Nonuniqueness of Representations of Wave Equations in Lorentzian Space-Times

TL;DR

This paper highlights the nonuniqueness of initial-boundary value problem formulations for wave equations in Lorentzian space-times, showing that different formulations can lead to different stability outcomes, with implications for various physically relevant spacetimes.

Contribution

It demonstrates that the formulation of initial-boundary value problems in Lorentzian space-times is not unique and can affect stability analysis, using the Rindler wedge as a case study.

Findings

Multiple inequivalent formulations exist for wave equations in Lorentzian space-times.

Different formulations can lead to different stability outcomes.

Methods can be extended to higher-dimensional space-times like Schwarzschild and Kerr.

Abstract

This brief note wants to bring to attention that the formulation of physically reasonable initial-boundary value problems for wave equations in Lorentzian space-times is not unique, i.e., that there are inequivalent such formulations that lead to a different outcome of the stability discussion of the solutions. For demonstration, the paper uses the case of the wave equation on the right Rindler wedge in 2-dimensional Minkowski space. The used methods can be generalized to wave equations on stationary globally hyperbolic space-times with horizons in higher dimensions, such as Schwarzschild and Kerr space-times.