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.

This is the correct non-relativistic limit. The

Jim Branson 2012-10-21