Yet another FTL question...

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Yet another FTL question...

Postby tomandlu » Sat Dec 30, 2017 5:40 pm UTC

If you could teleport instantly to any point in the universe, but didn't have any magical way to change your relative momentum, which you'd still have to handle conventionally, could you still potentially break causality?
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Re: Yet another FTL question...

Postby doogly » Sat Dec 30, 2017 5:50 pm UTC

Changing relative momentum doesn't require magic to do, just teleporting instantly. That breaks causality.
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Re: Yet another FTL question...

Postby eSOANEM » Sun Dec 31, 2017 12:21 am UTC

With enough time beforehand, any FTL travel can always be combined with prepared sublight systems to create the standard ansible paradoxes.
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Re: Yet another FTL question...

Postby ucim » Sun Jan 07, 2018 3:42 am UTC

jewish_scientist wrote:What does 'breaking causality' actually mean?
Causality: For two events A and B, "A can cause B" implies that "B cannot cause A".

Generally, A can cause B if B is in the light cone of A. They have a timelike separation. In such a case, if A is before B in one reference frame, it must be before B in all reference frames. ("There's not enough space between A and B to turn into enough time to change the ordering."). B could not cause A because B happens after A in all reference frames.

If A and B are separated by a spacelike interval, then there are reference frames in which A happens before B, and reference frames in which B happens before A. A and B are outside each other's light cone. A could not cause B, and B could not cause A, because in either case, A and B are too far apart for a message to pass between them, even at the speed of light.

But what if a message could get from A to B faster than light? Then the light cone "no longer matters". In one reference frame (where A happens before B), the message goes from A to B, (so A could cause B), but in another reference frame (where B happens before A) the message is seen to go from B to A. So B could cause A. And these two statements are both true at the same time, because there is no "preferred" reference frame. The "causality implication" (in blue, above) is no longer true. It is "broken".

Your dying of a bullet wound could conceivably cause somebody to shoot the gun that fires the bullet that already hit you. That seems nonsensical.

Of course, "seeming nonsensical" is not the same as "being false". Exhibit A: Quantum mechanics. :)

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Re: Yet another FTL question...

Postby doogly » Mon Jan 08, 2018 3:21 am UTC

Very precisely, it means [A(x_1),B(x_2)] = 0 if x_1 and x_2 are space-like separated. A and B are any observable operator. [ , ] is the commutator. Quantum mechanics obeys this condition, which is why the "spooky" action at a distance should not actually be considered spooky.
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Re: Yet another FTL question...

Postby ucim » Mon Jan 08, 2018 3:48 am UTC

I must be flying over Redmond. :)

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Re: Yet another FTL question...

Postby jewish_scientist » Thu Jan 11, 2018 11:36 pm UTC

Okay. I understand all of that. I had a different question though.

On a Penrose Diagram light travels at 45o, so anything moving FTL would be at >45o. Drawing these lines would create a FTL cone. Since an object's FTL cone is larger than its light cone, then two objects not in each others light cones still be in each other's FTL cones. Basically take all of the research done on causality and substitute 'light cone' with 'FTL cone'. Wouldn't causality not break now even though messages can travel FTL.

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Re: Yet another FTL question...

Postby DavidSh » Fri Jan 12, 2018 12:49 am UTC

I'm probably repeating what a previous poster has said, but I'll continue anyway.

If the path from one event A in spacetime to another event B is faster-than-light, then they have "space-like" separation. From some inertial frames, A occurs before B. From other inertial frames, B occurs before A. We can even find, for any faster-than-light speed S we like, an inertial frame from which the speed of travel from A to B is S, and another inertial frame from which the speed of travel from B to A is S. If the functioning of your FTL drive is the same in any inertial reference frame (principal of relativity), then you can use it to travel a loop in time from A to B back to A.

The general conclusion is (FTL, relativity, causality), pick at most two.

Not to say you couldn't create interesting fictional science with preferred reference frames that are hard to detect, just like the breaking of apparent symmetries in particle physics.

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Re: Yet another FTL question...

Postby gmalivuk » Fri Jan 12, 2018 1:45 am UTC

Yeah, there's no single agreed-upon "FTL cone" unless there's a single privileged rest frame for the universe. Otherwise any path that looks goes at all faster than light from one frame (even if only by 1% or something) will go back in time from another frame moving sufficiently fast relative to the first one.

The thing all inertial observers agree on is the spacetime interval between two events. This is the difference between the square of the spatial separation and the square of the (speed of light times) temporal separation.

If we measure time and distance in units where the speed of light is 1 (e.g. we use years and light-years), then we can say
Δs2 = Δt2 - Δx2, where Δt is the temporal separation between events and Δx is the spatial separation in one dimension.
(With this sign convention, Δs is the proper time experienced by an inertial observer that moves from the first event to the second.)

If I observe/calculate that event B happened 3 light-years from A, 5 years after A, then I calculate Δs2 to be 16. You must agree that Δs2 = 16, but maybe you see B as happening in the same location as A, 4 years after it (This happens if you move from where A happens to where B happens and experience 4 years while doing so). For you, a message sent at A and received at B would appear to have zero velocity. For someone else moving very fast relative to both of us, the message could appear to have any sublight speed, but they'd still calculate Δs2 = 16.

If I calculate that B happens 5 light-years from A, 5 years after A, then Δs2 = 0 and I conclude that if A caused B it did so at the speed of light. You must agree with this conclusion, but depending on your own motion relative to me, you could see B happening 1 light-year from A, 1 year later, or 100 light-years from A, 100 years later.

If I calculate that B happens 5 light-years from A, 3 years after A, then Δs2 = -16.
Depending on your motion, you might calculate that B happens 4 light-years from A at exactly the same time as A, or that B happens 5 light-years from A, 3 years before A. We (should) agree that neither A nor B could have caused the other, because to do so would have required moving faster than light, but we don't agree about which one happened first.
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Re: Yet another FTL question...

Postby Pfhorrest » Fri Jan 12, 2018 1:57 am UTC

To maybe simplify things for jewish_scientist: it's only the speed of light that has a constant angle on the Penrose diagram, because it's only the speed of light that's constant for all observers. You can't have an "FTL cone" for the same reason you can't have a "100mph cone" (of all events you could reach by traveling 100mph from a given event): because there's no such thing as absolute speed, for any speed other than c, either slower or faster.
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Re: Yet another FTL question...

Postby ucim » Fri Jan 12, 2018 5:45 am UTC

To simplify things even further ("as simple as possible, but no simpler"), there is something special about the light cone, and only the light cone. It is that it separates spacetime into two regions: one in which the temporal order of A and B is preserved in all reference frames, and another in which it is not. That's the key to the whole thing.

It's not whether something can travel (FTL) between the two... but rather, whether an observer can see A before B while a different observer sees B before A. Only when this is impossible can causality both exist and be preserved. And that only happens within the light cone, and if FTL travel is impossible. The light cone (and only the light cone) is related to the impossibility of temporal reversal, and FTL is related to the impossibility of A causing B if they are outside the light cone.

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Re: Yet another FTL question...

Postby tomandlu » Sun Jan 14, 2018 8:51 pm UTC

Okay, and in all honesty, although I believe in the principle that FTL breaks causality, I never get it.

So, given my original specification, how would I, for instance, arrange to give myself the combination of a safe when the only record of the combination was inside the safe?
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Re: Yet another FTL question...

Postby Eebster the Great » Sun Jan 14, 2018 10:08 pm UTC

But what if we increase the speed of light in 2208?

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Re: Yet another FTL question...

Postby gmalivuk » Mon Jan 15, 2018 1:29 pm UTC

tomandlu wrote:Okay, and in all honesty, although I believe in the principle that FTL breaks causality, I never get it.

So, given my original specification, how would I, for instance, arrange to give myself the combination of a safe when the only record of the combination was inside the safe?

When you receive the combination in a message from your friend, you open the safe and teleport to a distant location just as your friend passes by at a significant fraction of light speed (he left Earth some time ago under sublight propulsion). You send him a message with the combination and he instantly teleports to Earth, where he arrives before you left and sends your past self the password as he flies past at a significant fraction of light speed.

The trick lies in the fact that "instantaneous" means different things in different reference frames, and changing your own momentum isn't the only way to transfer information between frames.

(Of course, you also can't ban momentum changes. You yourself could gradually accelerate away from Earth when you arrive at the distant location, through conventional means, and eventually you'd be going fast enough that your "instantaneous" return hits Earth before you opened the safe.)

Edit: If you jump significantly far away, it only takes a walking speed. Jump a hundred million light-minutes away and then jump back when you're going a hundred millionth of the speed of light, and you'll arrive a minute earlier.
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Re: Yet another FTL question...

Postby tomandlu » Fri Jan 19, 2018 7:20 pm UTC

gmalivuk wrote:
tomandlu wrote:Okay, and in all honesty, although I believe in the principle that FTL breaks causality, I never get it.

So, given my original specification, how would I, for instance, arrange to give myself the combination of a safe when the only record of the combination was inside the safe?

When you receive the combination in a message from your friend...


Thanks for that. I've started doodling light cones - I'm determined to make it something I understand rather than just go along with IYSWIM.
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Re: Yet another FTL question...

Postby lightvector » Sat Jan 20, 2018 5:51 am UTC

Here's a graphical demonstration. In Newtonian physics, the following image shows how your spacetime coordinates transform when you begin at rest and then suddenly accelerate to start moving at a constant velocity to the right.
galilean.png

  • Upward is forward in time.
  • Right and left are forward and backward in space.
  • The thick horizontal black axis is is a line of constant time - the set of all points in spacetime whose time is the current time, i.e. "now".
  • The thick vertical black axis a line of constant position - the set of all points in spacetime that are at the same position as you, in the reference frame where you're at rest, i.e. "here". Or, you can think of it as the set of all points in spacetime that you will visit if you remain at your constant velocity and do not accelerate.
  • Initially, you start at rest relative to the grid. Things at rest relative to you trace out vertical blue lines as time progresses. Every horizontal line marks one unit of time passed.
  • Then, you suddenly accelerated to the right. Now the vertical lines slant to the left - this shows that as you move forward in time at your new constant velocity, things that used to be at rest relative to you are now moving leftward relative to you. But the horizontal lines haven't changed - what used to be 1 second when you were at rest is still 1 second now.

How does special relativity differ?

Go to this site, which has an interactive Minkowsky diagram: http://ibises.org.uk/Minkowski.html. Scroll down to where it says "Grid" and click "Make Grid". Then, go up, and in the "Boost to velocity" field, type in 0.15 and click "Boost".

That is how the grid transforms in special relativity when you accelerate to the right!
minkowsky.png

  • The vertical lines still slant to the left. Things that used to be at rest relative to you are still moving leftward relative to you, that's still true.
  • But the horizontal lines have slanted too! That means that "now" from your current perspective is a different set of points than "now" from the perspective you had before you accelerated to the right. Distant events that used to be in the past or in the future are still distant but are "now" instead of past or future.
  • Notice also that if you were to draw 45-degree lines, they would still perfectly pass through exactly the same corners of all the grid squares that they used to before the transformation. This shows that despite suddenly accelerating to the right, your light cone is actually still the same as before. If a photon initially travels rightward at the speed of light relative to you, tracing a 45 degree line, and you accelerate to the right, afterward it will *still* be travelling rightward at the full speed of light relative to you, still tracing the same fully-slanted 45 degree line, as opposed to the speed of light minus the speed added by your acceleration, which would be a less-slanted line.

Edit:
  • Also, the slanting of time across distance means that any form of instantaneous travel is also a form of time travel. For example, I placed a green and a blue "x" on the grid above.
  • Notice via the red grid how the green 'x" used to be at a point in spacetime at an earlier time from the origin (although distant in space), and the blue "x" at a yet earlier time. However, since we've boosted by 0.15c to the right, now the green "x" is at the same time, so if we were to instantaneously travel to the left, we would now be at the green x.
  • Now, fire your rockets in the other direction until you've gotten to a velocity of -0.15c relative to your original reference frame. Now the grid looks like this.
    minkowsky2.png
  • The blue x is now directly horizontal from the green x. They are at different positions in space, but they are at the same time. So activate your instantaneous travel device again and jump rightward, from the green x to the blue x. Then, fire your rockets until you are back at 0c relative to your original situation, and the grid is all nice and square again. You have now successfully traveled directly to the past from your starting point.


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