A Time Independent Energy Estimate for Outgoing Scalar Waves in the Kerr Geometry

TL;DR

This paper establishes a time-independent energy estimate for outgoing scalar waves in Kerr spacetime, providing uniform energy bounds for wave packets as their initial support moves to infinity.

Contribution

It introduces a novel time-independent energy estimate for outgoing waves in Kerr geometry, utilizing an integral representation of the wave propagator.

Findings

Derived a uniform outgoing energy estimate for wave packets

Applied the estimate to initial data with support at infinity

Utilized a new integral representation of the wave propagator

Abstract

The Cauchy problem for the scalar wave equation in the Kerr geometry is considered, with initial data which is smooth and compactly supported outside the event horizon. A time-independent energy estimate for the outgoing wave is obtained. As an application we estimate the outgoing energy for wave-packet initial data, uniformly as the support of the initial data is shifted to infinity. The main mathematical tool is our previously derived integral representation of the wave propagator.