Length Contraction © D Hoult 2011

Observer B has a long stick and decides to measure its length

Observer B has a long stick and decides to measure its length He / she has a great idea…

Observer B has a long stick and decides to measure its length He / she has a great idea… why not use the light beam clock possessed by A to do this ?

Observer B has a long stick and decides to measure its length He / she has a great idea… why not use the light beam clock possessed by A to do this ? We all agree on the speed of light so we just need to know our

Observer B has a long stick and decides to measure its length He / she has a great idea… why not use the light beam clock possessed by A to do this ? We all agree on the speed of light so we just need to know our relative speed

Observer B has a long stick and decides to measure its length He / she has a great idea… why not use the light beam clock possessed by A to do this ? We all agree on the speed of light so we just need to know our relative speed Here’s how it works…

Observer B holds the stick

Observer B holds the stick parallel to the direction of relative motion of A and B

Observer A (who has the torch) sends a pulse of light at the instant when the torch is near end 1 of B’s stick

The light is reflected by the mirror

The light is reflected by the mirror

and arrives at end 2 of B’s stick

The length of B’s stick as measured by B (L0) is given by

The length of B’s stick as measured by B (L0) is given by v × t where t is the time taken for light to follow the path shown below

Looking at the situation from A’s point of view

The length of B’s stick as measured by A (L) is

The length of B’s stick as measured by A (L) is given by v × t0

The length of B’s stick as measured by A (L) is given by v × t0 where t0 is the time taken for light to follow the shorter path shown below

Therefore we have L = v × t0 and L0 = v × t

Therefore we have L = v × t0 and L0 = v × t it has previously been shown that

Therefore we have L = v × t0 and L0 = v × t it has previously been shown that t = t0

Therefore we have L = v × t0 and L0 = v × t it has previously been shown that 1 t = t0 v2 1 – c2

Therefore L0 1 = L v2 1 – c2

Therefore L0 1 = L v2 1 – c2 or

Therefore L0 1 = L v2 1 – c2 or L =

Therefore L0 1 = L v2 1 – c2 or v2 L = L0 1 – c2

Conclusion: the observer moving relative to the stick measures it to be = L0 1 – c2

Conclusion: the observer moving relative to the stick measures it to be shorter than the measurement made by the person holding the stick v2 L = L0 1 – c2

The measurement made by the observer stationary relative to the stick (B in this case) is called the proper length (L0) of the stick v2 L = L0 1 – c2