A Roller Coaster Ride through Relativity/Appendix E

Travelling 'faster than light'

If you travel for a distance x at a speed v, owing to length contraction, the proper time interval (ie the number of years you age during the journey) between setting out and arriving will be:

$T=\frac{x}{\gamma v}=x\sqrt{\frac{1-v^2/c^2}{v}}=\frac{x\sqrt{1-v^2/c^2}}{cv}$
$T=\frac{x}{c}\sqrt{c^2/v^2-1}$

A light beam, on the other hand, will actually take:

$T=\frac{x}{c}$

If we put these two expression equal, we can find out at what speed it is necessary to travel in order to get the effect of travelling as fast as light.

$\frac{x}{c}\sqrt{c^2/v^2-1}=\frac{x}{c}$
$\sqrt{c^2/v^2-1}=1$
$c^2/v^2-1=1$
$c^2=2v^2$
$v=\frac{c}{\sqrt{2}}$

ie 71% of the speed of light.

Let me just say again what this means. You are not actually travelling faster than light – but you will reach Alpha Centauri, 4 light years away, in only 4 of your years.

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