Predictability of subluminal and superluminal wave equations

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.