Notes on Relativity and Cosmology - Physics Department, UCSB

6.5. ENERGY AND MOMENTUM FOR LIGHT 157
6.5 Energy and Momentum for Light
At this point we have developed a good understanding of energy and momentum
for objects. However, there has always been one other very important player
in our discussions, which is of course light itself. In this section, we’ll take a
moment to explore the energy and momentum of light waves and to see what it
has to teach us.
6.5.1 Light speed and massless objects
OK, let’s look one more time at the question: “What happens if we try to get an
object moving at a speed greater than c?” Let’s look at the formulas for both
energy and momentum. Notice that E = √ m0c2
becomes infinitely large as
1−v 2 /c2 v approaches the speed of light. Similarly, an object (with finite rest mass m 0 )
requires an infinite momentum to move at the speed of light. Again this tells
us is that, much as with our uniformly accelerating rocket from last week, no
finite effort will ever be able to make any object (with m 0 > 0) move at speed
c.
By the way, what happens if we try to talk about energy and momentum for
light itself? Of course, many of our formulas (such as the one above) fail to
make sense for v = c. However, some of them do. Consider, for example,
pc
E = v c . (6.16)
Since light moves at speed c through a vacuum, this would lead us to expect that
for light we have E = pc. In fact, one can compute the energy and momentum
of a light wave using Maxwell’s equations. One finds that both the energy and
the momentum of a light wave depend on several factors, like the wavelength
and the size of the wave. However, in all cases the energy and momentum
exactly satisfies the relation E = pc. As a result, we can consider a bit of light
(a.k.a., a photon) with any energy E so long as we also assign it a corresponding
momentum p = E/c. The energy and momentum of photons adds together in
just the way that we saw in section 6.4.3 for massive particles.
So, what is the rest mass of light? Well, if we compute m 2 0c 2 = E 2 − p 2 c 2 = 0,
we find m 0 = 0. Thus, light has no mass. This to some extent shows how light
can move at speed c and have finite energy. The zero rest mass ‘cancels’ against
the infinite factor coming from 1 − v 2 /c 2 in our formulas above.
By the way, note that this also goes the other way: if m 0 = 0 then E = ±pc
and so v c = pc
E
= ±1. Such an object has no choice but to always move at the
speed of light.
6.5.2 Another look at the Doppler effect
Recall that, for a massive particle (i.e., with m 0 > 0), if we are in a frame that is
moving rapidly toward the object, the object has a large energy and momentum