# Decay of Maxwell Fields on Reissner-Nordstr{\o}m-de Sitter Black Holes

## TL;DR

This paper proves decay of Maxwell fields in the exterior of Reissner-Nordström-de Sitter black holes using Morawetz estimates and vector field methods, demonstrating uniform and pointwise decay rates.

## Contribution

It introduces decay results for Maxwell fields on Reissner-Nordström-de Sitter black holes using geometric energy estimates and extends the analysis to more general spherically symmetric spacetimes.

## Key findings

- Uniform energy decay on achronal hypersurfaces
- Pointwise decay in time at rate t^{-1}
- Bounds on conformal energy via Morawetz vector field

## Abstract

In this paper we use Morawetz estimates with geometric energy estimates -the so-called vector field method- to prove decay results for the Maxwell field in the static exterior region of the Reissner-Nordstr{\o}m-de Sitter black hole. We prove two types of decay: The first is a uniform decay of the energy of the Maxwell field on achronal hypersurfaces as the hypersurfaces approach timelike infinities. The second decay result is a pointwise decay in time with a rate of $t^{-1}$ which follows from local energy decay by Sobolev estimates. Both results are consequences of bounds on the conformal energy defined by the Morawetz conformal vector field. These bounds are obtained through wave analysis on the middle spin component of the field. The results hold for a more general class of spherically symmetric spacetimes with the same arguments used in this paper.