Causality and superluminal behavior in classical field theories: Applications to $k$-essence theories and modified-Newtonian-dynamics-like theories of gravity

Field theories with Lorentz (or diffeomorphism invariant) action can exhibit superluminal behavior 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 propagation 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 stress the role played by the Cauchy problem and the notion of chronology. We show that, in general, superluminal behavior threatens causality only if one assumes that a prior chronology in spacetime exists. In the case where superluminal propagation occurs, however, there are at least two nonconformally related metrics in spacetime and thus two available notions of chronology. These two chronologies are on equal footing, and it would thus be misleading to choose ab initio one of them to define causality. Rather, we provide a formulation of causality in which no prior chronology is assumed. We argue that this is the only way to deal with the issue of causality in the case where some degrees of freedom propagate faster than others. In that framework, then, it is shown that superluminal propagation is not necessarily noncausal, the final answer depending on the existence of an initial data formulation. This also depends on global properties of spacetime that we discuss in detail. As an illustration of these conceptual issues, we consider two field theories, namely, $k$-essence 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, modified-Newtonian-dynamics-like theories of gravity, and varying speed of light theories.

Figure 1

The hatched part shows the extended future defined by two metrics (solid and dashed lines) in the case where one metric defines a wider cone than the other one (left) and vice versa (right).

Figure 2

The closed curve followed by the tachyonic signal viewed in the rest frame of $A$. The tachyon is sent by $A$ at time $t_0$ (event $E_0$) and received by $B$ at time $t_1$ (event $E_1$). Since $B$ moves with respect to $A$, the tachyonic signal he sends back to $A$ is actually received before it was sent, at time $t_2<t_0$ (event $E_2$). The dashed line represents the Minkowski cone, and horizontal and vertical lines are the space and time axes in the frame of $A$.