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Notes: The background for this question is working out details of a sci-fi story. Answers to the effect of "time travel isn't possible" or "FTL isn't possible" are therefore not helpful. I'm looking to ensure that I understand all of the relevant mathematical concepts properly under the assumption that there exists some method of traversing spacelike intervals. I am tagging this as "homework" because it seems like a homework style problem, despite not actually being a homework assignment.

The pilot of an FTL ship is trying to exploit FTL - Time Travel equivalence to travel into the past. The basic procedure is to travel to some distance away from Earth, then accelerate to change frames such that the ship's hyperplane of simultaneity intersects Earth at the desired time in the past (with padding to account for extra time taken in normal travel / accelerating to change frames), and then engage the FTL drive to travel back to Earth.

If the ship starts out in Earth's inertial frame starts out at time $t = 0$ at a distance x from Earth, we want the time coordinate $t'$ at length-contracted distance $x'$ from the ship (corresponding to the position of Earth) after accelerating to be a negative quantity, representing travel backwards in time from time 0. For simplicity, I assume instantaneous acceleration.

First, I calculate the $t'$ coordinate at distance $x'$ from the ship having accelerated to velocity $v$: $t' = \frac{t0 - vx'/c^2}{\sqrt{1-(v/c)^2}}$

The distance $x'$ is then given by Lorentz contraction: $x' = x\sqrt{1-(v/c)^2}$

And substitution of 0 for $t0$ and the definition for $x'$ gives $t' = \frac{-vx\sqrt{1-(v/c)^2}/c^2}{\sqrt{1-(v/c)^2}} = -\frac{vx}{c^2}$

with the interpretation that the ship must have velocity $v$ at position $x$ in Earth's frame, where velocity and position have the same sign, in order to achieve a displacement of $t'$ into the past when engaging the FTL drive to return to Earth.

The first question, of course, just to double-check, is: did I actually do that math correctly, or is there some misconception that makes my calculations so far invalid? If so, what is the correct derivation?

Assuming that I have a correct expression for how far back in time the ship's hyperplane of simultaneity intersects Earth, am I correct a) in interpreting the signs of position and velocity to mean that the ship must be traveling away from Earth to shift it into the past? and b) in assuming that the only velocity that matters is the projection of 3-velocity onto the $\hat{x}$ direction?

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    $\begingroup$ There's something fishy about asking if your derivations are correctly based on special relativity when the basis of the result you're trying to show is disallowed by the very same theory. Am I misinterpreting something? $\endgroup$ – BMS Sep 10 '14 at 21:53
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    $\begingroup$ @BMS - that's presumably the point of the question. We know the answer must be "nope, you're wrong". But can we figure out why this approach doesn't work? $\endgroup$ – Floris Sep 10 '14 at 21:54
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    $\begingroup$ @BMS Special relativity disallows spacelike traversals, but it doesn't disallow us from calculating planes of simultaneity and figuring out when "now" is at any given point in any given reference frame. If you like, you can ignore the actual FTL travel and just think of it as "how do I figure out what velocity and position I need to get an arbitrary time coordinate at position x?" $\endgroup$ – Logan R. Kearsley Sep 10 '14 at 22:03
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    $\begingroup$ @LoganR.Kearsley Next time, please just ask the essential question without the FTL part (or any other redundancies ;) ) attached - I certainly raised my eyebrows when I started reading your post and was about to vote to close before I saw the comments... $\endgroup$ – Danu Sep 10 '14 at 22:17
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    $\begingroup$ @LoganR.Kearsley (3 comments up) indeed, it would probably make your question a lot more palatable if you do that, i.e. drop any references to FTL travel or time travel and just ask about planes of simultaneity. I would also suggest removing your next-to-last paragraph, since we don't handle requests to check people's work. (See also this) $\endgroup$ – David Z Sep 10 '14 at 23:57
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There's nothing physically problematic with speeds, per se, that are faster than the speed of light. For example, if you quickly rotate an Earth-bound laser such that its beam crosses over the moon, the speed of the center of the beam as it travels across the moon's surface can easily be faster than $c$. There's no problem with a speed greater than $c$ in that case because you're just talking about the speed of an abstract set of events, not about the speed of an object.

There is, however, no inertial frame of reference that travels faster than $c$ relative to any other inertial frame of reference. In the equations you gave, if $v>c$, then $t'$ and $x'$ are both imaginary, which has no clear physical interpretation.

A number of famous physicists, starting with Einstein, have looked into how communication faster than the speed of light would amount to communication into the past. The way you investigate that without the equations immediately becoming meaningless is to have the equations involve at least two different relative speeds. The speed of the signal is just treated as the speed of an abstract set of events, not the speed of an inertial frame of reference. You then analyze that signal from the perspective of two different inertial frames of reference, which have a relative velocity $v<c$ with respect to one another.

Of course, faster-than-light signals in the real world aren't possible due to the lack of faster-than-light particles to convey those signals. Faster-than-light signals are just a thought experiment.

For an exposition of faster-than-light signals that avoids using faster-than-light inertial frames of reference, see the Tachyonic antitelephone Wikipedia article.

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