here - Department of Physics, HKU

CHAPTER 3. LORENTZ TRANSFORMATIONS 22
Hence, simultaneity depends on the observers. Recall that in Newtonian
mechanics, all observers would agree that either one event happened earlier
or the two events happened at the same time. (If the two events are light-like
related, then all observers will find that they are light-like, and we usually
do not discuss their chronological order.)
3.3 Addition of Velocities
If a bullet is moving with speed u relative to a train, and the train is moving
with speed v relative to the ground in the same direction, then in Newtonian
mechanics, the speed of the bullet relative to the ground is u + v. This is
not correct when u or v or both are near c. Let us consider the case that the
bullet is moving in general direction relative to the moving frame. Then, its
coordinates are
x ′ = u ′ x t′ (3.29)
y ′ = u ′ y t ′ (3.30)
z ′ = u ′ z t′ , (3.31)
where u ′ ≡ (u ′ x , u′ y , u′ z ) is the velocity of the bullet in the moving frame. By
Eq. (3.15) to Eq. (3.18), the coordinates of the bullet in the stationary frame
are
⎧
x = γ v (u ′ x ⎪⎨
t′ + vt ′ ) = γ v (u ′ x + v) t′
y = u ′ y t′
z = u
⎪⎩
′ z t′
t = γ v (t ′ + vu ′ x t ′ /c 2 ) = γ v (1 + vu ′ x/c 2 ) t ′
⎧
x =
u′ x + v
1 + vu
⎪⎨
′ x/c t 2
u ′ y
y =
γ v (1 + vu ′ x /c2 ) t
(3.32)
u ′ z
⎪⎩ z =
γ v (1 + vu ′ x/c 2 ) t
√
where γ v = 1/ 1 − v 2 /c 2 . Hence, the resulting velocity is
u ≡
( u
′
x + v u ′ y
1 + vu ′ x /c2, γ v (1 + vu ′ x /c2 ) , u ′ )
z
γ v (1 + vu ′ x /c2 )
. (3.33)
This is called the addition of velocities. One can easily check that if v = c
or the speed of the bullet √ u ′ x 2 + u ′ y 2 + u ′ z 2 = c, the resulting speed is still c.
This is consistent with the fact that speed of light is constant in any inertial
frame.