here - Department of Physics, HKU

CHAPTER 2. SPECIAL RELATIVITY 13
h
vt’
Figure 2.5: A moving light clock.
√
by the light pulse is 2 h 2 + (vt ′ /2) 2 . We have
Solving for t ′ ,
t ′ =
√
2 h 2 + (vt ′ /2) 2 = ct ′ . (2.13)
2h/c
√
1 − v 2 /c = t
(2.14)
√1 2 − v 2 /c 2
which is greater than t. Thus, a moving light clock is running slower. Notice
that not only the light clock, any clock will run slower by the same factor,
because, by the first premise, all physics is the same in any inertial frame.
2.5 Length Contraction
Length contraction is also called Lorentz contraction: the length of a
moving object will be shorter along the direction of motion. If the object is
of length l 0 at rest, then we will show that when it is moving with speed v,
the length becomes
l = l 0
√1 − v 2 /c 2 . (2.15)
We try to illustrate the length contraction as follows.
If the proper length of a rod is l 0 , and it is at rest relative to the observer,
the time taken for a photon to go from one end to another and back is
t 0 = 2l 0 /c.
Now, assume that the rod is moving with a speed v to the right, Fig. 2.6.
Again, we use the time of flight of a photon to measure its length l. In the
first half (top of Fig. 2.6), when the photon reaches the other end, the rod
has moved a distance vt ′ , where t ′ is the time of flight of the first half. We
have l + vt ′ = ct ′ . Similarly, in the second half (bottom of Fig. 2.6), while
the photon is coming back, we have l − v(t − t ′ ) = c(t − t ′ ), where t is the