Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 37A (Electrodynamics)

For a timelike worldline, $-{\eta }_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }>0$. Its square root is a Lorentz scalar, andThus the first term is the Lorentz-invariant free-particle action $-m{c}^2\int d\tau$. Written as a line element, it is also invariant under a change of the worldline parameter $\lambda$.

The interaction is the scalar line integralUnder an electromagnetic gauge transformationit changes only byThis endpoint term does not affect a fixed-endpoint variational problem. Hence both Lorentz invariance and gauge invariance select the displayed relativistic charged-particle action, with the expected free-particle and minimal-coupling terms.

WriteSinceits derivatives areThe Euler-Lagrange equation therefore givesUsing the electromagnetic field tensorand dividing by $d\tau /d\lambda$ givesRaising the first index yields the equivalent requested formThis is the Lorentz-force equation from the worldline action.

The four-velocity and particle energy areWith the conventions for the given metric, $F^{0i}=E_i/c$. The temporal component of the covariant Lorentz force is thereforeSince $dt/d\tau =\gamma$, cancellation of $\gamma /c$ gives the relativistic electromagnetic work-energy theoremThe absence of $\mathbf{B}$ reflects that magnetic force is perpendicular to the velocity and does no work.

The spatial part of the Lorentz force isPart (c), together with $\mathcal{E}=\gamma m{c}^2$, givesWriting $\mathbf{a}=d\mathbf{v}/dt$ and expanding the momentum derivative,Substitution of $d\gamma /dt$ gives the coordinate acceleration of a relativistic charged particle:

In the nonrelativistic limit $|\mathbf{v}|/c\to 0$, one has $\gamma \to 1$ and the final term is smaller by order $v^2/c^2$. Hencewhich is precisely Newton's second law with the ordinary Lorentz force.