Can we prove absolutely that FTL = causality violation

Question

I've been wondering about this for a long time. Given that special relativity is normally true, can we still prove that causality violation must occur if FTL is found? All proofs I can find, depend on it being absolutely true.

What occurs to me, is engaging an FTL drive might do something like prove the aether really does exist, but normal physics is disconnected from it. This model has a few problems, like for example anything transitioning to FTL ends up with an instantly high velocity relative to the aether.

Question almost matches Can FTL-Communication between two points in the same frame of reference break causality?, but the answers are opposed. The first (and most likely correct) answer doesn't apply; however the second one seems to apply but contradicts the first for that question's scenario.

I can prove the answer is NO by geometry if I make an assumption that we believe is false: attempting to cross a black hole event horizon is instantly fatal. A large enough (and galactic core black holes are large enough) black hole gives an absolutely backwards-pointing light cone that might be exploitable here (FTL drive usually means /can/ escape from black hole event horizon).

EDIT: Darn. Question appears to be meaningless without a particular theory to discuss. I was trying to ask if something similar to Noether's theorem for energy and momentum conservation is known under arbitrary laws of physics, but if I try to harden up the question to allow talking about drawing the parallel I get a contradiction in the question itself.

Answer 1

This is a perfectly good question; don't feel discouraged.

Coming up with a notion of 'causality' that doesn't refer to any physical theory is pretty hard. I'll discuss it here in the context of general relativity (GR). The basic entities in GR are events; i.e. spacetime points $(t, x, y, z)$.

Suppose we have two events $A$ and $B$ and that, according to some observer $\mathcal{O}$, $t_A > t_B$ - that is, $A$ occurs after $B$. We seek a necessary (not sufficient) condition under which $B$ can be said to have 'caused' $A$.

That condition is the following: $t_A > t_B$ must hold for all observers. If my turning on a stove ($B$) caused water to boil ($A$), there had better not be anyone for whom the water boiling happened before the stove turned on.

Because of the signature of the metric, temporal ordering is preserved under coordinate changes only for timelike separated events; i.e. only for events connected by a geodesic that goes "slower than the speed of light". See my answer here for an elaboration: What spacelike, timelike and lightlike spacetime interval really mean?.