The problem we have is how to take a time derivative if the time is the component of a 4-vector. We need some kind of scalar time to make sense of the equations we know and love. A well defined time, that does not need to be transformed, is the time in the rest frame of the particle. We call this the proper time . We will make use of it here, but later just try to rewrite our equations so that they are covariant in 4 dimensions.
The velocity 4-vector can be defined as.
We can dot the velocity 4-vector into itself.
To be consistent with non-relativistic equations we will define the momentum.
If we identify the time component as above, , we have the relation
A crucial test of this ``derived'' 4-vector is whether it gives the right physics in the non-relativistic limit. We did have some choice to make when inserting the energy into the momentum 4-vector. Start with the energy equation from above.
Jim Branson 2012-10-21