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.
dot product with itself is
This quantity should be a Lorentz scalar, which we will call
, and we get the equation.
Again the problem of
is vexing but we get the
basic Energy equation of Special relativity.
We understand this as the
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.
consider a boost in the minus
direction to have the particle moving in the plus
then the boost transformation gives.
These are very useful relations for many kinematic calculations.