Relativistic Energy

If the momentum needs a change of definition, then we can also expect the energy to be in need of revision.

In our non-relativistic considerations we had found that the total energy of a particle in the absence of an external potential was just its kinetic energy,

K_class. =

m v²

In the relativistic case we find that we have to consider the contribution of the mass to the energy of a particle. Einstein found that the energy of a particle with mass m at rest is:

E_o = m c²

(there it is, the most famous formula in all of science!)

If the particle is in motion, then the energy increases just like the time is dilated for a moving particle. The general case for the energy is then

E = $\gamma$ E_o = $\gamma$ m c²

The correct formula for the kinetic energy is then:

K = E - E_o = ($\gamma$-1) m c²

Classical approximation:

How can we recover the classical approximation above from this result?
Easy: for small speeds we had already said that

Insert this into the kinetic energy formula and get: