi and Maxwell equations carries the control up to the Teukolsky variables ${}^{(1)}b$, ${}^{(1)}f$, ${}^{(1)}\alpha$ and their negative-spin counterparts.

What would settle it

The central claim would be refuted by a smooth characteristic initial data set on the Reissner–Nordström exterior with $|Q|/M < \sqrt{15}/4$, obeying the paper's decay assumptions, whose outgoing energy flux for the Teukolsky variables ${}^{(1)}b$, ${}^{(1)}f$, ${}^{(1)}\alpha$ or their negative-spin counterparts diverges along a sequence of cones $C_{u_n}$. A more targeted check of the threshold: at $|Q|/M = \sqrt{15}/4$, one has $r_+ = r_c = 4Q^2/(3M)$, so the absorption coefficient $1 - 32Q^2/(24M r_+)$ in the proof of the coercive estimate vanishes; evaluating the final-sphere boundary term at this charge for the simplest nonzero angular mode would show whether coercivity genuinely degenerates there or whether a different weight function could absorb it.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: for any smooth solution of the linearised Einstein–Maxwell equations arising from characteristic initial data on the Reissner–Nordström exterior with $|Q|/M < \sqrt{15}/4$, the energy fluxes $E_u[\,^{(1)}b\,]$, $E_u[\,^{(1)}f\,]$, $E_u[\,^{(1)}\alpha\,]$ and their negative-spin counterparts on any outgoing null cone $C_u$ are bounded by a uniformly controlled initial-data energy $E_{\rm data}(u)$. Here the Teukolsky variables are gauge-invariant combinations of curvature, connection, and electromagnetic components that carry the physical gravitational and electromagnetic degrees of freedom. The proof establishes this from a canonical energy conservation law in double null gauge: the conserved fluxes are rewritten as manifestly positive gauge-invariant terms plus boundary terms, the boundary term on the final sphere is absorbed by a transport estimate exactly when $|Q| \le \sqrt{15}\,M/4$, and then a hierarchy of transport equations propagates control to the desired variables. No use is made of the Teukolsky equations or of any transformation to auxiliary wave-type variables; the boundedness statement is obtained purely from the conservation law and transport estimates. The theorem also holds with additional derivatives after commutation with the spacetime symmetries, giving pointwise bounds such as $\sup |r^4\Omega\,^{(1)}b|$, $\sup |r^2\Omega\,^{(1)}f|$, and $\sup |r\Omega^2\,^{(1)}\alpha|$ controlled by the initial energy.

Load-bearing premise

The argument assumes the perturbation falls off fast enough at large distances that certain weighted components and their first two derivatives have well-defined finite limits; if a finite-energy solution decays more slowly, the boundary terms in the conservation law need not vanish and the energy bound could fail.

Editorial extensions

If this is right

  • For every outgoing null cone $C_u$, the energy fluxes of the gauge-invariant Teukolsky variables ${}^{(1)}b$, ${}^{(1)}f$, ${}^{(1)}\alpha$ and their negative-spin counterparts are uniformly bounded by the initial-data energy whenever $|Q|/M < \sqrt{15}/4$.
  • These flux bounds imply uniform $L^2$ control on the cross-sectional spheres of each outgoing cone, and after commutation with the stationary and angular Killing fields, pointwise weighted bounds on the Teukolsky variables.
  • The boundedness statement is obtained without analysing the Teukolsky equations as hyperbolic equations and without transforming to auxiliary wave-type variables; the conservation law and transport estimates carry the whole argument.
  • Because the transport hierarchy itself remains valid for the full range $|Q| \le M$, any future proof of coercivity of the canonical flux beyond $\sqrt{15}/4$ would immediately upgrade the boundedness statement to the full subextremal range.
  • Commutation with the spacetime symmetries yields higher-derivative versions of the energy estimate, which turn sphere $L^2$ bounds into pointwise weighted bounds.

Reading between the lines

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

  • An extension left implicit in the paper: the boundedness statement does not by itself give decay of the perturbation toward infinity, and it is natural to ask whether the same conserved flux plus a monotonicity argument yields integrated or pointwise decay in $v$.
  • A testable consequence of the proof's structure: the charge threshold appears only in the absorption of the final-sphere boundary term, so one could look for a different weight function that absorbs that term beyond $|Q| = \sqrt{15}\,M/4$; failure would support the paper's suggestion that the threshold is genuine.
  • For a rotating charged black hole, the background is stationary but not static, so the double-null flux expressions would acquire additional terms; the paper's transport hierarchy would still be available, but the coercivity step would be the main obstacle.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. This paper studies the linearised Einstein–Maxwell system on a fixed subextremal Reissner–Nordström exterior. Working in double null gauge, the authors define flux integrals on characteristic cones, prove a conservation law by direct computation (Proposition 3.1), and rewrite the fluxes in a manifestly coercive form up to boundary terms. Under pointwise decay assumptions (61)–(62) at null infinity and for partially initial data normalised solutions supported on angular modes ℓ≥1, Theorem 3.1 establishes a coercive energy estimate whenever |Q|/M < √15/4. A hierarchy of transport estimates in Sections 4.1–4.2 then controls the energy fluxes of the gauge-invariant Teukolsky variables (1)b, (1)f, (1)α and their negative-spin counterparts along outgoing null cones, yielding the uniform boundedness statement of Theorem 1.1. Appendix A argues, via Sylvester's criterion, that the charge threshold is optimal for the method used here.

Significance. If correct, the paper provides a genuinely different route to boundedness estimates for linearised Einstein–Maxwell perturbations, avoiding the Teukolsky and Regge–Wheeler machinery and relying instead on a canonical-energy conservation law. This is a meaningful extension of the Holzegel–Collingbourne approach from Schwarzschild to a charged background. The proof is detailed: the conservation law is derived explicitly, the charge restriction appears as the transparent condition r_+ ≥ 4Q²/(3M), and the hierarchy estimates are written out. The authors are also explicit that the range |Q| < √15/4 M is suboptimal compared with known results in the full subextremal range, and Appendix A limits the optimality claim to the method. The main caveats are that Theorem 1.1 is stated more broadly than the hypotheses actually used in the proof, and that one step in the negative-spin hierarchy breaks down at Q=0 even though the theorem includes Q=0.

major comments (2)
  1. [§1.2 (Theorem 1.1), §2.7 ((61)–(62)), §3.2 (Theorem 3.1)] Theorem 1.1 is stated for “any smooth solution of the system of linearised perturbations arising from characteristic initial data,” but the proof of Theorem 3.1 requires the additional hypotheses explicitly listed in §2.7: partial initial data normalisation, support on ℓ≥1, and the pointwise decay assumptions (61)–(62). These are not cosmetic: (61)–(62) are used in the proof of Theorem 3.1 to show that the boundary term in (67)–(68) vanishes as v1→∞ and to identify the data boundary terms; without them the conservation law (65) does not lead to the coercive bound. The statement in §2.7 that these assumptions are “compatible” with the well-posedness theory of [4] does not imply that every smooth finite-energy characteristic data solution satisfies them, and no proof of such an implication is given. The theorem statement must either include (61)–(62), the partial normalisation and ℓ≥1 support as explicit hypotheses, or the authors must prove that all solutions covered by the current statement satisfy them.
  2. [§4.2, “Control of E_u[(1)α]”] The bound for the negative-spin variable (1)α is obtained by rewriting the Bianchi identity (56) as “Q r Ω ∇4(rΩ² (1)α) = …” and then applying Lemma 2.3.1. This identity divides by Q. For Q=0 the displayed equation degenerates and the argument does not apply, yet the stated range |Q|/M < √15/4 includes Q=0. No separate Schwarzschild treatment is supplied. Either add an argument covering Q=0 (for example by appealing to known vacuum results, if that is intended) or state the theorem for 0<|Q|<√15/4 M.
minor comments (3)
  1. [§3.2, Corollary 3.1] In the definition of E^{i,j}_{v0}[S](u0,u1), the displayed integral appears to be ∫_{v1}^{v0} … dv; since this is a flux on the initial outgoing cone C_{v0}, it should presumably be ∫_{u0}^{u1} … du.
  2. [§2.2 and §3.2] The notation “r{∇s” is used extensively in the master energy (66) and in the hierarchy estimates, but it is not formally introduced; the authors should define it explicitly alongside the other projected operators.
  3. [§2.7] The list in (61) includes both r^{2+s}(F)(1)β and r(F)(1)β as quantities with finite limits; for s>0 the second is redundant. This is harmless but could be streamlined.

Circularity Check

0 steps flagged · score 2.0 of 10

No exhibited circular step: the conservation law (65) is an explicit identity, the coercive Theorem 3.1 is proven under decay assumptions (61)-(62) that do not contain the target bound, and the Section 4 hierarchy genuinely derives the Teukolsky-flux bounds. The flagged gap is that Theorem 1.1 omits (61)-(62), which is a completeness issue, not circularity.

full rationale

Walking the claimed derivation chain—fluxes (63)-(64) to conservation law (65) to coercive estimate (Theorem 3.1) to the hierarchy of transport estimates (Section 4) to Theorem 1.1—I find no step in which a predicted quantity equals an input by construction. The flux integrands in (63)-(64) are explicit sums of linearised quantities, not defined as the conclusion; the conservation law is a derived identity ('Direct computation', Prop 3.1). Theorem 3.1's coercivity uses the decay assumptions (61)-(62) only to show that the v1-goes-to-infinity boundary term vanishes and to identify data terms; these are pointwise weighted-limit conditions that do not include the target flux bound, so the input does not contain the conclusion. The Teukolsky variables (1)f, (1)b, (1)alpha are defined explicitly in (52)-(53), the relations (55)-(56) used in Section 4 are restated in the paper, and the bounds (75), (78), (82), (84) are actually derived via Lemma 2.3.1 and the Codazzi/Bianchi equations, reducing to the master energy (66) plus initial-sphere norms on distinct null surfaces—genuine propagation, not renaming. The methodological template and gauge-invariant framework come from the authors' prior work ([1,2,3,19,27]), but the charged-case content is reproduced or checkable in the paper, so these citations are evidence rather than black boxes. The one flagged weakness is a hypothesis mismatch: Theorem 1.1 promises the bound for 'any smooth solution arising from characteristic initial data', while the proof of Theorem 3.1 requires the decay assumptions (61)-(62), partial initial-data normalisation, and support on l>=1, which are not stated in the theorem; per the review rule this is flagged (Section 2.7 versus Theorem 1.1 and Theorem 3.1) and weighed as a completeness and correctness risk, not as circularity, since (61)-(62) do not presuppose the boundedness being proved. Score 2, no circular steps exhibited.

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

The paper introduces no new physical entities. Its central claim rests on standard mathematical background (double null geometry), the assumed linearised Einstein-Maxwell system, and a set of technical conditions (partial initial data normalisation, ℓ>=1 support, and decay assumptions) that are imposed rather than derived. The charge restriction |Q|/M < sqrt(15)/4 is a theorem condition, not a free parameter fitted to data.

assumptions (5)
  • domain assumption The background Reissner-Nordström spacetime in double null coordinates (u,v,θ,φ) on the exterior region.
    Section 2.1 fixes the background metric and electromagnetic tensor; the proof operates entirely in this exterior region with the double null foliation.
  • standard math The linearised Einstein-Maxwell system in double null gauge, equations (7)-(44).
    Section 2.4 states the full system; the authors refer to [4,14] for its derivation, so the system is assumed to be the correct linearisation.
  • domain assumption Partial initial data normalisation conditions in Definition 2.1.
    The proof of Theorem 3.1 requires these conditions on the initial data; the authors assert they can be achieved by an appropriate pure gauge choice, so they are a valid gauge normalization.
  • domain assumption Support on angular modes ℓ ≥ 1.
    Electromagnetic perturbations contribute at ℓ=1; ℓ=0 modes are gauge degrees of freedom, so the restriction is necessary for the gauge-invariant quantities considered.
  • ad hoc to paper Decay assumptions (61)-(62) toward null infinity for all perturbed quantities and two derivatives.
    Theorem 3.1 relies on these pointwise decay/limit assumptions to make boundary terms vanish as v1 -> ∞ and to control horizon boundary terms; no theorem guarantees these for all finite-energy solutions, making this a load-bearing imposed assumption.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Conservation Laws and Boundedness for Linearised Einstein--Maxwell Equations on the Reissner--Nordstr\"om Black Hole." pith.science (2026). https://pith.science/paper/RPPS5TKR

@misc{pith2026250604812,
  author       = {Pith},
  title        = {Pith review of: Conservation Laws and Boundedness for Linearised Einstein--Maxwell Equations on the Reissner--Nordstr\"om Black Hole},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RPPS5TKR}},
  note         = {Machine review of arXiv:2506.04812}
}
abstract

We study the linearised Einstein--Maxwell equations on the Reissner--Nordstr\"om spacetime and derive the canonical energy conservation law in double null gauge. In the spirit of the work of Holzegel and the second author, we avoid any use of the hyperbolic nature of the Teukolsky equations and rely solely on the conservation law to establish control of energy fluxes for the gauge-invariant Teukolsky variables, previously identified by the third author, along all outgoing null hypersurfaces, for charge-to-mass ratio $\frac{|Q|}{M} < \frac{\sqrt{15}}{4}$. This yields uniform boundedness for the Teukolsky variables in Reissner--Nordstr\"om.

Figures

Figures reproduced from arXiv: 2506.04812 by the authors.

Figure 1
Figure 1. Initial data is prescribed on intersecting outgoing and ingoing null cones [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

29 extracted references · 24 canonical work pages

  1. [22]

    The linear stability of Reissner-Nordstr¨ om spacetime: the full sub-extremal range |Q| ăM ,

    ——, “The linear stability of Reissner-Nordstr¨ om spacetime: the full sub-extremal range |Q| ăM ,” Comm. Math. Phys. , vol. 380, 2020

    work page 2020

  2. [4]

    The linear stability of the Schwarzschild solution to gravitational perturbations,

    M. Dafermos, G. Holzegel, and I. Rodnianski, “The linear stability of the Schwarzschild solution to gravitational perturbations,” Acta Math., vol. 222, no. 1, pp. 1–214, 2019

    work page 2019

  3. [1]

    Uniform Boundedness for Solutions to the Teukolsky Equation on Schwarz- schild from Conservation Laws of Linearised Gravity,

    S. C. Collingbourne and G. Holzegel, “Uniform Boundedness for Solutions to the Teukolsky Equation on Schwarz- schild from Conservation Laws of Linearised Gravity,” Comm. Math. Phys. , vol. 405, no. 138, 2024

    work page 2024

  4. [2]

    Boundedness and Decay for the Teukolsky System of Spin ˘2 on Reissner–Nordstr¨ om Spacetime: The Case Qăă M ,

    E. Giorgi, “Boundedness and Decay for the Teukolsky System of Spin ˘2 on Reissner–Nordstr¨ om Spacetime: The Case Qăă M ,” Ann. Hen. Poin. , vol. 21, no. 8, pp. 2485–2580, 2020

    work page 2020

  5. [3]

    Boundedness and decay for the Teukolsky system of spin ˘1 on Reissner-Nordstr¨ om spacetime: theℓ“ 1 spherical mode,

    ——, “Boundedness and decay for the Teukolsky system of spin ˘1 on Reissner-Nordstr¨ om spacetime: theℓ“ 1 spherical mode,” Classical and Quantum Gravity , vol. 36, p. 205001, 2019

    work page 2019

  6. [5]

    Klainerman and J

    S. Klainerman and J. Szeftel, Global nonlinear stability of Schwarzschild spacetime under polarized perturbations, ser. Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 2020, vol. 210

    work page 2020

  7. [6]

    The non-linear stability of the Schwarzschild family of black holes,

    M. Dafermos, G. Holzegel, I. Rodnianski, and M. Taylor, “The non-linear stability of the Schwarzschild family of black holes,” 2021, arXiv:2104.08222

    arXiv 2021

  8. [7]

    Boundedness and Decay for the Teukolsky Equation on Kerr Spacetimes I: The Case |a|ăă M ,

    M. Dafermos, G. Holzegel, and I. Rodnianski, “Boundedness and Decay for the Teukolsky Equation on Kerr Spacetimes I: The Case |a|ăă M ,” Ann. PDE, vol. 5, no. 1, p. 2, 2019

    work page 2019

Show all 29 references
  1. [8]

    Uniform Energy Bound and Morawetz Estimate for Extreme Components of Spin Fields in the Exterior of a Slowly Rotating Kerr Black Hole II: Linearized Gravity,

    S. Ma, “Uniform Energy Bound and Morawetz Estimate for Extreme Components of Spin Fields in the Exterior of a Slowly Rotating Kerr Black Hole II: Linearized Gravity,”Comm. Math. Phys., vol. 377, no. 3, pp. 2489–2551, 2020

    2020

  2. [9]

    Boundedness and decay for the Teukolsky equation on Kerr in the full subextremal range |a|ă M : frequency space analysis,

    Y. Shlapentokh-Rothman and R. Teixeira da Costa, “Boundedness and decay for the Teukolsky equation on Kerr in the full subextremal range |a|ă M : frequency space analysis,” 2020, arXiv:2007.07211

    2020 arXiv

  3. [10]

    Boundedness and decay for the Teukolsky equation on Kerr in the full subextremal range |a| ă M : physical space analysis,

    ——, “Boundedness and decay for the Teukolsky equation on Kerr in the full subextremal range |a| ă M : physical space analysis,” 2023, arXiv:2302.08916

    2023 arXiv

  4. [11]

    Stability for linearized gravity on the Kerr spacetime,

    L. Andersson, T. B¨ ackdahl, P. Blue, and S. Ma, “Stability for linearized gravity on the Kerr spacetime,” Ann. PDE, vol. 11, no. 1, pp. Paper No. 11, 161, 2025

    2025

  5. [12]

    Kerr stability for small angular momentum,

    S. Klainerman and J. Szeftel, “Kerr stability for small angular momentum,” Pure Appl. Math. Q. , vol. 19, no. 3, pp. 791–1678, 2023

    2023

  6. [13]

    Wave equations estimates and the nonlinear stability of slowly rotating Kerr black holes,

    E. Giorgi, S. Klainerman, and J. Szeftel, “Wave equations estimates and the nonlinear stability of slowly rotating Kerr black holes,” Pure Appl. Math. Q. , vol. 20, no. 7, pp. 2865–3849, 2024

    2024

  7. [14]

    The linear stability of Reissner-Nordstr¨ om spacetime for small charge,

    E. Giorgi, “The linear stability of Reissner-Nordstr¨ om spacetime for small charge,” Ann. PDE , vol. 6, no. 8, 2020

    2020

  8. [15]

    Instability of Gravitational and Electromagnetic Perturbations of Extremal Reissner–Nordstr¨ om Spacetime,

    M. Apetroaie, “Instability of Gravitational and Electromagnetic Perturbations of Extremal Reissner–Nordstr¨ om Spacetime,” Ann. PDE, vol. 9, no. 22, 2023. 27

    2023

  9. [16]

    Boundedness and Decay for the Teukolsky system in Kerr-Newman spacetime: the case|a|,|Q|! M ,

    E. Giorgi, “Boundedness and Decay for the Teukolsky system in Kerr-Newman spacetime: the case|a|,|Q|! M ,” 2023, arXiv:2311.07408

    2023 arXiv

  10. [17]

    Linear stability of slowly rotating Kerr black holes,

    D. H¨ afner, P. Hintz, and A. Vasy, “Linear stability of slowly rotating Kerr black holes,”Invent. Math., vol. 223, no. 3, pp. 1227–1406, 2021

    2021

  11. [18]

    The linear stability of weakly charged and slowly rotating Kerr-Newman family of charged black holes,

    L. He, “The linear stability of weakly charged and slowly rotating Kerr-Newman family of charged black holes,” 2023, arXiv:2301.08557

    2023 arXiv

  12. [19]

    Conservation laws and flux bounds for gravitational perturbations of the Schwarzschild metric,

    G. Holzegel, “Conservation laws and flux bounds for gravitational perturbations of the Schwarzschild metric,” Class. Quantum Gravity , vol. 33, no. 20, p. 205004, 2016

    2016

  13. [20]

    Chandrasekhar, The Mathematical Theory of Black Holes

    S. Chandrasekhar, The Mathematical Theory of Black Holes . Oxford Univ. Press, New York, 1992

    1992

  14. [21]

    Electromagnetic-gravitational perturbations of Kerr-Newman spacetime: the Teukolsky and Regge- Wheeler equations,

    E. Giorgi, “Electromagnetic-gravitational perturbations of Kerr-Newman spacetime: the Teukolsky and Regge- Wheeler equations,” J. Hyperbolic Differ. Equ. , vol. 19, no. 1, 2022

    2022

  15. [23]

    Boundedness and Decay for the Teukolsky System in Kerr-Newman Spacetime II: The Case |a| ! M , |Q| ă M in axial symmetry,

    E. Giorgi and J. Wan, “Boundedness and Decay for the Teukolsky System in Kerr-Newman Spacetime II: The Case |a| ! M , |Q| ă M in axial symmetry,” Advances in Theoretical and Mathematical Physics , 2024, arXiv:2407.10750

    2024 arXiv

  16. [24]

    Generic instability of rotating relativistic stars,

    J. L. Friedman, “Generic instability of rotating relativistic stars,” Comm. Math. Phys. , vol. 62, no. 3, pp. 247–278, 1978

    1978

  17. [25]

    Stability of Black Holes and Black Branes,

    S. Hollands and R. M. Wald, “Stability of Black Holes and Black Branes,” Comm. Math. Phys. , vol. 321, no. 3, p. 629, 2013

    2013

  18. [26]

    Stability, instability, canonical energy and charged black holes,

    J. Keir, “Stability, instability, canonical energy and charged black holes,”Classical and Quantum Gravity, vol. 31, no. 3, p. 035014, Jan 2014

    2014

  19. [27]

    Coercivity properties of the canonical energy in double null gauge on the 4-dimensional Schwarzschild exterior,

    S. C. Collingbourne, “Coercivity properties of the canonical energy in double null gauge on the 4-dimensional Schwarzschild exterior,” Classical and Quantum Gravity , vol. 40, no. 22, p. 225013, 2023

    2023

  20. [28]

    Christodoulou and S

    D. Christodoulou and S. Klainerman, The Global Nonlinear Stability of the Minkowski Space . Princeton Uni. Press, 1993

    1993

  21. [29]

    Perturbations of a Rotating Black Hole. I. Fundamental Equations for Gravitational, Electro- magnetic, and Neutrino-Field Perturbations,

    S. A. Teukolsky, “Perturbations of a Rotating Black Hole. I. Fundamental Equations for Gravitational, Electro- magnetic, and Neutrino-Field Perturbations,” Astrophys. J., vol. 185, pp. 635–648, 1973. 28

    1973

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.