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

TL;DR

This paper investigates the behavior of Maxwell fields in Kerr black hole spacetimes, establishing boundedness and decay estimates for the middle components, with results depending on control over extremal components and the black hole's spin parameter.

Contribution

It introduces a new formulation of Maxwell equations in Kerr spacetime, decouples extremal components into Teukolsky equations, and derives decay estimates for middle components under certain conditions.

Findings

Uniform boundedness and decay of middle Maxwell components in Kerr spacetimes.

Decoupling of extremal components into Teukolsky equations.

Discovery of conservation laws for axisymmetric solutions at the horizon.

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.