In the low velocity limit, the proper time "tau" is just the coordinate time "t"
since v << c in the low velocity limit.
and hence the space component of the 4-velocity is just the 3-velocity
for i = x,y,z (space components)
Also the time component of the 4-momentum is relativistic energy. In the low velocity limit, the relativisitic energy is simply kinetic energy divided by speed of light which is constant.
Since
where E is the electric field.
this means
can be written as
which is equal to
Therefore the equation becomes
or
This means the rate of change of the total kinetic energy depends on the dot product of electric field and the particle's velocity. The kinetic energy remains constant if there is no electric field. And the kinetic energy does not change if only magnetic field is applied.
the same for the magnetic field vector B (in GR units, both vectors have the same...
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