Causality who needs it?

qraal

Hi All

I just don't get what "causality" means in relativity. Regular meaning: Cause before effect, sure, but what does it mean in relativity? Why should all observers have to agree about the order of events? Or have I missed something in the definition of relativity? Why does it matter?

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Rasalhague

Hi All

I just don't get what "causality" means in relativity. Regular meaning: Cause before effect, sure, but what does it mean in relativity? Why should all observers have to agree about the order of events? Or have I missed something in the definition of relativity? Why does it matter?
The same definition holds for relativity: cause comes before effect.

The quirk is that in relativity, if there isn't time for information to get from one event to another (given that nothing can travel faster than the speed of light in a vacuum), then there's no absolute way to determine which event happened first. If one happens first according to one spacetime coordinate system (reference frame), you could just as easily describe the same events using a different spacetime coordinate system moving at some constant velocity relative to the other system, and you can always find some relative velocity for which the events happen in a different order. But because information can't get from one such event to another, neither event can causally influence the other, and so no contradictions arise. By contrast, when there is time for information to get from one event to another, they always happen in the same order, no matter which spacetime coordinate system you chose to measure them in. So if it's possible for one event to cause another, there's no ambiguity about which order they happen in. If this wasn't the case, there could be no single consistent description of reality.

qraal

So how does FTL motion violate causality? It may violate the perceived order for distant observers, but why should that determine/constrain what happens locally?

The same definition holds for relativity: cause comes before effect.

The quirk is that in relativity, if there isn't time for information to get from one event to another (given that nothing can travel faster than the speed of light in a vacuum), then there's no absolute way to determine which event happened first. If one happens first according to one spacetime coordinate system (reference frame), you could just as easily describe the same events using a different spacetime coordinate system moving at some constant velocity relative to the other system, and you can always find some relative velocity for which the events happen in a different order. But because information can't get from one such event to another, neither event can causally influence the other, and so no contradictions arise. By contrast, when there is time for information to get from one event to another, they always happen in the same order, no matter which spacetime coordinate system you chose to measure them in. So if it's possible for one event to cause another, there's no ambiguity about which order they happen in. If this wasn't the case, there could be no single consistent description of reality.

A.T.

qraal said:
Causality... who needs it?
Not me. I'm perfectly fine with being my own grandfather.
qraal said:
So how does FTL motion violate causality? It may violate the perceived order for distant observers
Not the perceived order. The actual order in the observer's reference frame, which is on an equal footing with any other reference frame in relativity. The laws of physics would be different in that frame if the order of events is reversed.

Bob_for_short

Causality is an experimental fact and should be implemented in any theory.
In SR there is a clear distinction between causal and independent events.

Dale

Mentor
The laws of physics would be different in that frame if the order of events is reversed.
Hi AT, I know you know this already, but I just wanted to mention that this should read "The laws of physics would be different in that frame if the order of causally connected events is reversed."

qraal

There is a distinction? The impression I get is that observing an event means you're no longer distinct from it and its antecedents. What does 'causality' mean in SR? Why does it forbid FTL/space-like motion?

Causality is an experimental fact and should be implemented in any theory.
In SR there is a clear distinction between causal and independent events.

Bob_for_short

The impression I get is that observing an event means you're no longer distinct from it and its antecedents.
It's only an impression.
What does 'causality' mean in SR? Why does it forbid FTL/space-like motion?
Space-like events cannot be causally related because there is no way to influence in such conditions. A space-like interval is just (an invariant) distance between two instant but distant events.

qraal

Space-like events cannot be causally related because there is no way to influence in such conditions. A space-like interval is just (an invariant) distance between two instant but distant events.
Hmmm. I had thought 'space-like' meant the otherside of the line representing light in a Minkowski diagram. It's not? My confusion is worse than I realised!

DrGreg

Gold Member
So how does FTL motion violate causality? It may violate the perceived order for distant observers, but why should that determine/constrain what happens locally?
If faster-than-light transfer of "causal information" were possible, you could set up a "causal loop", a triangle of events A1, A2, B3 around which the information could circulate and end up back where it started.

Consider an inertial observer Alice and two events on her worldline A1 and A2 occurring in that order, so A1 is in the causal past of A2, and all observers agree on the order of the two events. At the later event A2, Alice sends an FTL signal to Bob who receives it at event B3. According to Alice, B3 occurs after A2. But if Bob is travelling fast enough, according to Bob B3 occurs before A2. In fact if Bob is travelling really fast enough, according to Bob B3 occurs before the earlier event A1. If Alice can send FTL signals, then so can Bob, and so there is nothing to stop Bob sending an FTL signal from B3 to A1.

With Bob's help acting as a relay, Alice has sent a message into her own past, from A2 to A1 (via B3). In principle, that could be a message to kill her own grandmother leading to a logical contradiction.

So the three concepts of
• relativity
• causality
• faster-than-light travel
are incompatible. You can have any two in the list, but not all three.

qraal

Hi DrGreg

Do you know of any good diagrammatic explanations around the Web for this one? I think I get what you're saying, but it still seems kind of odd. I've never gotten why FTL leads to backwards time-travel. Why does it? How can what Bob see be anything other than light arriving at his eyes? Why should he be looking into the past, past when Alice sent the signal? Why should his perceptions have an apparently causal effect???

Hmmm.

If faster-than-light transfer of "causal information" were possible, you could set up a "causal loop", a triangle of events A1, A2, B3 around which the information could circulate and end up back where it started.

Consider an inertial observer Alice and two events on her worldline A1 and A2 occurring in that order, so A1 is in the causal past of A2, and all observers agree on the order of the two events. At the later event A2, Alice sends an FTL signal to Bob who receives it at event B3. According to Alice, B3 occurs after A2. But if Bob is travelling fast enough, according to Bob B3 occurs before A2. In fact if Bob is travelling really fast enough, according to Bob B3 occurs before the earlier event A1. If Alice can send FTL signals, then so can Bob, and so there is nothing to stop Bob sending an FTL signal from B3 to A1.

With Bob's help acting as a relay, Alice has sent a message into her own past, from A2 to A1 (via B3). In principle, that could be a message to kill her own grandmother leading to a logical contradiction.

So the three concepts of
• relativity
• causality
• faster-than-light travel
are incompatible. You can have any two in the list, but not all three.

Rasalhague

There is a distinction? The impression I get is that observing an event means you're no longer distinct from it and its antecedents. What does 'causality' mean in SR? Why does it forbid FTL/space-like motion?
Suppose you could travel at a constant velocity greater than c (the speed of light in a vacuum) from one point in spacetime to another (call them A and B). It's possible to chose an inertial reference frame (call it F) in which A precedes B. But it will also be possible to chose another, equally valid inertial reference frame (call it F'), moving at some constant velocity relative to F, in which your arrival at B precedes your departure from A. Suppose you arrive at B, find a bomb with a lit fuse and put it out. According to F', the bomb has already exploded when you arrive at B. So will observers see an explosion or not? If FTL travel was possible there'd be a contradiction.

At least, that's what I've been assuming. I'll have to read the article Count Iblis posted a link to...

DrGreg

Gold Member
Do you know of any good diagrammatic explanations around the Web for this one? I think I get what you're saying, but it still seems kind of odd. I've never gotten why FTL leads to backwards time-travel. Why does it? How can what Bob see be anything other than light arriving at his eyes? Why should he be looking into the past, past when Alice sent the signal? Why should his perceptions have an apparently causal effect???
Have a look at this thread. There's a diagram in post #32, and I gave another numerical example in post #42 (knowledge of Lorentz transform required).

Jonathan Scott

Gold Member
If it turned out (for example from some future form of quantum theory) that there were some "preferred frame" (often called "subspace" in science fiction) which in some sense happened to have an absolute velocity of zero, then sending signals at up to infinite speed relative to that frame would not violate causality. However, the transformations of Special Relativity mean that relative to some other frames that "infinite speed" would only be a tiny amount more than the speed of light, whereas in some frames the signal would be appear to be travelling so fast it would go backwards in time (but less in time than in space, so overall the motion gives a spacelike separation).

This method is of course incompatible with the principle of relativity, in that it is impossible for have some method which allows signals to be sent a finite amount faster than light from any arbitrary frame of reference without violating causality.

qraal

Hi DrGreg

Makes more sense now, thanks! The signalling diagram between Bob and Alice stumped me a bit at first, but then I realised they were relativistic mirror-images of each other's motion from the other's point-of-view. Light dawned. And - damn! - the signals really do seem to go back in time. If an inertial observer was merely bouncing super-luminal signals off a mirror that's travelling at the same speed there would be no causality issues would there? The trick is when observers are in relative motion to each other... then causality rears its complicated head.

Have a look at this thread. There's a diagram in post #32, and I gave another numerical example in post #42 (knowledge of Lorentz transform required).

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colin456

Just to elaborate on the two great answers given by Rasalhague slightly. Rasalhague stated that the fact that given a frame where an event A precedes an event B, it is only possible to find a different inertial frame where B precedes A if the faster than light transmission of information is possible. This can be seen simply as a result of assuming causality and one of the lorentz transformations.

Suppose there is an inertial frame, F, such that event A precedes event B and the two have a time seperation in F of dt and a spatial separation of dx (these are not necessarily infinatesimals). Take another inertial frame, F', such that F' is traveling at a speed of u with respect to F. The time interval between events A and B in F', call it dt', will be given by the well known lorentz transformation

dt'=(dt-u*dx/c^2)/(1-u^2/c^2)^0.5

Lets assume for a second that dt' is negative such that in frame F', B precedes A. The question is, in what conditions is this possible?

If dt' < 0 then clearly that means (dt-u*dx/c^2) < 0 and thus you get the condition to be

c*dt < (u/c)*dx

It is important to notice that c*dt is the distance a light beam would travel between events A and B in frame F. We now have a condition for which the temporal order of two events in one reference frame can be reversed simply by switching to a different reference frame. The ways in which this can happen are that either dx > cdt, i.e. event B is not within the light-cone of event A, or if dx<cdt then (u/c)>1 in which case u>c.

If the condition for causality to hold is states as follows, "it is not possible for an event A to be the cause of an event B if a reference frame can be found such that B precedes A in time", then it can be seen from what was given above that we have to exclude all reference frames where u>c and state the following condition "it is not possible for an event to cause another event which is outside its lightcone, i.e. where dx>cdt". Thus, information cannot travel faster than the speed of light.

Rasalhague

Rasalhague stated that the fact that given a frame where an event A precedes an event B, it is only possible to find a different inertial frame where B precedes A if the faster than light transmission of information is possible.
Hi Colin, did you accidentally miss something out of this sentence? I don't think this is quite what I said. We could say, given an inertial frame where A precedes B, it's only possible to find a different inertial frame where B precedes A if the separation between A and B is spacelike - in other words, if information can't be transmitted from A to B or vice versa without exceding the speed of light. But if there is a spacelike separation between A and B, and one inertial frame where A precedes B, then it's always possible to find a different interial frame in which B precedes A.

colin456

Oh right. I must have skimmed it a bit too fast. Well thats what I was arguing anyway.

Rasalhague

Oh right. I must have skimmed it a bit too fast. Well thats what I was arguing anyway.
Yeah, I guessed that's what you must have meant.

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