The question arose in an earlier discussion as to how faster than light (FTL) travel and relativity combine to violate causation. That involves an understanding that relativity gives observers going at different velocities different understanding of when two distant events are simultaneous.
What I called "different velocities" is better termed a non-zero velocity with respect to each other, and the relative velocity should be a significant fraction of the speed of light to see measurable effects.
Special Relativity requires three rules:
1) The speed of light is the same whatever the source and whatever the observer.
2) The speed of any observer with respect to another is perceived to be the same by both observers.
3) When any observer sees two events as occurring at the same time and in the same place, so will any other observer.
Let us consider two spaceships, the Smith with captain Smith, and the Jones with captain Jones. Each is a light-second long. They are traveling through space and they pass each other. Jones sees the Smith heading (galactic) north at half the speed of light relative to him, and Smith sees the Jones heading (galactic) south at half the speed of light relative to him.
On each ship, there is a light at the stern which flashes once every second and a light at the bow which flashes when the light from the stern reaches it.
Jones sees the light from the Jones's stern take a second to reach the bow, and the light from the bow take a second to reach the stern. The lights blink simultaneously, and the light waves reach the other light simultaneously.
Smith sees the light from the Jones's stern travel at the speed of light, but the bow receding at half the speed of light. If he sees the Jones as L long, he sees the light travel 2L before it reaches the bow. He sees the light from the bow travel at the speed of light but the stern approach it at half the speed of light. It only travels 2/3 L before it meets the stern. So Smith sees the light going from the Jones's stern to the bow traveling 3 times as far -- and, thus, taking 3 times as long -- as the light traveling from the stern to the bow. The light from the stern reaching the bow, and the light at the bow flashing is seen as happening at the same place and the same time by Jones; thus they are seen as happening at the same time by Smith (rule 3, above).
While Jones sees the two lights flashing simultaneously, Smith sees the light at the bow flashing significantly later than the light at the stern, since the light from the bow needs only one third the time to reach the stern.
Follow me below the orange spiral nebula to see how this disagreement about simultaneity results in the other strange reports from relativity and means that FTL communication violates causality.
The three italicized paragraphs below are merely details to convince you that the same principles apply when Jones looks at what happens on the Smith.
Smith sees the light from the Smith's stern take a second to reach the bow, and the light from the bow take a second to reach the stern. The lights blink simultaneously, and the light waves reach the other light simultaneously.
Jones sees the light from the Smith's stern travel at the speed of light, but the bow receding at half the speed of light. If he sees the Smith as L long, he sees the light travel 2L before it reaches the bow. He sees the light from the bow travel at the speed of light but the stern approach it at half the speed of light. It only travels 2/3 L before it meets the stern. So Jones sees the light going from the Smith's stern to the bow traveling 3 times as far -- and, thus, taking 3 times as long -- as the light traveling from the stern to the bow. The light from the stern reaching the bow, and the light at the bow flashing is seen as happening at the same place and the same time by Smith; thus they are seen as happening at the same time by Jones (rule 3, above).
While Smith sees the two lights flashing simultaneously, Jones sees the light at the bow flashing significantly later than the light at the stern, since the light from the bow needs only one third the time to reach the stern.
Now the disagreement about what is simultaneous can explain the symmetric oddities of what is generally reported about relativity.
Smith sees the Jones as shrunk along the north-south axis because he sees that the moment that the Smith's bow passes the Jones's stern, the Jones's bow has not yet neared the Smith's stern. Jones sees Smith as shrunk on the north-south axis because he sees that the moment that the Smith's bow passes the Jones's stern, the Jones's bow long since passed the Smith's stern. They can make such contradictory observations because they do not agree what is "the moment" if it refers to events at a distance.
Smith sees the clocks on the Jones as running slower than his, and Jones sees the clocks on the Smith as running slower than his. If each has a set of clocks along the edge of his ship, synchronized as far as he is concerned, then the other sees the clocks towards the stern of the other ship as running earlier than the clocks toward the bow. Smith and Jones both see Smith's clock at the stern record 2 seconds later when Jones's bow passes it than Smith's clock at the bow records when bow passes it. Since Jones sees Smith's clock at the stern as running earlier, he sees that clock pass less time from the time that the his bow passed Smith's bow.
CAUSATION
Two space-faring empires, the Chatons and the Chienners, are hostile but nor at war. Each has the Ansible, instant communication. Fleets from the two empires pass each other heading in opposite directions; their relative velocity is c/2, half the speed of light. Each fleet travels in two waves, one two light days behind the other. The leading waves of the two fleets pass warily but peacefully.
When the Chaton leading wave nears the Chienner trailing wave, the Chienners fire at them. They fire back and report to the Chaton trailing wave by ansible.
"Well," the Chaton trailing wave responds, "we haven't come up to their leading wave yet. When we do, we will fire as soon as we are in range."
The do, The Chienner leading wave is surprised. They, however, fire back and contact the Chienner trailing wave by ansible to report.
"Well," the Chienner trailing wave responds, "we haven't come up to their leading wave yet. When we do, we will fire as soon as we are in range."
So, who started the fight? From the Chaton perspective, the Chienners fired first, and the Chatons merely returned fire. From the Chienner perspective, the Chatons fired first, and the Chienners merely returned fire.
Okay, instant communication destroys cause and effect. Actually, any communication faster than light from the perspective of one frame of reference is communication backwards in time from another perspective. I haven't proven this here, merely asserted it. I believe, though, that the discussion above makes this assertion believable.
Quantum Entanglement
If you have two electrons which are entangled, however distant, and both are tested for spin on the North-South axis, one will test North, and the other will test South.
One explanation for this is that somehow, if electron A tests North, it communicates to electron B that fact, and electron B obediently tests South. Notice, however, that this is an explanation of the observed fact that one of the two will test North and the other will test South. It is not, itself, an observation. (Indeed, if you are enough of a crackpot, the explanation is that every single subatomic particle in the cosmos is in instant communication with every other one.)
Consider, for a moment, the thought experiment of two quite distant electrons which are still entangled. (They must begin in close proximity to be entangled, but they can be separated as distant as you have time and resources to take them.) They are tested close enough to "at the same time" that light starting out from neither test can reach the other particle until after it has been tested.
From Smith's perspective, A is tested first. If it tests North, then B will consequently test South. If it tests South, then B will consequently test North.
From Jones's perspective, B is tested first. If it tests North, then A will consequently test South. If it tests South, then A will consequently test North.
Neither perspective -- not even the interaction between them -- can possibly create a conflict with causation.