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:

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

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

$T={x \over c}$

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.

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

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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