Notes on Relativity and Cosmology - Physics Department, UCSB

6.6. DERIVING THE EXPRESSIONS 159
6.6 Deriving the Relativistic expressions for Energy
and Momentum
Due to its more technical nature and the fact that this discussion requires a
more solid understanding of energy and momentum in Newtonian physics, I
have saved this section for last. It is very unlikely that I will go over this in
class and you may consider this section as optional reading. Still, if you’re
inclined to see just how far logical reasoning can take you in this subject, you’re
going to really enjoy this section.
It turns out that the easiest way to do the derivation is by focusing on momentum
11 . The energy part will then emerge as a pleasant surprise. The argument
has four basic inputs:
1. We know that Newtonian physics is not exactly right, but it is a good
approximation at small velocities. So, for an object that moves slowly, it’s
momentum is well approximated by p = mv.
2. We will assume that, whatever the formula for momentum is, momentum
in relativity is still conserved. That is, the total momentum does not
change with time.
3. We will use the principle of relativity; i.e., the idea that the laws of physics
are the same in any inertial frame of reference.
4. We choose a clever special case to study. We will look at a collision of two
objects and we will assume that this collision is ‘reversible.’ That is, we
will assume that it is possible for two objects to collide in such a way that,
if we filmed the collision and played the resulting movie backwards, what
we see on the screen could also be a real collision. In Newtonian physics,
such collisions are called elastic because energy is conserved.
Let us begin with the observation that momentum is a vector. In Newtonian
relativity, the momentum points in the same direction as the velocity vector.
This follows just from symmetry considerations (in what other direction could
it point?). As a result, it must also be true in relativistic physics. The only
special direction is the one along the velocity vector.
It turns out that to make our argument we will have to work with at least two
dimensions of space. This is sort of like how we needed to think about sticks
held perpendicular to the direction of motion when we worked out the time
dilation effect. There is just not enough information if we stay with only one
dimension of space.
So, let us suppose that we are in a long, rectangular room. The north and south
walls are fairly close together, while the east and west walls are far apart:
11 I first learned this argument by reading Spacetime Physics by Taylor and Wheeler.