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The Maxwell equations on full sub-extremal and extremal Kerr spacetimes

Gabriele Benomio, Rita Teixeira da Costa

TL;DR

This paper establishes uniform boundedness and decay for Maxwell fields on the Kerr exterior across the full range |a|<=M by exploiting a tensorial formulation in the algebraically special frame. Extremal components obey decoupled Teukolsky equations, while the middle components satisfy a coupled transport-elliptic system that is recast into a decoupled, modified form; horizon charges read off initial data to yield stationary contributions. In the sub-extremal case, existing Teukolsky results provide unconditional bounds for all unknowns; in the extremal case, the results are conditional on a conjectured Teukolsky energy estimate for extremal components. The analysis uncovers horizon-conservation laws for axisymmetric middle components and connects tensorial Maxwell dynamics to spin-weighted Teukolsky theory, building a framework potentially extensible to nonlinear Einstein-Maxwell settings. Overall, the work advances a robust, derivative-preserving stability theory for Maxwell fields on Kerr spacetimes, with horizon phenomena and mode-decomposed analyses playing central roles.

Abstract

We study the Cauchy problem for the Maxwell equations in the exterior region of Kerr black hole spacetimes. The equations are formulated for components of the Maxwell field relative to the algebraically special frame of Kerr, with the unknowns treated as tensorial quantities associated with a non-integrable horizontal distribution. The extremal Maxwell components decouple into Teukolsky equations, whereas the middle Maxwell components form a coupled system of transport and elliptic equations. Assuming control over the extremal components, we prove uniform boundedness (without loss of derivatives) and decay estimates for the middle components in the full |a|<=M range of spacetime parameters. Our analysis relies on (i) deriving a decoupled system of transport and elliptic equations for two modified middle Maxwell components and (ii) decomposing general solutions into a dynamical and stationary part, the latter determined by two real (electric and magnetic) charges which are entirely read off from the initial data at the event horizon. In the sub-extremal |a|<M case, works of Shlapentokh-Rothman and the second author provide the necessary control over the extremal components, yielding unconditional boundedness and decay results for all the unknowns in the equations. In the extremal |a|=M case, we formulate a conjectural boundedness and decay statement for the extremal components, motivated by work of Casals, Gralla and Zimmerman on fixed azimuthal mode solutions compactly supported away from the event horizon. Our boundedness and decay results for all the unknowns in the equations remain, therefore, conditional. We show that the complicated dynamics of the extremal components at the event horizon is inherited by the middle components; in particular, we uncover novel conservation laws for the middle components of axisymmetric solutions.

The Maxwell equations on full sub-extremal and extremal Kerr spacetimes

TL;DR

This paper establishes uniform boundedness and decay for Maxwell fields on the Kerr exterior across the full range |a|<=M by exploiting a tensorial formulation in the algebraically special frame. Extremal components obey decoupled Teukolsky equations, while the middle components satisfy a coupled transport-elliptic system that is recast into a decoupled, modified form; horizon charges read off initial data to yield stationary contributions. In the sub-extremal case, existing Teukolsky results provide unconditional bounds for all unknowns; in the extremal case, the results are conditional on a conjectured Teukolsky energy estimate for extremal components. The analysis uncovers horizon-conservation laws for axisymmetric middle components and connects tensorial Maxwell dynamics to spin-weighted Teukolsky theory, building a framework potentially extensible to nonlinear Einstein-Maxwell settings. Overall, the work advances a robust, derivative-preserving stability theory for Maxwell fields on Kerr spacetimes, with horizon phenomena and mode-decomposed analyses playing central roles.

Abstract

We study the Cauchy problem for the Maxwell equations in the exterior region of Kerr black hole spacetimes. The equations are formulated for components of the Maxwell field relative to the algebraically special frame of Kerr, with the unknowns treated as tensorial quantities associated with a non-integrable horizontal distribution. The extremal Maxwell components decouple into Teukolsky equations, whereas the middle Maxwell components form a coupled system of transport and elliptic equations. Assuming control over the extremal components, we prove uniform boundedness (without loss of derivatives) and decay estimates for the middle components in the full |a|<=M range of spacetime parameters. Our analysis relies on (i) deriving a decoupled system of transport and elliptic equations for two modified middle Maxwell components and (ii) decomposing general solutions into a dynamical and stationary part, the latter determined by two real (electric and magnetic) charges which are entirely read off from the initial data at the event horizon. In the sub-extremal |a|<M case, works of Shlapentokh-Rothman and the second author provide the necessary control over the extremal components, yielding unconditional boundedness and decay results for all the unknowns in the equations. In the extremal |a|=M case, we formulate a conjectural boundedness and decay statement for the extremal components, motivated by work of Casals, Gralla and Zimmerman on fixed azimuthal mode solutions compactly supported away from the event horizon. Our boundedness and decay results for all the unknowns in the equations remain, therefore, conditional. We show that the complicated dynamics of the extremal components at the event horizon is inherited by the middle components; in particular, we uncover novel conservation laws for the middle components of axisymmetric solutions.

Paper Structure

This paper contains 50 sections, 26 theorems, 270 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Overview of the proofs
  3. Related works and outlook
  4. Outline of the paper
  5. The Kerr exterior manifold
  6. Definition
  7. The Kerr-star and Boyer--Lindquist differentiable structures
  8. A hyperboloidal differentiable structure
  9. The algebraically special horizontal distribution
  10. The algebraically special frame
  11. Induced metric and connection coefficients
  12. Tensor products and horizontal differential operators
  13. Notable geometric identities
  14. Induced structure and norms over foliation spheres
  15. Foliation sphere-projection on the event horizon
  16. ...and 35 more sections

Key Result

Theorem 1.1

Fix $M>0$. For any $|a|< M$, solutions of the Maxwell equations eq:Maxwell-eqs-intro on the Kerr exterior arising from suitably regular initial data

Figures (1)

  • Figure 1: $\Sigma_\tau$ are asymptotically null, smooth and foliate $\mathcal{M}$.

Theorems & Definitions (68)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Proposition 2.1: $\mathbb{S}^2$-projection formulae
  • Proposition 3.1: Tensorial Teukolsky equations
  • Lemma 3.2
  • proof
  • Proposition 3.3: Modified Maxwell equations
  • Remark 3.4
  • proof : Proof of Proposition \ref{['prop:reduced-system']}
  • ...and 58 more