pith. sign in

arxiv: 2604.09298 · v1 · submitted 2026-04-10 · 🧮 math.AP · gr-qc· hep-th

Massless Maxwell fields on sub-extremal Kerr-Newman exteriors

Bobby Eka Gunara , Mulyanto , Emir Syahreza Fadhilla , Fiki Taufik Akbar This is my paper

Pith reviewed 2026-05-10 17:44 UTC · model grok-4.3

classification 🧮 math.AP gr-qchep-th
keywords Kerr-Newman spacetimeMaxwell fieldsenergy decay estimatesscattering theoryblack hole exteriorsMorawetz estimatesr^p hierarchypolynomial decay
0
0 comments X

The pith

The stationary-subtracted massless Maxwell field on sub-extremal Kerr-Newman exteriors satisfies non-degenerate boundedness, Morawetz estimates, polynomial decay, and two-sided scattering when rotation and charge parameters are weak.

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

The paper establishes decay and scattering properties for the massless Maxwell system on fixed sub-extremal Kerr-Newman black hole exteriors. In the weak regime where both the rotation parameter a and charge Q are small fractions of the mass M, it proves boundedness without degeneracy, integrated local energy decay, an outgoing r^p hierarchy, radiation fields, polynomial decay, and two-sided scattering. These conclusions also hold for the axisymmetric sector under slow rotation without smallness on Q, and extend to the full sub-extremal range once one order-zero radiative local-energy estimate is added to the master system. The results are obtained inside one reconstruction and scattering framework that separates the unconditional weak case from the conditional full-range case. A reader would care because these estimates supply the linear theory needed to understand electromagnetic wave propagation around charged rotating black holes.

Core claim

For the stationary-subtracted Maxwell field on Kerr-Newman exteriors in weak regimes with 0 < |a| ≤ c_a M and 0 < |Q| ≤ c_Q M, non-degenerate boundedness, Morawetz/integrated local energy decay, the outgoing r^p hierarchy, radiation fields, polynomial decay, and two-sided scattering are proved. The same conclusions apply to the axisymmetric sector in the relative slow-rotation regime 0 < |a| ≤ ε_KN |Q| < M, and for the full sub-extremal range a² + Q² < M² when supplemented by one order-zero radiative local-energy estimate for the master system.

What carries the argument

The stationary-subtracted Maxwell field reduced to a master system on the Kerr-Newman exterior, which supports energy estimates, r^p hierarchies, and scattering construction.

If this is right

  • The field and its derivatives remain bounded in a non-degenerate norm for all time in the weak regimes.
  • Integrated local energy decay controls the solution away from the event horizon and spatial infinity.
  • Radiation fields exist at null infinity, allowing the definition of scattering data at both ends.
  • Polynomial decay rates hold for the field and its derivatives as time goes to infinity.
  • Two-sided scattering operators are constructed that map initial data to radiation fields at past and future infinity.

Where Pith is reading between the lines

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

  • Proving the single extra local-energy estimate independently would remove the conditional restriction and give unconditional results for all sub-extremal parameters.
  • The same reconstruction and scattering framework could be applied to other linear fields, such as gravitational perturbations, on the same background.
  • These linear decay and scattering results supply a necessary step toward nonlinear stability questions for the Kerr-Newman family under electromagnetic matter.

Load-bearing premise

The rotation parameter a and charge Q must remain small enough relative to the mass M for the perturbative energy estimates to close without an extra local-energy bound.

What would settle it

An explicit or numerically generated solution of the Maxwell equations on a weakly charged, weakly rotating Kerr-Newman background that grows unbounded or lacks radiation fields at future and past null infinity would disprove the boundedness and scattering statements.

read the original abstract

We study the massless Maxwell system on fixed sub--extremal Kerr-Newman exteriors. In the weak Kerr--Newman regimes with $0<|a|\le c_a M$, $0<|Q|\le c_Q M$, and $c_a,c_Q \in (0,1) $, we prove non--degenerate boundedness, Morawetz / integrated local energy decay, the outgoing $r^p$ hierarchy, radiation fields, polynomial decay, and two--sided scattering for the stationary-subtracted Maxwell field. In the axisymmetric relative slow--rotation regime $0<|a|\le \varepsilon_{\mathrm{KN}}|Q|<M$ with $\varepsilon_{\mathrm{KN}} \in (0,1)$, the same conclusions hold on the axisymmetric sector, with no perturbative restriction on $|Q|/M$ beyond sub--extremality. For the full sub-extremal range $a^2+Q^2<M^2$, by adding one order--zero radiative local--energy estimate for this master system, we could obtain similar results as the weak regimes. This separates the unconditional weak theory from the conditional full--range theory within one reconstruction/scattering framework.

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.

Referee Report

1 major / 0 minor

Summary. The manuscript establishes a reconstruction and scattering framework for the stationary-subtracted massless Maxwell field on fixed sub-extremal Kerr-Newman exteriors. It proves non-degenerate boundedness, Morawetz/integrated local energy decay, the outgoing r^p hierarchy, radiation fields, polynomial decay, and two-sided scattering unconditionally in the weak regimes 0<|a|≤c_a M and 0<|Q|≤c_Q M, and in the axisymmetric slow-rotation sector 0<|a|≤ε_KN |Q|<M. For the full sub-extremal range a²+Q²<M², the same conclusions are stated to follow upon the addition of one order-zero radiative local-energy estimate for the master system.

Significance. If the unconditional results hold, the work advances the linear theory of electromagnetic fields on charged rotating black-hole backgrounds by extending perturbative scattering results from Kerr to include nonzero charge, while the conditional full-range statement usefully isolates the remaining technical obstacle. The separation of weak-regime proofs from the conditional extension provides a clear modular framework that could support future nonlinear applications.

major comments (1)
  1. Abstract: the statement that 'by adding one order-zero radiative local-energy estimate for this master system, we could obtain similar results' is load-bearing for any full-range claim, yet the manuscript provides neither a derivation nor a justification of this estimate (nor its integration into the existing energy identities and Morawetz multipliers). The weak-regime methods are explicitly stated to be insufficient without it, so the full sub-extremal conclusions remain conditional rather than established.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address the major comment below, clarifying the scope and conditional nature of our results.

read point-by-point responses

  1. Referee: Abstract: the statement that 'by adding one order-zero radiative local-energy estimate for this master system, we could obtain similar results' is load-bearing for any full-range claim, yet the manuscript provides neither a derivation nor a justification of this estimate (nor its integration into the existing energy identities and Morawetz multipliers). The weak-regime methods are explicitly stated to be insufficient without it, so the full sub-extremal conclusions remain conditional rather than established.

    Authors: We agree that the manuscript neither derives nor justifies the order-zero radiative local-energy estimate for the full sub-extremal range, and that the weak-regime methods are insufficient without it. The abstract statement is intended only to indicate the modularity of the reconstruction/scattering framework: the unconditional results are confined to the weak regimes, while the full-range extension is explicitly conditional on this single additional estimate. We do not claim to prove the full sub-extremal results in the present work. To address the concern, we will revise the abstract to state explicitly that the full-range conclusions are conditional and not established here. We will also add a clarifying remark in the introduction emphasizing that the weak-regime techniques do not extend to the full range without this estimate. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations rely on standard energy identities without reduction to inputs

full rationale

The paper explicitly separates unconditional proofs in weak Kerr-Newman regimes (using Morawetz multipliers and energy estimates on the Maxwell system) from conditional full-range results that require an additional unproven order-zero radiative local-energy estimate. No quoted steps reduce by construction to fitted parameters, self-definitions, or self-citation chains; the framework is presented as self-contained for the proven regimes with standard methods.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the standard Kerr-Newman metric solving the Einstein-Maxwell equations, the sub-extremality condition to ensure a regular event horizon, and the applicability of energy methods and Morawetz estimates to the Maxwell system on asymptotically flat Lorentzian manifolds.

axioms (2)
  • domain assumption The Kerr-Newman metric is a solution of the Einstein-Maxwell equations with parameters M, a, Q satisfying a² + Q² < M².
    Invoked throughout to define the background spacetime.
  • standard math Standard Sobolev embeddings and integration-by-parts identities hold on the exterior region.
    Used implicitly in deriving energy estimates.

pith-pipeline@v0.9.0 · 5535 in / 1497 out tokens · 35217 ms · 2026-05-10T17:44:49.229076+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

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

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

12 extracted references · 12 canonical work pages

  1. [1]

    Sterbenz and D

    J. Sterbenz and D. Tataru, Local energy decay for Maxwell fields part I: Spherically symmetric black-hole backgrounds,Int. Math. Res. Not. IMRN2015, no. 11, 3298–3342

    work page

  2. [2]

    Benomio and R

    G. Benomio and R. Teixeira da Costa, The Maxwell equations on full sub-extremal and extremal Kerr spacetimes, preprint (2025), arXiv:2512.08917

    work page arXiv 2025

  3. [3]

    Giorgi, Electromagnetic–gravitational perturbations of Kerr–Newman spacetime: the Teukolsky and Regge–Wheeler equations,J

    E. Giorgi, Electromagnetic–gravitational perturbations of Kerr–Newman spacetime: the Teukolsky and Regge–Wheeler equations,J. Hyperbolic Differ. Equ.19(2022), 1–139

    work page 2022

  4. [4]

    Giorgi, The Carter tensor and the physical–space analysis in perturbations of Kerr–Newman spacetime,J

    E. Giorgi, The Carter tensor and the physical–space analysis in perturbations of Kerr–Newman spacetime,J. Differential Geom.127(2024), 277–371

    work page 2024

  5. [5]

    Hollands and V

    S. Hollands and V. Toomani, On the radiation gauge for spin–1 perturbations in Kerr–Newman spacetime,Class. Quantum Grav.38(2021), 025013

    work page 2021

  6. [6]

    Giorgi, Boundedness and decay for the Teukolsky system in Kerr–Newman spacetime I: the case|a|,|Q| ≪M, preprint (2023), arXiv:2311.07408

    E. Giorgi, Boundedness and decay for the Teukolsky system in Kerr–Newman spacetime I: the case|a|,|Q| ≪M, preprint (2023), arXiv:2311.07408

    work page arXiv 2023

  7. [7]

    Giorgi and J

    E. Giorgi and J. Wan, Boundedness and decay for the Teukolsky system in Kerr–Newman spacetime II: the case|a| ≪M,|Q|< Min axial symmetry, preprint (2024), arXiv:2407.10750

    work page arXiv 2024

  8. [8]

    Jezierski and T

    J. Jezierski and T. Smołka, A geometric description of Maxwell field in a Kerr spacetime,Class. Quantum Grav.33(2016), no. 12, 125035

    work page 2016

  9. [9]

    Dafermos and I

    M. Dafermos and I. Rodnianski, The red-shift effect and radiation decay on black hole spacetimes, Comm. Pure Appl. Math.62(2009), no. 7, 859–919

    work page 2009

  10. [10]

    Dafermos and I

    M. Dafermos and I. Rodnianski, A new physical-space approach to decay for the wave equation with applications to black hole spacetimes, inXVIth International Congress on Mathematical Physics, World Scientific, 2010, pp. 421–432

    work page 2010

  11. [11]

    Bound edness and decay for the Teukolsky equation on Kerr in the full subextremal range |a|< m: physical 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|< M : physical space analysis, preprint (2023), arXiv:2302.08916

    work page arXiv 2023

  12. [12]

    Civin,Stability of charged rotating black holes for linear scalar perturbations, Ph.D

    D. Civin,Stability of charged rotating black holes for linear scalar perturbations, Ph.D. thesis, University of Cambridge, 2015. 83

    work page 2015