1. Doesn't it only appear to violate causality if Charlie is observing things at the speed of light? If Alice sent two signals, one to Bob and one to Charlie, wouldn't Charlie then get notification of sending before observing or being notified of receipt by Bob (Assuming he doesn't pass Bob before the signal would arrive)?
Well...the use of 'appear to violate causality' is a bit deceptive here. You're assuming that one reference frame describes how the events
really happens, and another (Charlie's) describes how the events
appear to happen. In this model, some people just see the 'appearance' of an event, like the way a distant observer can think that a flash of lightning precedes a blast of thunder, even though they both come from the same event.
It doesn't really work that way.
First off, just take a glance at the second postulate of relativity.
As measured in any inertial frame of reference, light is always propagated in empty space with a definite velocity c that is independent of the state of motion of the emitting body.
Imagine that you and I are going to have a duel. To have a fair duel, we need a signal to draw and fire at the
exact same moment. We know that the speed of light is always measured the same in all reference frames, so we decide to use a flash of light from a pile of gunpowder exploding to signal the start of our duel.
Now here's the key. Our duel is going to happen on a moving train car. We are inside the car, at opposite ends, and the gunpowder is in the middle. We are equidistant from it.
We have brought in a judge, The_E, to ensure that our duel is fair.
As our train car rushes past a station, the gunpowder goes off, light is emitted, and it moves away from the pile of gunpowder towards both of us at the speed of light. We are equidistant from the gunpowder, and since the speed of light is a constant, the flash of light reaches us each at the same time.
The_E declares that the duel is fair, since he concludes that the flash of light reached each of us simultaneously. (Think of the light reaching you as Event A, and the light reaching me as Event B.)
However, JeffVader is standing on the train platform that we passed just as the gunpowder went off. Now, from
his perspective, one of us (the one at the back of the train car) is moving
into the light emitted from the gunpowder...and the other one (the one at the front) is moving
away.
Because, remember, JeffVader measures the speed of light to be a constant! Which means the speed of the light emitted from the pile of gunpowder
is not affected by the movement of the pile of gunpowder...which, to him, is moving along with the train, even though
to us it was stationary.As a result, he declares that the person moving
towards the light, drawn along by the motion of the train, sees it first, and the person moving
away saw it second. He declares the duel unfair!
Who is correct? The_E, who is aboard the train car with us, or JeffVader, on the platform? The_E claims the duel was fair, JeffVader doesn't, one of them has to be correct, no?
Right?
Wrong. They are
both correct. Both of their versions of reality are equally true. It is equally valid to say that the duel was fair as to say that it was unfair, depending on your reference frame.
And this means that Charlie's description of what happens in the signaling scenario is, unfortunately, just as valid as Alice and Bob's.
Which leads to something very important: the first postulate of relativity.
The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems of coordinates in uniform translatory motion.
This means that if you can demonstrate that a superluminal signal can move back in time in just
one reference frame, it can move back in time in
all reference frames.
So if Charlie sees that signal as moving back in time, then his perception can hold true for any reference frame, and causality violations can occur to any observer. (I would struggle to do the math to prove this, unfortunately.)
On to question #2.
2. Does the scenario placing the effect before the cause still apply if Charlie is only moving at 2/3c?
Redoing the math quickly:
t' = 5/4(.5 - (2/3)*1/1)
Yes. Charlie will place event B as occuring about 12 seconds before event A.
Reduce to 1/3 C and Charlie will again see event B after event A, however.
Remember, though, what holds in one reference frame must hold in all, so really we only need to demonstrate time travel in one IRF.
3. Why does it matter if one observer sees causality violated? We see the effects of things before the causes all the time, like a bullet impact being seen before the gunshot is heard. If the effect arrives faster than the cause, what does it matter if anybody thinks it's backwards? In other words, just because you observe an effect before a cause, why should it indicate that time has flip-flopped, wouldn't it make more sense that your observation is just slow?
See the First Postulate of Relativity above. The laws of physics hold equally in all reference frames. If they didn't, then you would have one privileged reference frame that was 'more true' than others, and you'd be right back to Newton.
The analogy you brought up is a useful one, however.
When you see a bullet impact before the gunshot, you are actually describing two different events: event A is
bullet arrival and event B is
sound arrival. Both of these propagate from event C, which was
bullet fired.All observers will always agree that event A precedes event B, because the bullet is faster than the sound. No matter what,
everyone will agree on this order. If you stood sufficiently close to event C you might see events A and B as very nearly simultaneous.
But you will
never, ever, ever see event B occur before event A (assuming constant bullet and sound velocity.) The sound will never reach you before the supersonic bullet.
Observations matter when they're based on physical law.
And here's why, put in a very (I hope) simple nutshell.
If two events take place in space-time in such a way that a beam of light could travel between them, then all observers will agree on the order of the events.
If two events take place in such a way that a beam of light
could not possibly have traveled between them, something very special happens.
These events could not
possibly be causally connected. Nothing goes faster than light.
This means that observers can safely disagree on the ordering of the events.
But a superluminal signal could causally connect the events.
Imagine that we take the Alice/Charlie/Bob problem above. Alice has a bomb under her chair. She sends the detonation signal for the bomb to Bob via a 2xC signal, as above. Bob sends it back, to the bomb under Alice's chair. The bomb blows up, killing poor suicidal Alice.
Everything seems fine.
But for Charlie, cruising past at 3/5C - who, remember, will see the signal as reaching Bob many seconds before it is
ever sent...Bob will receive the detonation signal before it is sent, and send it back to Alice and the bomb
even earlier, thus detonating the bomb before Alice sends the detonation signal in the first place. It's a paradox!
You might ask why this is paradoxical. Surely Charlie's perceptions don't matter.
But remember, Charlie's reference frame is
equally valid. The laws of physics have to be the same in all reference frames. The fact that Alice and Bob are stationary with respect to each other doesn't give them special privilege to be right!
Put differently: how can Charlie possibly exist in a universe where a bomb detonates for no reason? Bombs don't detonate for no reason. How can Charlie's reference frame be correct if he sees Alice blow up (event B) but never witnesses event A (Alice sending the signal) because Alice has already exploded? That universe just can't exist, it makes no physical sense.
Which leads nicely to the extra credit...
Extra credit: And doesn't this negate the example of the superluminal billiard ball, and the self exploding bomb paradoxes? Since the bomb and detonator occupy the same frame of reference, and the billiard ball carries it's frame with it.
Ah, but that frame is not privileged as the 'correct' one. What happens in one frame must make physical sense in all frames.
If something makes sense in one frame, but does not make sense in others, it is physically impossible, because the laws of physics must hold equally across all frames.
To explode, the bomb must receive a detonation signal. If there exists a reference frame wherein the bomb could have exploded before it ever received the detonation signal,
thereby preventing the signal from being sent, then the whole system is impossible and paradoxical.
