Notes on Relativity and Cosmology - Physics Department, UCSB

162 CHAPTER 6. DYNAMICS: ENERGY AND ...
a vector. If this notation bothers you, just replace all my p A ’s with p Ay . Here,
|⃗p B | is the usual length of this vector, and similarly for x⃗
B .
Technically speaking, what we will do is to rearrange this formula as
⃗p B = (p A)(∆x⃗
B )
, (6.18)
L
where now we have put the direction information back in. We will then compute
⃗p B in the limit as the vertical velocity of particle A (and thus p A , in Alice’s
frame) goes to zero. In other words, we will use the idea that a slowly moving
particle (in Alice’s frame) could have collided with particle B to determine
particle B’s momentum.
Let us now take a moment to calculate p A . In the limit where the velocity of
particle A is small, we should be able to use p A = m 0 dy/dt after the collision.
Now, we can calculate dy/dt by using the time ∆t A that it takes particle A to
go from the collision site (in the center of the box) to the north wall. In this
time, it travels a distance L, so p A = m 0 L/∆t A . Again, the distance is L in
Alice’s frame, Bob’s frame, or the Box’s frame of reference since it refers to a
direction perpendicular to the relative motion of the frames.
Thus we have
⃗p B = lim
v A→0
m 0 (∆x⃗
B )
. (6.19)
∆t A
This is a somewhat funny formula as two bits (p B and ∆⃗x B ) are measured in
the lab frame while another bit t A is measured in Alice’s frame. Nevertheless,
the relation is true and we will rewrite it in a more convenient form below.
Now, what we want to do is in fact to derive a formula for the momentum
of particle B. This formula should be the same whether or not the collision
actually took place. Thus, we should be able to forget entirely about particle
A and rewrite the above expression purely in terms of things having to do with
particle B. We can do this by a clever observation.
Recall that we originally set things up in a way that was symmetric with respect
to particles A and B. Thus, if we watched the collision from particle A’s
perspective, it would look just the same as if we watched it from particle B’s
perspective. In particular, we can see that the proper time ∆τ between the
collision and the event where particle A hits the north wall must be exactly the
same as the proper time between the collision and the event where particle B
hits the south wall.
Further, recall that we are interested in the formula above only in the limit of
small v A . However, in this limit Alice’s reference frame coincides with that of
particle A. As a result, the proper time ∆τ is just the time ∆t A measured by
Alice. Thus, we may replace ∆t A above with ∆τ.
⃗p B = m 0(∆x⃗
B )
. (6.20)
∆τ
Now, this may seem like a trivial rewriting of (6.19), but this form is much
more powerful. The point is that ∆τ (proper time) is a concept we undersand