Notes on Relativity and Cosmology - Physics Department, UCSB

6.4. MORE ON MASS, ENERGY, AND MOMENTUM 153
6.4 More on Mass, Energy, and Momentum
In the last section we saw that what we used to call mass and energy can be
converted into each other – and in fact are converted into each other all of the
time. Does this mean that mass and energy really are the same thing? Well,
that depends on exactly how one defines mass and energy.... the point is that,
as with most things in physics, the old (Newtonian) notions of mass and energy
will no longer be appropriate. So, we must extend both the old concept of
mass and the old concept of energy before we can even start talking. There are
various ways to extend these concepts. I’m going to use a more modern choice
which is standard in the technical literature. Unfortunately, this modern choice
seems to be less common in the popular literature and may therefore seem to
contradict things you have read elsewhere.
6.4.1 Energy and Rest Mass
My notion of mass will be independent of reference frame. This is not the case
for an older convention which has a closer tie to the old F = ma. This older
convention then defines a mass that changes with velocity. However, for the
moment, let me skirt around this issue by talking about the “rest mass” (m 0 ,
by definition an invariant) of an object, which is just the mass (inertia) it has
when it is at rest 8 . In particular, an object at rest has inertia m 0 c 2 .
Recall that, in Newtonian physics, an object also has an energy 1 2 m 0v 2 due to its
motion. Almost certainly, this expression will need to be modified in relativity,
but it should be approximately correct for velocities small compared with the
speed of light. Thus, for a slowly moving object we have
E = m 0 c 2 + 1 2 m 0v 2 + small corrections. (6.5)
Note that we can factor out an m 0 c 2 to write this as:
E = m 0 c 2 (1 + 1 2
v 2
+ small corrections). (6.6)
c2 In section 6.6 we will derive the precise form of these small corrections. However,
this derivation is somewhat technical and relies on a more in-depth knowledge of
energy and momentum in Newtonian physics than some of you will have. Since
I do not want it to obscure the main points of our discussion, I have relegated
the derivation to a separate section (6.6) at the end of the chapter. For the
moment, we will content ourselves with a well-motivated guess.
8 I suspect that the linguistic evolution is as follows: People first used the old convention
where mass meant inertia. They then introduced this separate notion of ‘rest mass.’ As time
passed, physicists found that the only time they ever used the word mass was in the phrase
‘rest mass,’ as the other concept was in fact better served by the term energy (which, as we
have seen, carried inertia). After awhile, they got tired of inserting the word ‘rest’ and writing
the subscript 0.