Environmental Engineering Reference
In-Depth Information
This is a very interesting result indeed. The first term in parentheses vanishes
automatically since momentum is conserved in
S
. However the second term, in
square brackets is not
a priori
zero. Therefore, if we are to have any chance of
salvaging momentum conservation in Special Relativity we need also to insist that
the term in square brackets also vanishes. This is equivalent to saying that the
quantity
γ(v)m
should also be conserved, i.e. we require
γ(v
A
)m
A
+
γ(v
B
)m
B
=
γ(v
C
)m
C
+
γ(v
D
)m
D
.
(7.24)
Remarkably, this apparently new conservation law is nothing other than the
relativistic manifestation of the law of conservation of energy. To make this more
explicit, for a particle of mass
m
and speed
u
let us define the quantity
γ(u)mc
2
.
E
=
(7.25)
Eq. (7.24) then takes the form
E
A
+
E
B
=
E
C
+
E
D
.
This quantity has the units of energy, and it is conserved, but apart from that it
is far from clear at this stage that this has anything at all to do with the kinetic
energy of a classical non-relativistic particle.
7.1.1 The equivalence of mass and energy
In order the gain more insight, it makes sense for us to explore Eq. (7.25) in
the limit that
u
c
. In this limit, we can express
γ(u)
as a Taylor series about
u
=
0, i.e.
u
2
2
c
2
.
γ(u)
u
c
1
+
(7.26)
Substituting this into Eq. (7.25) yields the much more revealing
1
2
mu
2
.
u
c
mc
2
E
+
(7.27)
This is simply the sum of the non-relativistic kinetic energy and a static term, i.e.
mc
2
is the energy associated with a particle at rest. If we were to make the addi-
tional assumption that mass is conserved then the conservation of
E
is equivalent
to the conservation of kinetic energy. This is what we usually do in non-relativistic
mechanics, although the conservation of mass is often assumed to be self evident
and is rarely explicitly stated. For example, if we go back to our process AB
→
CD then the conservation of
E
implies, in the non-relativistic limit, that
1
2
m
A
v
A
+
1
2
m
B
v
B
=
1
2
m
C
v
C
+
1
2
m
D
v
D
,
(7.28)
m
A
c
2
m
B
c
2
m
C
c
2
m
D
c
2
+
+
+
+
