here - Department of Physics, HKU

CHAPTER 3. LORENTZ TRANSFORMATIONS 24
conservation of momentum does not hold in the laboratory frame.
2mv
1 + v 2 /c 2 ≠ 2mv fx = 2mv . (3.37)
To retain the conservation of momentum (and experiments verify that this
is the correct way), we define the momentum as
p ≡
mv
√1 − v 2 /c 2 . (3.38)
Note that in this definition, v is always the velocity of the particle relative
to the frame, and v is its speed. Then, momentum is conserved in our case
because the x-component of the initial total momentum is
=
=
=
=
mv i
√
1 − vi 2 /c 2
2mv 1
1 + v 2 /c 2 √
1 −
4v 2
(1+v 2 /c 2 ) 2 c 2
2mv
√
(1 + v 2 /c 2 ) 2 − 4v 2 /c 2
2mv
√
(1 − v 2 /c 2 ) 2
2mv
1 − v 2 /c 2 , (3.39)
and the x-component of the final total momentum is
2
mv fx
√
1 − (v
2
fx + v 2 fy )/c2
=
=
=
2mv
√
1 − (v 2 + v 2 (1 − v 2 /c 2 ))/c 2
2mv
√
1 − 2v 2 /c 2 + v 4 /c 4
2mv
1 − v 2 /c . (3.40)
2
(The factor of 2 is due to the fact that there are two particles.)
Let us assume that the correct generalization of the Newton’s second law
is
F = dp
dt = d mv
dt
√1 − v 2 /c . (3.41)
2