It is obviously important it determine how Energy and Momentum transform in Special Relativity.
A reasonable guess is that momentum is a 3-vector conjugate to position, so we need to find what
the fourth component is to make a 4-vector.
We again have the problem of the speed of light not being equal to one in our units.
The answer, which we will derive below, is that
**the Momentum-Energy 4-vector is**

where the choice of where to put the
could be made by dimensional analysis.
The
**dot product with itself** is

This quantity should be a Lorentz scalar, which we will call
, and we get the equation.

Multiplying by
and rearranging.

Again the problem of
is vexing but we get the
**basic Energy equation of Special relativity**.
We understand this as the
**rest energy
** added in quadrature with
**
**.
For a particle at rest we get the
**rest energy equation**.

Of course any 4-vector
**transforms like a 4-vector** so we have the transformation equations for momentum

Lets start in the rest frame and do a transformation.

If we
**consider a boost in the minus
direction** to have the particle moving in the plus
direction afterward,
then the boost transformation gives.
**These are very useful relations for many kinematic calculations**.

**Subsections**
Jim Branson
2012-10-21