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

TL;DR

The paper addresses whether superluminal propagation in Lorentz-invariant field theories undermines causality. It introduces a mixed, chronology-free framework in which causality is defined by well-posed Cauchy problems across all propagating sectors, rather than a single preferred spacetime chronology. The authors show that superluminal signals do not inherently threaten causality, provided global hyperbolicity holds in the extended sense and initial data are specified on appropriate surfaces; they illustrate this with k-essence and bimetric gravity and discuss quantum-induced propagation. The work has implications for dark energy models, MOND-like gravity, and VSL theories by clarifying when causality constraints impose genuine restrictions on model building and how to assess their physical viability.

Abstract

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.

Figures (2)

• 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 in the opposite case (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 thin line represents the Minkowski cone, and horizontal and vertical lines are the space and time axis in the frame of $A$.