Relativistic Lorentz force

$md{u}^{\mu }/d\tau =q{F}^{\mu }{}_{\nu }{u}^{\nu }$ gives $d\mathbf{p}/dt=q(\mathbf{E}+\mathbf{v}\times \mathbf{B})$ and $d(\gamma m{c}^2)/dt=q\mathbf{E}\cdot \mathbf{v}$.

For a constant electromagnetic field tensor, the relativistic Lorentz force is a constant-coefficient linear system on four-velocity, with solutionThe electric-type blocks of $F$ generate hyperbolic rotations, while magnetic-type blocks generate ordinary rotations.

When uniform $\mathbf{E}$ and $\mathbf{B}$ are parallel, the relativistic Lorentz force splits into independent hyperbolic motion along the fields and circular motion in the perpendicular plane. The circle is periodic in proper time, while electric acceleration makes the helix's longitudinal pitch grow exponentially from one proper-time cyclotron period to the next.

A particle released from rest in a constant electric field parallel to its motion follows a spacetime hyperbola and has rapidity proportional to proper time.

Perpendicular fields with $\mathbf{E}\cdot \mathbf{B}=0$ and $E^2-c^2{B}^2=0$ form a null electromagnetic field. The constant-field Lorentz-force generator is nilpotent, so its exponential terminates as a polynomial.

ChooseA particle released from rest at the origin has the proper-time trajectoryEliminating proper time gives the semicubical parabolaThis signed coefficient assumes that the positive $x$ direction is chosen along $\mathbf{E}$; reversing that axis changes the signs of both $x^3$ and its coefficient.