Superluminal motion and closed signal curves

We discuss some properties of Lorentz invariant theories which allow for superluminal motion. We show that, if signals are always sent forward in time, closed curves along which signals propagate can be formed. This leads to problems with causality and with the second law of thermodynamics. Only if one singles out one frame with respect to which all signals travel forward in time, the formation of 'closed signal curves' can be prevented. However, the price to pay is that in some reference frames perturbations propagate towards the past or towards the future, depending on the direction of emission.