here - Department of Physics, HKU

CHAPTER 4. RELATIVISTIC KINEMATICS 29
Their relative speed in their frame is given by
Hence, the observed frequency is
v 1 + v 2
1 + v 1 v 2 /c 2 . (4.8)
⎛
f 1
⎝ 1 + (v ⎞
1+v 2 )/c 1/2
1+v 1 v 2 /c 2
⎠
1 − (v 1+v 2 )/c
1+v 1 v 2 /c 2
= f 1
( 1 + v1 /c + v 2 /c + v 1 v 2 /c 2
1 − v 1 /c − v 2 /c + v 1 v 2 /c 2 ) 1/2
= f 1
( (1 + v1 /c)(1 + v 2 /c)
(1 − v 1 /c)(1 − v 2 /c)
) 1/2
. (4.9)
Another way to explain this formula is that, in the frame of the other spaceship,
the observed frequency is just the Doppler shifted frequency of the one
in Eq. (4.7) of speed v 2 , and this is exactly given by the above formula.
There are many applications of the Doppler effect. For example, police
uses radar to monitor speed of vehicles by emitting radio wave of known
frequency and detect the frequency of the reflected wave. The two are related
by the Doppler effect. Also, astronomers detect the speeds of galaxies by
measuring the shift of the spectrum of light emitted.
4.2 Conservation of Energy and Momentum
We are going to prove that energy and momentum are conserved in any frame
if they are in one frame. Let the two frames, one moving one stationary, be
related as in Eq. (3.15) to Eq. (3.18), as usual, and let the velocity of a particle
be u and u ′ relative to the stationary and moving frames as in Eq. (3.29) to
Eq. (3.33). We first prove an useful relation
1 − u2
= 1 c 2 ( c2
=
=
c 2 − (u′ x + v)2
(1 + vu ′ x /c2 ) 2
u ′ 2
y
u ′ 2
z
−
(1 + vu ′ x /c2 ) (1 − 2 v2 /c 2 ) −
(1 + vu ′ x /c2 ) (1 − 2 v2 /c 2 ) ⎠
1 (
c 2 (1 + vu ′
c 2 (1 + vu ′ x/c 2 ) 2 x /c2 ) 2 − (u ′ x + v)2
−u ′ y2 (1 − v 2 /c 2 ) − u ′ 2 z (1 − v 2 /c 2 ) )
1 (
c 2 + 2vu ′
c 2 (1 + vu ′ x /c2 ) 2 x + v2 u ′ 2 x /c 2 − u ′ x2 − 2u ′ x v − v2
⎞