The twin paradox explains, using relativity theory, what happens to a person travelling close to the speed of light. When he returns to Earth, he finds his twin brother much older because the traveller's clock moved slower. If we took this example to an extreme, what would happen if the traveller went FTL speed? Would he get younger? How do I write down, mathematically, the equations to solve this problem? (I know it is physically impossible, but it should be mathematically possible.)

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    $\begingroup$ So you wish to apply relativity to a domain relativity strictly prohibits? $\endgroup$ – lemon Feb 18 '16 at 9:29
  • $\begingroup$ It is mathematically impossible, that is what is shown by the equations you ask for. $\endgroup$ – RedGrittyBrick Feb 18 '16 at 9:57
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    $\begingroup$ You will get useless results if you attempt to do these calculations. It will yield answers like infinity or imaginary numbers for gamma. $\endgroup$ – Jaywalker Feb 18 '16 at 10:25
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    $\begingroup$ I find myself impatient with comments bleating you can't do that without explaining why. The calculation is perfectly possible, and actually quite straightforward, and explaining how the calculation is done shows very clearly why it yields a physically meaningless result. $\endgroup$ – John Rennie Feb 18 '16 at 11:17

If we fix a coordinate frame on the Earth, where the stationary twin is living, then the trajectory of the travelling twin in four-dimensional spacetime forms a line called the world line. If we assume the travelling twin heads out then back along the $x$ axis (so we can ignore $y$ and $z$) then the world line looks something like:


The elapsed time measured by the travelling twin is called the proper time, $\tau$, and is simply the length of the world line i.e. the length of the blue line. However we have to calculate this length using the Minkowski metric (again supressing the $y$ and $z$ axes):

$$ c^2d\tau^2 = c^2dt^2 - dx^2 $$

We need to integrate this equation to get $\tau$, and we do it by rearranging to give:

$$ \tau = \int_A^B \sqrt{1 - \frac{v^2(t)}{c^2}} \, dt \tag{1} $$

where we have used the fact that $dx/dt$ is simply the coordinate velocity $v$ - note that the velocity is a function of time. To calculate the elapsed time for the twin simply choose your form for $v(t)$ and do the integral.

By now (I hope) you're thinking that this turned out to be surprisingly simple, and indeed it is pretty straightforward. It also shows us immediately why you can't have a coordinate velocity faster than light. If $v \gt c$ then $v^2/c^2 \gt 1$ and consequently $1-v^2/c^2 \lt 0$. This means you have to take the square root of a negative number and the elapsed time becomes imaginary. But the elapsed time is a real scalar - it is literally just the time shown on a clock carried by the travelling twin - so it can't be imaginary.

It is, as you say, mathematically possible to do the calculation for an FTL trajectory, but it gives a physically meaningless result.

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    $\begingroup$ That's a very nice answer... but if I was the OP (or a very smart physicist, like Dirac, which I am obviously not, of course) I would probably ask if there is a possibility that the definition of proper time is incomplete and if, maybe, a redefinition can be found (e.g. by squaring the equation or by interpreting it with non-scalar quantities?) that gives identical results for all scenarios with v<c but somehow algebraically extends proper time for v>c so that it retains some physical sense. In other words... how can we extend special relativity such that FTL is meaningful? $\endgroup$ – CuriousOne Feb 18 '16 at 11:46
  • $\begingroup$ @CuriousOne Yes, that is a good comment. There may even be a discontinuity at v = c. But John Rennie gives a good and complete answer that is theoretically sound (which I am accepting as the correct answer). I guess anything else would only be speculation at this point. I was mistakenly assuming that at v > c time would be negative, not imaginary as relativity is predicting, and I am glad he corrected me. $\endgroup$ – user Feb 18 '16 at 13:36
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    $\begingroup$ @user: I didn't suggest that John Rennie is wrong. Of course he isn't. I would point out, though, that the program I was hinting at was actually carried out (and is still being investigated) by theorists. It is speculation, but if you are actually curious about FTL and you want something other than the standard argument, then you might want to check out physics with multiple time dimensions. While everything I have seen so far runs into difficulties, so unlike the Dirac equation it's not really opening up new horizons, it's still worth looking at it technically, just for the fun of it. $\endgroup$ – CuriousOne Feb 18 '16 at 21:41
  • $\begingroup$ @CuriousOne do you have any actual link or reference? I am curious too (no pun intended) $\endgroup$ – user Feb 18 '16 at 22:34
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    $\begingroup$ @user: If you do an arxiv search for "two time physics", you will find a number of articles that contain the references to a non-trivial number of related papers. There is an easy to read overview talk called "Physics with Two Time-like Dimensions" by Berndt Mùˆller. It also talks about the consequences for closed timelike curves and I think it does contain some good references to higher quality papers related to the topic. $\endgroup$ – CuriousOne Feb 18 '16 at 22:57

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