22
3 The Motion of Particles
Fig. 3.2 The interaction of particles illustrates conservation of 4-momentum
Let us see if this definition of energy and momentum makes physical sense. If
4-momentum is conserved in an interaction in one reference frame, as in Fig. 3.2,
we may express the fact as
p
μ
(total in) =
i
p
μ
i = k
μ
(total out) =
j
k
μ
j .
(3.11)
That is the total energy and momentum in are equal to the total energy and momentum
out. Since both sides are 4-vectors they transform the same way in going to another
reference frame, and the same equation holds; that is energy and momentum are
conserved in the other frame. It is moreover nice that both energy and momentum
conservation are contained in a single 4-vector equation. This is what we mean when
we say that an equation or a law is covariant or form invariant: it has the same form
in any reference frame. Clearly it is reasonable to expect that the fundamental laws
of nature should be covariant, and in relativity this is indeed a basic postulate.
Example 3.1 Let us evaluate the invariant p
σ p σ . This is most easily done in
the proper frame of the particle, where we know from (3.6) that the 4-velocity
is u
μ
= (c, 0, 0, 0); hence
p
σ p σ = m
2 u
σ u σ = m
2 c
2
.
(3.12)
This is also obvious since the square of the 4-velocity is c
2 as we noted after
(3.6).
Example 3.2 There is always a reference frame in which a system of particles
has a net momentum of zero, naturally called the center of momentum frame.
To see that it exists first choose any inertial frame, and calculate the total
energy E and momentum
P of the particles. Then orient the x axis along the
momentum so the momentum 4-vector is P
μ
= (E/c, P, 0, 0). In a prime
frame moving at v along that x axis the energy and momentum are, from the
Lorentz transformation (1.18),
3 The Motion of Particles
Fig. 3.2 The interaction of particles illustrates conservation of 4-momentum
Let us see if this definition of energy and momentum makes physical sense. If
4-momentum is conserved in an interaction in one reference frame, as in Fig. 3.2,
we may express the fact as
p
μ
(total in) =
i
p
μ
i = k
μ
(total out) =
j
k
μ
j .
(3.11)
That is the total energy and momentum in are equal to the total energy and momentum
out. Since both sides are 4-vectors they transform the same way in going to another
reference frame, and the same equation holds; that is energy and momentum are
conserved in the other frame. It is moreover nice that both energy and momentum
conservation are contained in a single 4-vector equation. This is what we mean when
we say that an equation or a law is covariant or form invariant: it has the same form
in any reference frame. Clearly it is reasonable to expect that the fundamental laws
of nature should be covariant, and in relativity this is indeed a basic postulate.
Example 3.1 Let us evaluate the invariant p
σ p σ . This is most easily done in
the proper frame of the particle, where we know from (3.6) that the 4-velocity
is u
μ
= (c, 0, 0, 0); hence
p
σ p σ = m
2 u
σ u σ = m
2 c
2
.
(3.12)
This is also obvious since the square of the 4-velocity is c
2 as we noted after
(3.6).
Example 3.2 There is always a reference frame in which a system of particles
has a net momentum of zero, naturally called the center of momentum frame.
To see that it exists first choose any inertial frame, and calculate the total
energy E and momentum
P of the particles. Then orient the x axis along the
momentum so the momentum 4-vector is P
μ
= (E/c, P, 0, 0). In a prime
frame moving at v along that x axis the energy and momentum are, from the
Lorentz transformation (1.18),