Conformal scattering of the wave equation in the Vaidya spacetime

TL;DR

This paper develops a conformal scattering operator for scalar waves in Vaidya spacetime, demonstrating its mathematical properties and establishing a framework for understanding wave behavior in dynamic black hole backgrounds.

Contribution

It introduces a novel construction of the conformal scattering operator in Vaidya spacetime using vector field methods and decay estimates, extending previous results to a more general setting.

Findings

The scattering operator is an isomorphism.

Energy estimates ensure the operator's properties.

Dense range of the trace operator established.

Abstract

We construct the conformal scattering operator for the scalar wave equation on the Vaidya spacetime using vector field methods. The spacetime we consider is Schwarzschild, near both past and future timelike infinities, in order to use existing decay results for the scalar field, ensuring our energy estimates. These estimates guarantee the injectivity of the trace operator and the closure of its range. Finally, we solve a Goursat problem for the scalar waves on null infinities, demonstrating that the range of the trace operator is dense. Consequently, this implies that the scattering operator is an isomorphism.