Environmental Engineering Reference
In-Depth Information
7.2.2 Two useful relations and massless particles
γ mc
2
A pair of particularly useful relations can be derived using
E
=
and
p
=
γ mv
. The first of them is obtained simply by taking the ratio:
cp
E
v
c
.
=
(7.35)
The usefulness of this equation lies in the fact that if we are given the energy and
momentum then it is possible to compute the speed without first computing
γ
.The
second equation takes a little more effort to derive but will turn out to be very
useful indeed. Let us consider the combination
E
2
c
2
p
2
:
−
E
2
c
2
p
2
(γ mc
2
)
2
(γ mv)
2
c
2
−
=
−
m
2
c
4
m
2
v
2
c
2
−
=
1
−
v
2
/c
2
E
2
c
2
p
2
m
2
c
4
.
−
=
(7.36)
Why is this so interesting? Well the main value arises because the right-hand-side
is the same in all inertial frames. We say that the combination
E
2
c
2
p
2
is Lorentz
invariant. Quantities such as this, whose values all inertial observers agree upon,
arise most naturally in the Part IV of this topic where we explore more fully the
symmetry alluded to at the end of the previous section. For now we note the result,
its utility in helping us solve problems will be apparent when we come to tackle
some of the later examples.
The formula for the total energy of a particle
E
−
γ mc
2
tells us that a massive
particle has an energy which approaches infinity as the particle's speed approaches
c
. Practically, this means that it is impossible to accelerate a massive particle in
such a way that its speed exceeds
c
, for to do so would require an infinite amount of
work. This is a much celebrated prediction of Einstein's theory and it is certainly in
accord with experiments. For example, to accelerate the protons which will circulate
at CERN's LHC to within a few metres per second of light speed requires a power
input comparable to that of the whole of the city of Geneva. The power costs
are in fact so prohibitive that CERN has to shut down over the winter months.
Interestingly, Einstein's theory does not however exclude the possible existence of
particles which travel at exactly the speed of light. According to Eq. (7.35) such
particles would have an energy and momentum related by
=
E
=
cp.
(7.37)
You may be worrying that for these particles
γ
and therefore they have
infinite energy and momentum. This problem can be avoided but only if the particles
carry zero mass. In which case
E
→∞
γ mc
2
γ mv
are simply no longer
well defined equations. The very existence of massless particles may sound like
a contradiction but in Special Relativity the counter intuitive equivalence of mass
and energy provides the loophole which allows for their being, providing that
they travel at light speed. In fact, we now know that the wave-particle duality of
=
and
p
=
