## The Momentum-Energy 4-Vector

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