# 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:

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

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

${\displaystyle T={x \over c}}$

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.

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

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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