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

Abstract

Field theories with Lorentz (or diffeomorphism invariant) action can exhibit superluminal behavior through the breaking of local Lorentz invariance. Quantum induced superluminal velocities are wellknown 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.

Formation of caustics in k-essence and Horndeski theory

“…It should be noted that both sub-and super-liminal propagation of perturbations may exist, depending on the model and the background solution. As it was argued in [25][26][27], this does not necessarily lead to the acausality problem in k-essence. Similar arguments can be given for the galileon model [28].…”

Causality and the speed of sound

“…[8,9,10,11]) that there need not be problems with either issue, if one approaches the problem in an open-minded way. The key point is the following: we usually associate causality with the light cone, but any suitable set of cones can be used to define the macroscopic limiting speed, including sound cones that lie outside the light cones.…”

“…For example, Ref. [8] encapsulates this view by stating that: "causality should not be expressed in terms of the chronology induced by the gravitational field . .…”

“…[8,9,10,11]), and propose matter models that allow superluminal signal propagation, which can lead to interesting effects in cosmological and astrophysical contexts. (These are not the first such models: see e.g.…”

Beyond the cosmological standard model

“…A theorem due to Leray 66 says that the causal structure is set by an effective metric [149,1014,1016,1017]…”