Predictability of subluminal and superluminal wave equations

TL;DR

This paper compares subluminal and superluminal Lorentz invariant wave equations, revealing that superluminal equations generally have better predictability with unique maximal developments, contrary to common claims.

Contribution

It demonstrates that superluminal wave equations always have unique maximal globally hyperbolic developments, improving predictability over subluminal equations.

Findings

Superluminal equations have unique maximal developments.

Subluminal equations can have multiple maximal developments.

Theoretical conditions for predictability are established.

Abstract

It is sometimes claimed that Lorentz invariant wave equations which allow superluminal propagation exhibit worse predictability than subluminal equations. To investigate this, we study the Born-Infeld scalar in two spacetime dimensions. This equation can be formulated in either a subluminal or a superluminal form. Surprisingly, we find that the subluminal theory is less predictive than the superluminal theory in the following sense. For the subluminal theory, there can exist multiple maximal globally hyperbolic developments arising from the same initial data. This problem does not arise in the superluminal theory, for which there is a unique maximal globally hyperbolic development. For a general quasilinear wave equation, we prove theorems establishing why this lack of uniqueness occurs, and identify conditions on the equation that ensure uniqueness. In particular, we prove that superluminal equations always admit a unique maximal globally hyperbolic development. In this sense, superluminal equations exhibit better predictability than generic subluminal equations.