Fundamentals of Time and Relativity

Let us assume that in the inertial frame, characterized by the future-pointing timelike unit vector ${\text{e}}_0$, a charge $q$ is moving with velocity $v$. Then the four-velocity of the particle has the components $u^{\mu }=(\gamma ,\gamma v)$. In this frame of reference, we consider the invariant expression ${\text{e}}_{\mu }{F}^{\mu \nu }{u}_{\nu }$, with $F^{\mu \nu }$ given by (9.3). Writing out in terms of the 3-vector $E$ and $B$ fields we obtain

At the right-hand side of (9.5), one recognizes the well-known Lorentz force depending on the velocity of the particle in this inertial frame. Apart from the Lorentz factor, the temporal component (9.6) equals the work $qE\cdot v$ done by the electric field on the particle per unit time; see Relativistic Force. The magnetic field does not contribute because the magnetic force at the right-hand side of (9.5) is always perpendicular to the velocity.