Notes on Relativity and Cosmology - Physics Department, UCSB

156 CHAPTER 6. DYNAMICS: ENERGY AND ...
Furthermore, we see that this ‘size’ does not depend on the frame of reference
and so does not depend on how fast the object is moving. However, for a
rapidly moving object, both the time part (the energy) and the space part (the
momentum) are large – it’s just that the Minkowskian notion of the size of a
vector involves a minus sign, and these two parts largely cancel against each
other.
6.4.3 How about an example?
As with many topics, a concrete example is useful to understand certain details
of what is going on. In this case, I would like to illustrate the point that while
energy and momentum are both conserved, mass is not conserved.
Let’s suppose we take two electrons and places them in a box. Suppose that
both electrons are moving at 4/5c, but in opposite directions. If m e is the rest
mass of an electron, each particle
|p| =
m e v
√
1 − v2 /c 2 = 4 3 m ec, (6.12)
and an energy
E =
m e c 2
√
1 − v2 /c 2 = 5 3 m ec 2 . (6.13)
We also need to consider the box. For simplicity, let us suppose that the box also
has mass m e . But the box is not moving, so it has p Box = 0 and E Box = m e c 2 .
Now, what is the energy and momentum of the system as a whole? Well, the
two electron momenta are of the same size, but they are in opposite directions.
So, they cancel out. Since p Box = 0, the total momentum is also zero. However,
the energies are all positive (energy doesn’t care about the direction of motion),
so they add together. We find:
So, what is the rest mass of the systemas a whole?
p system = 0,
E system = 13
3 m ec 2 . (6.14)
E 2 − p 2 c 2 = 169
9 m2 e c4 . (6.15)
So, the rest mass of the positronium system is given by dividing the right hand
side by c 4 . The result is 13
3 m e, which is significantly greater than the rest mass
of the Box plus twice the rest mass of the electron!
Similarly, two massless particles can in fact combine to make an object with a
finite non-zero mass. For example, placing photons in a box adds to the mass
of the box. We’ll talk more about massless particles (and photons in particular)
below.