# Uniform energy bound and Morawetz estimate for extreme components of spin fields in the exterior of a slowly rotating Kerr black hole I: Maxwell field

## TL;DR

This paper establishes energy bounds and decay estimates for extreme spin components of Maxwell fields in the exterior of a slowly rotating Kerr black hole, advancing understanding of their stability and behavior.

## Contribution

It introduces a novel approach to derive energy and Morawetz estimates for extreme Newman-Penrose components of Maxwell fields in Kerr spacetime.

## Key findings

- Proved energy estimates for inhomogeneous spin-weighted Fackerell-Ipser equations.
- Established uniform energy bounds for extreme Maxwell components.
- Demonstrated Morawetz decay estimates for these components.

## Abstract

This first part of the series treats the Maxwell equations in the exterior of a slowly rotating Kerr black hole. By performing a first-order differential operator on each extreme Newman-Penrose (N-P) scalar in a Kinnersley tetrad, the resulting equation and the Teukolsky master equation for the extreme N-P component are both in the form of an inhomogeneous \textquotedblleft{spin-weighted Fackerell-Ipser equation\textquotedblright} (SWFIE) and constitute a weakly coupled system. We first prove energy estimate and integrated local energy decay (Morawetz) estimate for this type of inhomogeneous SWFIE following the method in (Dafermos and Rodnianski in Decay for solutions of the wave equation on Kerr exterior spacetimes I-II: the cases $|a|\ll M$ or axisymmetry, 2010, arXiv:1010.5132), and then utilize these estimates to achieve both a uniform bound of a positive definite energy and a Morawetz estimate for the coupled system of each extreme N-P component. The same type of estimates for the regular extreme N-P components defined in the regular Hawking-Hartle tetrad is also proved. The hierarchy here is generalized in our second part (Ma in Uniform energy bound and Morawetz estimates for extreme components of spin fields in the exterior of a slowly rotating Kerr black hole II: linearized gravity, 2017, arXiv:1708.07385) of this series to treat the extreme components of linearized gravity.