Are there any objective wavefunction collapse theories which are local and forbid superluminal signalling? Are there any objective wavefunction collapse theories which are local and forbid superluminal signalling? GRW is nonrelativistic and nonlocal.
 A: No, and there cannot be. Imagine entangling two particles, sending one of them off to a colleague on Mars, and then measuring them both at almost the same time (according to Earth's reference frame, for the sake of argument). An objective collapse theory would say that whoever measures their particle first "collapses" the joint wavefunction of the two particles, putting the other person's particle into a definite state, which they then measure. Whatever speed the wavefunction collapse "propagates" at, it must be fast enough to reach the other experimenter before she makes her measurement. Since the two measurements are at a space-like separation, this speed has to be faster than light, there just isn't any way around it.
Note that in any objective collapse theory, the state that the second particle ends up in has to depend on the action taken by the first experimenter, otherwise it's impossible to explain the results of entanglement experiments. This means that different interpretations of what "measurement" is, and what causes the "collapse" cannot change the conclusion that the "collapse" has to happen superluminally.
