here - Department of Physics, HKU

CHAPTER 2. SPECIAL RELATIVITY 14
ct ′
✲
vt ′
✲✛
l
✲
v(t − t ′ ) c(t − t ′ )
✲✛
Figure 2.6: Length Contraction.
total time of flight. Eliminate t ′ , we have
2l = (1 − v 2 /c 2 )ct . (2.16)
The times of flight in the two reference frames, t 0 and t, are related by
Eq. (2.14). Finally, we have
l = (1 − v 2 /c 2 ) ct/2 =
√
√
1 − v 2 /c 2 ct 0 /2 = 1 − v 2 /c 2 l 0 . (2.17)
Notice that the transverse direction will not be contracted. Therefore,
the shape of a fast moving object will be distorted.
Now, we will give an example that shows the various concepts in this
chapter. When some cosmic particles enter our atmosphere and collide with
the molecules, many other unstable particles will be created. Let us assume
the lifetime of such an unstable particle be T, also assume for simplicity the
thickness of the atmosphere be L. What will be their minimum speed if we
detect those particles on the Earth surface?
There are two equivalent ways to solve this problem. Let the minimum
speed of the particle be v. From the point of view of the particle, due to
length contraction, the thickness of the atmosphere is shorten to
√
L 1 − v 2 /c 2 . (2.18)
In order to reach the Earth surface, vT must be equal to or larger than this
length. Hence, we have
√
vT = L 1 − v 2 /c 2
c
v = √1 + (cT/L) . (2.19)
2