Relativistic Form of The Lorentz Force
Because the electric and magnetic fields are dependent on the velocity of an observer, the relativistic form of the Lorentz force law can best be exhibited starting from a coordinate-independent expression for the electromagnetic and magnetic fields, and an arbitrary time-direction, where
and
is a space-time plane (bivector), which has six degrees of freedom corresponding to boosts (rotations in space-time planes) and rotations (rotations in space-space planes). The dot product with the vector pulls a vector from the translational part, while the wedge-product creates a space-time trivector, whose dot product with the volume element (the dual above) creates the magnetic field vector from the spatial rotation part. Only the parts of the above two formulas perpendicular to gamma are relevant. The relativistic velocity is given by the (time-like) changes in a time-position vector, where
(which shows our choice for the metric) and the velocity is
Then the Lorentz force law is simply (note that the order is important)
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