A Roller Coaster Ride through Relativity/Appendix E

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Travelling 'faster than light'[edit]

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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