For instance, MOMENTUM CONSERVATION must still hold, or else we would be able to tell one reference frame from another (in an absolute sense) by seeing which one got less than its share of momentum in a collision. To pursue this example, we invoke MOMENTUM CONSERVATION in a glancing collision between two identical billiard balls, as pictured in Fig. 24.1:
[Get ready to keep track of a lot
of subscripts and primes! If you want to avoid the tedium of
paying close attention to which quantity is measured in whose
rest frame, skip to the formal derivation in terms of
LORENTZ INVARIANTS and the 4-MOMENTUM . . . . ]
Now, each of A and B is at rest in its own reference frame
before the collision (A sees B approaching from the right
at -u whereas B sees A approaching from the left at +u);
after the collision, each measures24.1
its own final velocity transverse
(perpendicular) to the initial direction of motion of the other.
Out of courtesy and in the spirit of scientific cooperation,
each sends a message to the other reporting this measurement.
By symmetry, these messages must be identical:
Meanwhile, MOMENTUM CONSERVATION must still hold
for the transverse components in each frame:
We may now apply the LORENTZ VELOCITY TRANSFORMATION to the
transverse velocity component of A:
Combining Eq. (1) with Eq. (3) gives which, combined with Eq. (5), gives or . Similarly, combining Eq. (2) with Eq. (4) gives or .
We can express both results in a general form without any subscripts: