As you may know, like we can combine position and time in one four-vector , we can also combine energy and momentum in a single four-vector, . From the Lorentz transformation property of time and position, for a change of velocity along the -axis from a coordinate system at rest to one that is moving with velocity we have
(10.1) |
we can derive that energy and momentum behave in the same way,
To understand the context of these equations remember the definition of
(10.3) |
In Eq. ( 10.2 ) we have also re-expressed the momentum energy in terms of a velocity . This is measured relative to the rest system of a particle, the system where the three-momentum .
Now all these exercises would be interesting mathematics but rather futile if there was no further information. We know however that the full four-momentum is conserved, i.e., if we have two particles coming into a collision and two coming out, the sum of four-momenta before and after is equal,