Radiation fields on Schwarzschild spacetime

TL;DR

This paper defines and analyzes the radiation field for the wave equation on Schwarzschild spacetime, establishing regularity, unitarity, and support properties of the radiation components related to black hole horizons and null infinity.

Contribution

It introduces a rigorous definition of the radiation field on Schwarzschild spacetime and proves key properties like regularity, unitarity, and support theorems for the wave equation solutions.

Findings

Radiation field components are regular and well-defined at the horizon and null infinity.

The radiation field is unitary with respect to the conserved energy.

Support theorems characterize the radiation field's behavior based on initial data.

Abstract

In this paper we define the radiation field for the wave equation on the Schwarzschild black hole spacetime. In this context it has two components: the rescaled restriction of the time derivative of a solution to null infinity and to the event horizon. In the process, we establish some regularity properties of solutions of the wave equation on the spacetime. In particular, we prove that the regularity of the solution across the event horizon and across null infinity is determined by the regularity and decay rate of the initial data at the event horizon and at infinity. We also show that the radiation field is unitary with respect to the conserved energy and prove support theorems for each piece of the radiation field.