  Field theories whose full action is Lorentz invariant (or diffeomorphism
invariant) can exhibit superluminal behaviors through the breaking of local
Lorentz invariance. Quantum induced superluminal velocities are well-known
examples of this effect. The issue of the causal behavior of such propagations
is somewhat controversial in the literature and we intend to clarify it. We
provide a careful analysis of the meaning of causality in classical
relativistic field theories, and we stress the role played by the Cauchy
problem and the notions of chronology and time arrow. We show that superluminal
behavior threaten causality only if a prior chronology on spacetime is chosen.
In the case where superluminal propagations occur, however, there is at least
two non conformally related metrics on spacetime and thus two available notions
of chronology. These two chronologies are on equal footing and it would thus be
misleading to choose \textit{ab initio} one of them to define causality.
Rather, we provide a formulation of causality in which no prior chronology is
assumed. We argue this is the only way to deal with the issue of causality in
the case where some degrees of freedom propagate faster than others. We
actually show that superluminal propagations do not threaten causality. As an
illustration of these conceptual issues, we consider two field theories, namely
k-essences scalar fields and bimetric theories of gravity, and we derive the
conditions imposed by causality. We discuss various applications such as the
dark energy problem, MOND-like theories of gravity and varying speed of light
theories.
