Hi Pat,
Good question! This is an example of the kind of question that
arises when we use E=mc^2 without looking at how it was derived and
what restrictions there are.
E=mc^2 describes the energy of a massive particle when it is at
rest. Sometimes we call this E the rest energy, and m the rest mass,
sometimes writing it as m_0. If a massive particle is moving, its
energy increases with its speed. Some people like E=mc^2 so much that
they define "m" so that it, too, depends on the speed. I don't like
this because it removes one of two useful concepts -- energy and mass,
by equating them. It turns out that the rest mass m_0 is a useful
property of objects, even when they are moving at very high speeds, and
a slightly more complicated version of Einstein's famous equation is
much more useful in practice:
E^2 = (p*c)^2 + (m_0*c^2)^2
where p is the momentum of the object, in appropriate units I write
m_0 to make the old-style people happy, but people who use relativity
every day just call it "m", and use the name "invariant mass" because
it doesn't depend on the motion. This relation is true for a collection
of particles, too, not just single particles.
For photons, m_0 = 0. This means that E=p*c, and photons have both
energy and momentum, but not mass ("rest mass"). We have an answer
somewhere on this site which explains why massless objects must travel
at the speed of light.
One of Albert Einstein's great achievements was the reconciliation
of special relativity (which is required if the laws of electricity and
magnetism is to remain consistent in all frames of reference) and the
observation of things which obey Newton's law of gravitation. General
relativity is the outcome of that process, and it's pretty wild.
Perhaps the best way to talk about light being "bent" by a
gravitational field, is the attitude taken by Misner, Thorne and
Wheeler in their fine book "Gravitation". Light always travels in
straight lines. What gravity does is it bends space around so that
straight lines are the curved paths we observe light to follow, in a
set of coordinates we find convenient.
You can do a rough calculation of the bending of light in a
gravitational field by using E=mc^2 for light and treating everything
as if relativity otherwise didn't exist. But the answer you get is a
factor of two off.
But the Newtonian picture, which includes the idea of
gravitaitonal potential energy, must remain a good explanation of
phenomena, at least in its domain of applicability. If you shine lots
of light from the surface of the earth to an object way up high on top
of a tower, you increase the energy of the object on the tower, and
hence its mass (provided its momentum remains zero). Drop this object
down to the surface, and its change in potential energy, and hence its
kinetic energy, depends on its mass, and therefore on how hot it is,
and so somehow that light must have contributed to the gravitational
potential energy somehow. To make the relativitstic description work
out consistently with this Newtonian picture, light must lose some
energy as it goes up in a gravitational field. But its energy is
proportional to its frequency, so the frequency of a light ray must
decrease as it goes up. There's a real problem here, since an observer
on the ground watching a train of light waves go past counting each
crest in a unit of time may compare his answer with another similar
observer way up high, watching the same light beam go past (but with a
lower frequency). In the same amount of time, the higher observer must
count fewer crests of the light wave. Where do the extra crests go?
It turns out you cannot make wave crests go away. But you can
explain that the clocks used by the ground-based observer run at a
different speed from the observer up high. So the claim that a photon
gains potential energy as it loses kinetic energy as it travels up out
of a gravitational field is in a Newtonian language which isn't a full
explanation of what's going on. What's going on is that the light is
traveling from one place to another where the geometry of space and
time is different, and the clocks run at different speeds.
Tom
(republished on 07/23/06)
