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2.1. Relativistic classical mechanics
The proof of (2.15) is left to the reader in the problems.
We define the three-vector kinetic momentum p as
p = γ m v.
(2.18)
Equation (2.15) is an expression of Newton’s law of motion for a
charged particle, where the left side is the time rate of change of
the kinetic momentum, and the right side is known as the Lorentz
force.
In principle it is possible to calculate all of particle optics by solving (2.15), for the position x and the velocity v as functions of
time t, but further considerations will lead to a more detailed understanding, and to greater computational efficiency.
Problems
1. An arbitrary four-vector A µ = (A 1 , A 2 , A 3 , A 4 ) is defined in
terms of its four components. For two reference frames in relative
uniform motion with velocity v along the z-direction, the components of A µ are related in the two frames by
A
�
1 = A 1
A
�
2 = A 2
A
�
3 = γ (A 3 + iβA 4 )
A
�
4 = γ (−iβA 3 + A 4 ),
where γ is given by (2.9) and β = v/c. This is known as a Lorentz
transformation. Show that the inner product of any two fourvectors A µ and B µ satisfies
4
4
4
4
A
� B
� =
A µ B µ .
µ µ
µ=1
µ=1
An inner product of two four-vectors is thus said to be invariant
under a Lorentz transformation.