Relativity/FTL/Causality Question

[Cycloneman]

If one travels at a speed greater than c at a particular point, then returns back, one travels through time into the past. For purposes of this discussion, we'll assume that we're using something like an Alderson Drive; we can only travel between star systems in predefined ways.

According to this site (written by an astrophysicist, so I presume it's pretty accurate), one would only observe causality paradoxes if an object moved between two frames of reference with a difference of velocity u at velocity v if v*u > (c^2). Is this true? If it is, does that mean that, if one had a "velocity" of precisely v=(c^2)/u, would the apparent time it took to travel between two systems be instantaneous (or however long it takes to cycle your drive), and thus prevent time travel?

Sorry if this is a dumb question.

[adam_grif]

Probably going to be talking straight out my ass here. I also can't comment on whether what he's saying is true, but assuming it is...

If the time-reversal threshold varies based on the relative speed of observers, in order to maintain causality your speed limit would vary wildly based on the relative velocity of the observers in the place you're going to, at the distance you're traveling. This implies that in order to work out a safe speed to travel out without violating causality, your engine has to know (or physics has to know, if that's what's enforcing the upper limit) the velocity of all objects that can see you when you get there, and the drive must also know precisely how far you will go when you turn it off. This is to say, your FTL drive must see into the future of a distant location in space in order to safely get there. What happens if the drive explodes halfway and you drop out of hyperspace in the middle of a system where the observers are all going off at completely different relative velocities than your intended destination? Did the drive factor that in when calculating your speed?

I'm also not sure you can only consider the observers who can see you shortly after you drop out of FTL/hyperspeed/warp/whatever. AFAIK since no frames are privileged, you have to make sure that causality is not violated for every frame of reference, not just the ones looking at you.


You probably won't get a good answer until Surlethe/Kurneko/Wong/Nyrath show up, if they do at all.

[Cycloneman]

I'm also not sure you can only consider the observers who can see you shortly after you drop out of FTL/hyperspeed/warp/whatever. AFAIK since no frames are privileged, you have to make sure that causality is not violated for every frame of reference, not just the ones looking at you.

Yeah, I guess I should clarify. It seems like FTL that operates in this way would make you travel through time, but not in a useful way. If you go from A to B and then back from B to A at c^2/u, and you appear, to local observers, to take no time at all, thus preventing transmission of data backwards in time. If you go from A to B and then travel from B to A in realspace, even at c, you will not arrive before you left, no matter how close or distant A and B are. So while from certain frames of reference you are violating causality, you do not violate causality in a useful way.

[Bakustra]

There are still problems with causality because you're carrying information from the future even if you're restricted to v*u <= c^2. If you travel from say, Earth to Alpha Centauri, you are carrying information from Earth that would take 4 years to arrive at Alpha Centauri. This doesn't result in obvious paradoxes if you do it slowly enough, but you can then turn around from Alpha Centauri and carry information from Alpha Centauri to Earth. Say you witness some catastrophe visible from Earth at Alpha Centuari, and that you move at 4c. This is much less than c^2 when multiplied by the relative velocity of the Alpha Centauri system. It takes one year to get back, and when you do, you have knowledge of events that will happen in three years. You've just carried information from the future to the past. The same with the initial trip; if your launch is visible from Alpha Centauri, you can carry the information of that from the future to the past. While you don't go into your own past at any given time, you still carry information into the past of others. (e.g., you violate causality from those reference frames). Either relativity or causality still must go. Remember that information is what's critical in relativity.

[Freefall]

According to this site (written by an astrophysicist, so I presume it's pretty accurate), one would only observe causality paradoxes if an object moved between two frames of reference with a difference of velocity u at velocity v if v*u > (c^2). Is this true? If it is, does that mean that, if one had a "velocity" of precisely v=(c^2)/u, would the apparent time it took to travel between two systems be instantaneous (or however long it takes to cycle your drive), and thus prevent time travel?

Uh, maybe sort of. Let's see.

The Lorentz transformation for velocity is
v' = (v-u)/(1-uv/c^2)

If you plug in v=c^2/u, then you get

v' = (c^2/u - u)/(1-1) = a/0 (where a is whatever; doesn't really matter here)

Well, fairly standard math would lead you to conclude that a/0 = infinity, meaning the ship's velocity in the moving frame would be infinite, which would suggest it would get there in zero time.

The problem is, a/0 isn't actually infinity, it is only infinity in the limit as the denominator goes to zero. If it's actually zero, then the result is undefined, meaning who knows what happens there?

What if we try to translate back into the original coordinate system?

v = (v'+u)/(1+uv'/c^2), where we have v' = infinity

So we get inf/inf. That's not good. I think that's "undefined" too. I guess that's at least consistent. Well, except that we were supposed to get v=c^2/u. Oh well.

Hm, what kind of kinetic energy would a ship with infinite velocity have?

K = mc^2/(1-v^2/c^2)^(1/2), so if v is infinite, you actually get the squareroot of negative infinity in the denominator. So it looks like K = mc^2/i*inf, whatever that is. Complex kinetic energy can't be good.

It might just be zero; 1/i*inf = (1/i)(1/inf) = 1/i*0 = 0. I forget what all the special rules with i are, but that seems reasonable. Not sure if I buy a ship with infinite velocity having zero energy though.

Well, except that, like I said before, really the velocity is undefined, and as far as I'm aware, you can't do math with "undefined."

I myself am leaning towards the idea of "can't happen." The thing with uv=c^2 is that it requires v>c, which requires infinite energy to happen, so it is sort of impossible from that point of view. It's true that mathematically you could also have u=v=c, or u>c, but physically neither situation would work in the way you want because if u=v, then the ship is moving as fast as the coordinate frame, and will never get there anyway, and if u>c while v<c, then the ship will actually lag further and further behind. So if you want uv=c^2, then v>c is necessary if you want it to ever actually reach the moving frame. But v>c is kind of a problem for various reasons.

Anyway, yeah, the serious physics guys here should be able to give you a more meaningful answer. I think most people tend to dislike my position on relativity, because when they ask me, "what happens if someone is on a plane going the speed of light, and they step forward?" I just say, "the plane can't ever reach the speed of light, and it's squished into a pancake before it gets there anyway."

I think a lot of people find that explanation boring though. I could also just be wrong.

[sirocco]

Well it's just that I find all this time travel and FTL issues troubling since we'd have first to experiment with near lightspeed travel first.

I remember reading somewhere that time paradoxes are just impossible as in "Time is already the result of all the time travel and FTL trips ever done". It's the paradoxes in themselves that can't happen not the time travel.

I think it is linked to Stable time loop.

Possible?

[Freefall]

It would basically mean everything is predetermined, but yeah, it makes some sense.

The funny thing with paradoxes is that they are basically just contradictions. It's sort of like working with a bunch of equations and coming up with 1=2; if that happens, most people are going to figure they messed up somewhere along the line and start over. But when it comes to temporal/physics paradoxes, all of a sudden people like to imagine that the universe just gets destroyed, as opposed to figuring it probably just means you can't do that. Or that we are just missing certain vital knowledge on the subject.

Aren't topics like this usually put in SLAM?

[Kuroneko]

If it is, does that mean that, if one had a "velocity" of precisely v=(c^2)/u, would the apparent time it took to travel between two systems be instantaneous (or however long it takes to cycle your drive), ...

That's exactly what it means. I'm going to lift an image from another website to elaborate:

This animation shows the effect of acceleration on velocity vectors in spacetime (or a continuous Lorentz transformation), which is that they trace out a hyperbola. If the upper vertical axis is time and right horizontal space, then one can represent any subluminal velocity v as the vector [cosh α; sinh α] with α such that v = sinh α/cosh α = tanh α, or any multiple of that vector, since the ratio of space to time would be the same. Velocity addition just adds to the hyperbolic angle α (aka rapidity).

Note that for subluminal speeds (the top and bottom branches), you can't change your orientation in time: if you've got a positive t-component, then so it will remain no matter what angle you add. On the other hand, a superluminal velocity needs |x|>|t|, so that its velocity vector will be on either the left or right branches of the hyperbola, depending on which direction in space the spaceship is going. On the right branch, a velocity vector can be taken to be [sinh β; cosh β] instead, with v = coth β (or again, any multiple of it).

But as you can see in the diagram that for the left and right branches, it's quite possible to go from negative-t to positive-t and vice versa, as the rotation crosses the horizontal axis representing infinite velocity (some distance in t = 0). When does this happen? Well, you reach infinite speed when you the angle you're adding is α = -β, and since tanh and coth are recipricols, this means recipricol speed: |uv| = 1, i.e., c² in normal units.

... and thus prevent time travel?

Well, that's a bit different. If our criteria is that you don't observe a a backward-time-velocity of the ship, then yes. But that's not very significant by itself, since you're not the only observer around. If the ship's local velocity is superluminal, then there its velocity will point backward in time for some observers.

Does that break physics? If we're talking about flat spacetime, then even one superluminal signal, carried by a ship or otherwise, makes it possible for someone to arrange other subluminal relays that send information back in time. All you would really need are highly-relativistic flybys of the endpoints of the journey. So it's not enough to say no one in the SW galaxy is actually doing that; one should also worry about why it's not being exploited.

It's possible to handwave hyperspace by supposing that it isn't part of normal spacetime and some mechanism makes attempts to break causality fail, e.g., feedback of virtual particles breaking the scheme. Alternative, 'interdiction fields' that prevent hyperspace jumps actually work by attempting the FTL signal (the ship) to send information to the past, and this mechanism breaks the attempted jump instead. Perhaps potential causal conflicts is also why FTL speeds seem to be highly variable in the SW galaxy. That's more than a bit ad hoc, though.