1.5 μs at the rest frame, but they are observed by detectors on the grand. Although
the length by the product of the life time and the speed of light are only 450 m, they
can arrive at the grand after traveling about 20 km of the atmosphere. This is the
evidence of the delay of the time in the frame traveling with the speed of the muon.
This phenomenon is also understood by use of the contraction of the space. Since
the velocity of the muon is almost the speed of light, the thickness of the atmosphere
is roughly less than hundreds of meter in the muon frame. The space and time
contractions are consistent like this example.
5.2.3 Lorentz Transformation of Velocities
It is easy to obtain the relation of Lorentz transformation of velocities:
v ¼
dx
dt
, v
0
¼
dx
0
dt 0
ð5:2:33Þ
Using (5.2.29) it is easy to obtain the transformation of the velocity in the parallel
direction at first:
v k ¼
dx k
dt
¼
dx k
0
=dt
0
þ V 0
1 þ V 0 =c 2 dx k
0
=dt 0
À
Á
Then, the perpendicular velocity vector is also obtained, and they are written in the
form:
v k ¼
v k
0
þ V 0
1 þ V 0 v k
0 =c 2
v ⊥ ¼
v ⊥
0
γ 0 1 þ V 0 v k
0 =c 2
À
Á
ð5:2:34Þ
Note that the velocity transformation in the parallel direction has no Lorentz factor
dependence, while the perpendicular velocity reduced by a factor of Lorentz
contraction.
(5.2.34) can also be applicable to light, and consider the light is irradiated
isotropic in S
0 frame. For a light emitted with an angle θ
0 in S
0 frame, it is observed
to propagate in θ direction in S frame with the following relation:
tanθ ¼
1
γ 0
sin θ
0
β 0 þ cos θ
0
ð
Þ
ð5:2:34aÞ
This is shown in Fig. 5.2 schematically. The uniform beam emission is elongated in
the direction of the motion of the light source as seen in Fig. 5.2. This means that
174
5 Relativistic Laser-Electron Interactions
the length by the product of the life time and the speed of light are only 450 m, they
can arrive at the grand after traveling about 20 km of the atmosphere. This is the
evidence of the delay of the time in the frame traveling with the speed of the muon.
This phenomenon is also understood by use of the contraction of the space. Since
the velocity of the muon is almost the speed of light, the thickness of the atmosphere
is roughly less than hundreds of meter in the muon frame. The space and time
contractions are consistent like this example.
5.2.3 Lorentz Transformation of Velocities
It is easy to obtain the relation of Lorentz transformation of velocities:
v ¼
dx
dt
, v
0
¼
dx
0
dt 0
ð5:2:33Þ
Using (5.2.29) it is easy to obtain the transformation of the velocity in the parallel
direction at first:
v k ¼
dx k
dt
¼
dx k
0
=dt
0
þ V 0
1 þ V 0 =c 2 dx k
0
=dt 0
À
Á
Then, the perpendicular velocity vector is also obtained, and they are written in the
form:
v k ¼
v k
0
þ V 0
1 þ V 0 v k
0 =c 2
v ⊥ ¼
v ⊥
0
γ 0 1 þ V 0 v k
0 =c 2
À
Á
ð5:2:34Þ
Note that the velocity transformation in the parallel direction has no Lorentz factor
dependence, while the perpendicular velocity reduced by a factor of Lorentz
contraction.
(5.2.34) can also be applicable to light, and consider the light is irradiated
isotropic in S
0 frame. For a light emitted with an angle θ
0 in S
0 frame, it is observed
to propagate in θ direction in S frame with the following relation:
tanθ ¼
1
γ 0
sin θ
0
β 0 þ cos θ
0
ð
Þ
ð5:2:34aÞ
This is shown in Fig. 5.2 schematically. The uniform beam emission is elongated in
the direction of the motion of the light source as seen in Fig. 5.2. This means that
174
5 Relativistic Laser-Electron Interactions