On causality and superluminal behavior in classical field theories. Applications to k-essence theories and MOND-like theories of gravity

摘要

Field theories whose full action is Lorentz invariant (or diffeomorphisminvariant) can exhibit superluminal behaviors through the breaking of localLorentz invariance. Quantum induced superluminal velocities are well-knownexamples of this effect. The issue of the causal behavior of such propagationsis somewhat controversial in the literature and we intend to clarify it. Weprovide a careful analysis of the meaning of causality in classicalrelativistic field theories, and we stress the role played by the Cauchyproblem and the notions of chronology and time arrow. We show that superluminalbehavior threaten causality only if a prior chronology on spacetime is chosen.In the case where superluminal propagations occur, however, there is at leasttwo non conformally related metrics on spacetime and thus two available notionsof chronology. These two chronologies are on equal footing and it would thus bemisleading to choose \textit{ab initio} one of them to define causality.Rather, we provide a formulation of causality in which no prior chronology isassumed. We argue this is the only way to deal with the issue of causality inthe case where some degrees of freedom propagate faster than others. Weactually show that superluminal propagations do not threaten causality. As anillustration of these conceptual issues, we consider two field theories, namelyk-essences scalar fields and bimetric theories of gravity, and we derive theconditions imposed by causality. We discuss various applications such as thedark energy problem, MOND-like theories of gravity and varying speed of lighttheories.