Environmental Engineering Reference
In-Depth Information
12.3 THE ENERGY-MOMENTUM FOUR-VECTOR
Now we turn our attention to energy and momentum. In Part II we had to
work rather hard in order to motivate the relativistic equations for momentum and
energy. Recall that we considered a particular scattering process and viewed it in
two different inertial frames with the goal of finding a definition of momentum
which was consistent with a universal law for the conservation of momentum.
Energy conservation then arose almost as if by accident. Finally we can present a
much more transparent account of the underlying physics.
Given the velocity four-vector presented in Eq. (12.7) we can define what we
shall call the momentum four-vector:
P
=
m
U
=
γ(u)m(c,
u
).
(12.23)
This is evidently a four-vector since
U
is a four-vector and
m
is a four-scalar. If
we should seek to generalise the law of conservation of momentum so that it holds
in all inertial frames then we need look no further than Eq. (12.23) and invoke the
law of conservation of four-momentum. The new conservation law states that for
an isolated system of particles the quantity
i
P
i
is fixed where the summation is
over all particles in the system. Given Eq. (12.23) it follows that
γ(u
i
)m
i
c
(12.24)
i
and
γ(u
i
)m
i
u
(12.25)
i
are seperately conserved. These are none other than the new laws for energy and
momentum conservation which we worked so hard to determine in Part II (see
Eq. (7.15) and Eq. (7.25)). Defining the energy and momentum of a particle of
mass
m
moving with velocity
u
to be
γ(u)mc
2
E
=
and
(12.26)
p
=
γ(u)m
u
(12.27)
it follows that the momentum four-vector can be written
P
=
(E/c,
p
).
(12.28)
Now since
P
is a four-vector the quantity
P
P
is a four-scalar. As such it can be
evaluated in any inertial frame. If we evaluate it in the inertial frame in which the
particle is a rest we find
·
m
2
c
2
P
·
P
=
(12.29)
