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

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.Comment: 15 pages, 2 figures; minor changes, references added, submitted to Phys.Rev.

Causality and Superluminal Fields

The expression of causality depends on an underlying choice of chronology. Since a chronology is provided by any Lorentzian metric in relativistic theories, there are as many expressions of causality as there are non-conformally related metrics over spacetime. Although tempting, a definitive choice of a preferred metric to which one may refer to is not satisfying. It would indeed be in great conflict with the spirit of general covariance. Moreover, a theory which appear to be non causal with respect to (hereafter, w.r.t) this metric, may well be causal w.r.t another metric. In a theory involving fields that propagate at different speeds (e.g. due to some spontaneous breaking of Lorentz invariance), spacetime is endowed with such a finite set of non-conformally related metrics. In that case one must look for a new notion of causality, such that 1. no particular metric is favored and 2. there is an unique answer to the question : ``is the theory causal?''. This new causality is unique and defined w.r.t the metric drawing the wider cone in the tangent space of a given point of the manifold. Moreover, which metric defines the wider cone may depend on the location on spacetime. In that sense, superluminal fields are generically causal, provided that some other basic requirements are met.Comment: 3 pages, Prepared for the Proceedings of the Eleventh Marcel Grossmann Meeting on General Relativity, Berlin, Germany, 23-27 July 2006; document class change

The two-body problem: analytical results in a toy-model of relativistic gravity

The two body problem in a scalar theory of gravity is investigated. We focus on the closest theory to General Relativity (GR), namely Nordstr\"om's theory of gravity (1913). The gravitational field can be exactly solved for any configuration of point-particles. We then derive the exact equations of motion of two inspiraling bodies including the exact self-forces terms. We prove that there is no innermost circular orbit (ICO) in the exact theory whereas we find (order-dependent) ICOs if post-Newtonian (PN) truncations are used. We construct a solution of the two body problem in an iterative (non-PN) way, which can be viewed as a series in powers of $(v/c)^{5}$. Besides this rapid convergence, each order also provides non-perturbative information. Starting from a circular Newtonian-like orbit, the first iteration already yields the 4.5 PN radiation reaction. These results not only shed light on some non-perturbative effects of relativistic gravity, but may also be useful to test numerical codes.Comment: 7 Figures, To appear in the proceedings of Albert Einstein's Century International Conference, Paris, France, 18-22 Jul

Field-theoretical formulations of MOND-like gravity

Modified Newtonian dynamics (MOND) is a possible way to explain the flat galaxy rotation curves without invoking the existence of dark matter. It is however quite difficult to predict such a phenomenology in a consistent field theory, free of instabilities and admitting a well-posed Cauchy problem. We examine critically various proposals of the literature, and underline their successes and failures both from the experimental and the field-theoretical viewpoints. We exhibit new difficulties in both cases, and point out the hidden fine tuning of some models. On the other hand, we show that several published no-go theorems are based on hypotheses which may be unnecessary, so that the space of possible models is a priori larger. We examine a new route to reproduce the MOND physics, in which the field equations are particularly simple outside matter. However, the analysis of the field equations within matter (a crucial point which is often forgotten in the literature) exhibits a deadly problem, namely that they do not remain always hyperbolic. Incidentally, we prove that the same theoretical framework provides a stable and well-posed model able to reproduce the Pioneer anomaly without spoiling any of the precision tests of general relativity. Our conclusion is that all MOND-like models proposed in the literature, including the new ones examined in this paper, present serious difficulties: Not only they are unnaturally fine tuned, but they also fail to reproduce some experimental facts or are unstable or inconsistent as field theories. However, some frameworks, notably the tensor-vector-scalar (TeVeS) one of Bekenstein and Sanders, seem more promising than others, and our discussion underlines in which directions one should try to improve them.Comment: 66 pages, 6 figures, RevTeX4 format, version reflecting the changes in the published pape

Non-standard baryon-dark matter interactions

After summarizing the respective merits of the Cold Dark Matter (CDM) and Modified Newtonian Dynamics (MOND) paradigms in various stellar systems, we investigate the possibility that a non-standard interaction between baryonic and dark matter could reproduce the successes of CDM at extragalactic scales while making baryonic matter effectively obey the MOND field equation in spiral galaxies.Comment: 10 pages, to appear in World Scientific, proceedings of DARK 200

Escaping from MOND

We present a new test of modified Newtonian dynamics (MOND) on galactic scales, based on the escape speed in the solar neighbourhood. This test is independent from other empirical successes of MOND at reproducing the phenomenology of galactic rotation curves. The galactic escape speed in MOND is entirely determined by the baryonic content of the Galaxy and the external field in which it is embedded. We estimate that the external field in which the Milky Way must be embedded to produce the observed local escape speed of 550 km/s is of the order of a_0/100, where a_0 is the dividing acceleration scale below which gravity is boosted in MOND. This is compatible with the external gravitational field actually acting on the Milky Way.Comment: 4 pages, 1 figure; accepted for publication in MNRA

Fab Four: When John and George play gravitation and cosmology

Scalar-tensor theories of gravitation have recently regained a great interest after the discovery of the Chameleon mechanism and of the Galileon models. The former allows, in principle, to reconcile the presence of cosmological scalar fields with the constraints from experiments at the Solar System scale. The latter open up the possibility of building inflationary models that, among other things, do not need ad hoc potentials. Further generalizations have finally led to the most general tensor-scalar theory, recently dubbed the "Fab Four", with only first and second order derivatives of the fields in the equations of motion and that self-tune to a vanishing cosmological constant. This model has a very rich phenomenology that needs to be explored and confronted with experimental data in order to constrain a very large parameter space. In this paper, we present some results regarding a subset of the theory named "John", which corresponds to a non-minimal derivative coupling between the scalar field and the Einstein tensor in the action. We show that this coupling gives rise to an inflationary model with very unnatural initial conditions. Thus, we include a non-minimal, but non-derivative, coupling between scalar field and Ricci scalar, a term named "George" in the Fab Four terminology. In this way, we find a more sensible inflationary model, and, by performing a post-newtonian expansion of spherically symmetric solutions, we derive the set of equations that constrain the parameter space with data from experiments in the solar system.Comment: Minor changes, references added. Version accepted for publication in Advances in Astronom

Dynamics of a lattice Universe

We find a solution to Einstein field equations for a regular toroidal lattice of size L with equal masses M at the centre of each cell; this solution is exact at order M/L. Such a solution is convenient to study the dynamics of an assembly of galaxy-like objects. We find that the solution is expanding (or contracting) in exactly the same way as the solution of a Friedman-Lema\^itre-Robertson-Walker Universe with dust having the same average density as our model. This points towards the absence of backreaction in a Universe filled with an infinite number of objects, and this validates the fluid approximation, as far as dynamics is concerned, and at the level of approximation considered in this work.Comment: 14 pages. No figure. Accepted version for Classical and Quantum Gravit