1. Length Contraction
© D Hoult 2011

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

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

4. 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 ?

5. 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

6. 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

7. 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…

8. Observer B holds the stick

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

10. 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

11. The light is reflected by the mirror

12. The light is reflected by the mirror

13. and arrives at end 2 of B’s stick

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

15. 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

16. Looking at the situation from A’s point of view

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

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

23. 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

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

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

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

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

28. Therefore L0 1 = L v2
1 – c2

29. Therefore L0 1 = L v2
1 – c2 or

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

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

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

33. 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

34. 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