Any worldline you draw outside of the light cone means v>c. Now let's look at what comes out of the Lorenz tranformation in that case:

View attachment 57263
(Source:

https://en.m.wikipedia.org/wiki/Lorentz_transformation)

Since the Lorenz factor becomes an imaginary number (a multiple of i, with i-square equals -1), both time and distance become an imaginary number, which does not make sense at all.

That is why Wikipedia states: "The value of

*v* must be smaller than

*c* for the transformation to make sense."

So any worldline you draw outside of the light cone lies outside of the applicable area of SR, and in the "twilight zone". You cannot casually use them to demonstrate a violation of causality, which is what many scientists tend to do (!).

Of course, Lorentz transformations make no sense when the velocity is greater than c. That is understood. That is why the argument that gives violations of causality from superluminal motion makes no use of such a thing!

The argument goes like this: imagine you have some 'object' moving with some speed v0 > c. In these vague enough terms, this is not yet an impossibility: the 'object' might not be a physical object, but rather a shadow, the end of a laser pointer beam, etc. Say we're in a reference frame where the Earth is at rest at the origin (which I'll call "the Earth's reference frame"), and that at t=0 the object passes x=0. Then at t=4s, it passes by Alpha Centauri, x=~4 light years.

Clearly, I cannot describe what happens

*from the perspective of the object*. As far as I know, such a perspective might not even exist or make sense as a concept. But, can I describe what happens from the perspective of other,

*subluminal* reference frames? You bet. Consider what happens from the perspective of Alice, who's on a spaceship heading towards Alpha Centauri with a speed of v=0.98c (γ ≈ 5), passing by Earth (x'=x=0) at t'=t=0 (following wikipedia's convention, primed coordinates are in Alice's reference frame). When does the object reach Alpha Centauri, according to her? We only need one of the Lorentz transformation equations, but I'll write them both for completeness:

t' = γ ((4 s) - v (4 ly) / c²)

x' = γ ((4 ly) - v (4 s))

t' ≈ 5 * ((4 s) - 0.98 * (4 ly) / c) ≈ 5 * (4 s - 0.98 * 4 years) ≈ -19.6 years

x' ≈ 5 * ((4 ly) - 0.98c * (4 s)) ≈ 5 * (4 ly - 3.92 ls) ≈ 20 ly

So Alice would see the object arrive at Alpha Centauri almost 20 years before it departed Earth.

This is not that surprising, in fact, it's just the familiar argument about relativity of simultaneity wearing a fake mustache. The Lorentz transformations, as FatPhil said, map spacelike ("faster than light") intervals to spacelike intervals and timelike ("slower than light") intervals to timelike intervals, but only in the latter case is the relative order of events preserved. Therefore, the only kinds of trajectories allowed to real particles and objects carrying information are timelike (and lightlike).

I emphasize that in this argument I only made use of special relativity in regimes where it is applicable, well understood, and tested -- no superluminal Lorentz transformations here. If we take the humble interpretation that the "object" is the beam of a laser pointer, all this holds and is perfectly meaningful, yet causes no paradox: just because someone points a laser pointer at me doesn't mean I can use it to send a message to the next victim.

But say that it is a real object, carrying information. You might posit, as DavidB66 did, that Alice is just "wrong" here -- that the object

*really* arrives after it departed, and that any perception to the contrary is merely some kind of perspective illusion. Well, as long as you're unwilling to reject special relativity altogether, that doesn't help either. Consider the following scenario:

1. Bob, on Earth, sends a superluminal (1 light-year / second) message to Alpha Centauri.

2. Alice, now at Alpha Centauri (x = 4 ly), moving with the same velocity of 0.98c along the direction from Earth to Alpha Centauri, receives the message at t' ≈ -19.6 years.

3. Alice replies immediately by superluminal message (1 light-year / second

*from her perspective*), which arrives at Earth at t' ≈ -19.6 years. When's that, according to Bob? We use the inverse transformation:

t = γ (t' + v x' / c²)

x = γ (x' + v t')

Earth is at x = 0:

0 = γ (x' + v t') -> x' = -vt' (as expected)

Plugging into the equation for t,

t = γ (t' - v² t' / c²) = γ (1 - v² / c²) t' = t' / γ = -19.6 years / 5 = -3.92 years

Bob receives the message 3.92 years before he sent it.

This is now a bona-fide closed timelike curve, a fact which is independent of reference frame and agreed upon by all observers (though Alice and Bob disagree on which is leg is the one that's backwards in time).

Note also that it doesn't matter what the putative mechanism might be for the superluminal messaging -- it doesn't matter if it's a warp drive, a wormhole, tachyons, some kind of ansible, whatever. As long as spacetime far away from the object remains undeformed, special relativity applies and a version of this argument can be constructed. The key thing is the ability to do superluminal round trips while preserving the principle of relativity, that is, the idea that there's nothing special about Earth's reference frame when compared to Alice's.

The last sentence also indicates the limits of applicability for this argument. If the principle of relativity fails to hold, it may be impossible to traverse one or both of those superluminal legs, which could save causality after all. For example, if there's some secret reference frame with respect to which superluminal travel takes place, we're morally back to Newton: time is once more absolute. Or maybe you forbid superluminal travel unless it's going from left to right, etc. The important point here is that in all cases we're rejecting relativity in a pretty big way. It's not just some illusion or cosmic clerical error.

Hence the dictum: relativity, causality, FTL. Pick at most two.