The Helically-Reduced Wave Equation as a Symmetric-Positive System

TL;DR

This paper demonstrates that the helically-reduced wave equation, arising in Einstein equation reductions, can be formulated as a symmetric-positive system, ensuring unique solutions under certain boundary conditions in 2+1D Minkowski spacetime.

Contribution

It introduces a symmetric-positive formulation of the helically-reduced wave equation, enabling the proof of existence and uniqueness of solutions for mixed-type PDEs.

Findings

The reduced equation can be cast into symmetric-positive form.

Unique, strong solutions exist for a class of boundary conditions.

The formulation applies to PDEs with elliptic and hyperbolic regions.

Abstract

Motivated by the partial differential equations of mixed type that arise in the reduction of the Einstein equations by a helical Killing vector field, we consider a boundary value problem for the helically-reduced wave equation with an arbitrary source in 2+1 dimensional Minkowski spacetime. The reduced equation is a second-order partial differential equation which is elliptic inside a disk and hyperbolic outside the disk. We show that the reduced equation can be cast into symmetric-positive form. Using results from the theory of symmetric-positive differential equations, we show that this form of the helically-reduced wave equation admits unique, strong solutions for a class of boundary conditions which include Sommerfeld conditions at the outer boundary.