Yesterday, I was reading a paper by Ellis, Maartens and MacCallum on "Causality and the Speed of Sound" which made me rethink a few things about causality which I thought I new and now would like to share. See also an old post on faster than light communication.
First of all, there is the connection between causality and special relativity: It is a misconception that a relativistic theory is automatically causal. Just because you contracted all Lorentz indices properly does not mean that in your theory there is no propagation faster than light. There is an easy counter-example: Take the Lagrangian . where f is a smooth function (actually quadratic is enough for the effect) of the usual kinetic term of a scalar phi. I have already typed up a brief discussion of this theory but then I realised that this might actually be the basis of a nice exam problem (hey guys, are you reading this???) for the QFT course I am currently teaching. So just a sketch at this point: The equation of motion allows for solutions of the form and when you now expand small fluctuations around this solution you see that they propagate with an adjustable speed of sound depending on f and V.
Obviously, this theory is Lorentz invariant, it's only the solution which breaks this invariance (as most interesting solutions of field theories do).
The next thing is how you interpret this result: For suitably chosen V and f you can communicate with space-time points which are space-like to you. So is that really bad? If you think about it (or read the above mentioned paper) you find that this is not necessarily so: You really only get into trouble with causality if you have the possibility to call yourself in the past and tell you the lottery numbers of a drawing in the future of your past self.
If you can communicate with space-points, this can happen: If you send a signal faster than the speed of light to a point P which is space like to you, then from there it can be sent to your past, part of which is again space-like to P. If the sender at P (a mirror is enough) is moving the speed of communication (as measured by the respective sender) has to be only infinitesimally faster than the speed of light (if the whole set-up is Lorentz invariant).
In the theory above, however, this cannot happen: The communication using the fluctuations of the field phi is always to the future as defined by the flow of the vector field V (which we assume to be time-like and future directed). Thus you cannot send signals to points which are upstream in that flow and all of your past is. And using light (according to the usual light-cones) does not help either.
This only changes if you have two such field with superluminus fluctuations: Then you can use one field's fluctuations to send to P (which has to be downstream for that field) and the other field to send the signal from P to your past. So strictly speaking, only if you have two such fields, there is potential for sci-fi stories or get rich fast (or actually: in the past) schemes. But who stops you to have two such fields if one is already around?
At this point, it might be helpful to formalise the notion of "sending signals" a bit further. This also helps to better understand the various notions of velocity which are around when you have non-trivial dispersion relations: As an undergrad you learn that there is the phase velocity which is and that there is the group velocity but at least to me nobody really explained why the later one is important. It was only claimed that it is this velocity which is responsible for signal propagation.
Anyway, what you probably really want is the following: You have some hyperbolic field equation which you solve for some Cauchy data. Then you change the Cauchy data in a compact region K and solve again. Hopefully, the two solution differ only in the region causally connected to K. For this, it is the highest derivative term in the field equation (the leading symbol) which matters and if you Fourier transform you see this is actually the group velocity.
Formulating this "sending to P and back" procedure is a bit more complicated. My suspicion is that it's like the initial value problem when you have closed time-like loops: Then not all initial data is consistent: If my time fore example is periodic with a period of one year I should only give initial data which produces a solution with the same periodicity. But how exactly does this work for the two superluminal fields?
There is one further complication: If gravity is turned on and I have to give initial data for it as well, things get a lot more complicated as the question of a point with given coordinates is space-like to me depends on the metric. But my guess would be that also changes in the metric propagate only with maximally the speed of light in the reference metric.
And finally, there is the problem that the theory above (for non-linear f) is a higher derivative theory. Thus the initial value problem in that theory is likely to require more than phi and its time derivative to be given on the Cauchy surface.