is the only 4-vector which can be formed from the velocity components of the material particle which are defined in three dimensions by
We therefore see that
must be that 4-vector which is to be equated to the 4-vector of momentum and energy whose existence we have proved above. By equating the components, we obtain, in three-dimensional notation,
We recognize, in fact, that these components of momentum agree with those of classical mechanics for velocities which are small compared to that of light. For large velocities the momentum increases more rapidly than linearly with the velocity, so as to become infinite on approaching the velocity of light.
If we apply the last of equations (43) to a material particle at rest (
