Well, it's important to note that special relativity doesn't say there is no absolute time, but rather that if there is, it is undetectable in any inertial reference frames.
> special relativity doesn't say there is no absolute time, but rather that if there is, it is undetectable in any inertial reference frames.
No, SR says there is no absolute time, because such a thing is undetectable in any inertial reference frame. The theory that makes all the same predictions as SR, but also says that there is such a thing as "absolute time", but it's undetectable in any inertial reference frame (so it doesn't change any experimental predictions), is Lorentz Ether Theory (LET), not SR.
My understanding is that SR doesn't go quite as far as saying 'there is no absolute time', rather it dismisses the concept as undetectable and unnecessary (and useless).
In other words, it doesn't disprove absolute time, it just dispenses with it.
"Absolute time" is supposed to be a physical concept, right? If it is, then showing that it is undetectable and unnecessary is "disproving" it. And the post of yours that I originally responded to certainly seemed to indicate that it was a physical concept--you were saying that if there is absolute time, it can restrict what kinds of FTL travel are possible, which is a physical restriction.
If "absolute time" is just a philosophical concept with no physical consequences, then no physical theory, SR or otherwise, is going to have anything to say about it anyway. But then, it can't be used to resolve the "paradox" you were claiming it could resolve, because it can't restrict what kinds of FTL travel are possible.
Well, the overall point is:
If FTL communication is possible, then you get grandfather type paradoxes, unless you have an absolute time (or it is resolved some other way). And SR does not rule out absolute time, it just discards it.
If absolute time has physical effects, which it must if it can prevent FTL grandfather paradoxes, then yes, SR does rule it out. SR is Lorentz invariant, and absolute time preventing FTL grandfather paradoxes would break Lorentz invariance.
That doesn't mean you couldn't construct a theory that included absolute time and in which FTL travel was allowed but grandfather paradoxes were prevented. But such a theory would not be consistent with SR. To be viable, such a theory would have to match the predictions of SR in the domain in which those predictions have been verified, but it would necessarily make different predictions from SR about FTL travel. (No such theory exists that I'm aware of; I've seen hand-waving about it in discussions of LET, but never any concrete theory.)
My personal prediction would be that such a theory would end up being falsified (either that or rendered meaningless by experiments showing that no type of FTL travel is possible at all). But that's a separate question.
You're right, it would violate Lorentz invariance, but only for FTL travel. In that sense it is consistent with SR for speeds <= c.
I think it's quite easy to make such a theory, I'll get around to writing a blog post on it some day :)
The main result is that for communication (or travel) faster than c, invariance is broken, and it becomes possible to distinguish the absolute reference frame.
Which, according to our best current theories, we don't.