Uniqueness of Solutions to the Helically Reduced Wave Equation with Sommerfeld Boundary Conditions

TL;DR

This paper proves a uniqueness theorem for solutions to the helical reduction of the wave equation with Sommerfeld boundary conditions in Minkowski space, relevant to modeling binary inspiral in general relativity.

Contribution

It establishes a uniqueness result for a class of mixed elliptic-hyperbolic equations with Sommerfeld conditions, including nonlinear cases in three dimensions.

Findings

Uniqueness of solutions under specified boundary conditions

Applicability to nonlinear problems in binary inspiral modeling

Extension of results to higher-dimensional Minkowski spaces

Abstract

We consider the helical reduction of the wave equation with an arbitrary source on $(n+1)$-dimensional Minkowski space, $n\geq2$. The reduced equation is of mixed elliptic-hyperbolic type on ${\bf R}^n$. We obtain a uniqueness theorem for solutions on a domain consisting of an $n$-dimensional ball $B$ centered on the reduction of the axis of helical symmetry and satisfying ingoing or outgoing Sommerfeld conditions on $\partial B\approx S^{n-1}$. Non-linear generalizations of such boundary value problems (with $n=3$) arise in the intermediate phase of binary inspiral in general relativity.