External-field charged-particle action

Adopt the following motion principle for the massive charged test body of Charged and spinning test-body quantities. Let $a<b$ be finite real parameters and let $z:[a,b]\to O$ be a $C^2$ future-directed timelike segment in a local potential domain, with its endpoint events fixed in the variations. A change from this parameter interval to another compact interval $[\tilde{a},\tilde{b}]$ is admissible when it is a $C^2$ diffeomorphism $\varphi :[\tilde{a},\tilde{b}]\to [a,b]$ with ${\varphi }^{\mathrm{\prime }}(\lambda )>0$ throughout; the same oriented segment is then represented by $\tilde{z}=z\circ \varphi$. In any chosen representation write $\gamma :[\alpha ,\beta ]\to O$ and let $\lambda$ denote its parameter; the choice $\lambda =\tau$ is included when the curve is parameterized by proper time as in Charged and spinning test-body quantities. Use seconds for the parameter in SI evaluation.

For each such parameterized segment, the physical trajectories of this idealized model are the stationary points of

$S[\gamma ]=-mc{\int }_{\alpha }^{\beta }\sqrt{-g({\gamma }^{\mathrm{\prime }},{\gamma }^{\mathrm{\prime }})}d\lambda +q{\int }_{\alpha }^{\beta }A({\gamma }^{\mathrm{\prime }})d\lambda ,$

where $A$ is the prescribed $C^2$ external potential, $F=dA$, $m>0$ is constant rest mass, $q$ is constant charge, and $c>0$ is the SI conversion constant. A local fixed-endpoint variation is a $C^2$ family ${\gamma }_{ϵ}:[\alpha ,\beta ]\to O$, defined for $ϵ$ near $0$, with ${\gamma }_0=\gamma$, each ${\gamma }_{ϵ}$ future timelike, and ${\gamma }_{ϵ}(\alpha )=\gamma (\alpha )$ and ${\gamma }_{ϵ}(\beta )=\gamma (\beta )$; the local variation field is supported in a coordinate chart. The segment is stationary when ${\frac{d}{dϵ}S[{\gamma }_{ϵ}]\mid }_{ϵ=0}=0$ for every such variation. A reparameterization acts on the whole family by ${\gamma }_{ϵ}\mapsto {\gamma }_{ϵ}\circ \varphi$. With $\lambda =\tau$ in seconds, coordinates in metres and $A$ in tesla metres, $S$ has units joule seconds. The potential and metric are fixed inputs, independent of the particle path. This is a local action postulate: no global potential is assumed, and a global field equation or global trajectory is not part of the postulate. Reparameterization invariance and its implication for stationarity are proved separately in Stationary charged action and Lorentz force.

This model describes a test body: it does not evaluate the body's own field, include self-gravity, or adopt radiation reaction, variable mass or charge, electric dipole forces, or spin-dependent translational forces. The postulate is an adopted model assumption, not a deduction from the electromagnetic field equations or a claim that nature follows this action.

For dependent deductions, the adopted premise is that on a finite fixed-endpoint timelike segment in a local potential domain, the model trajectory is stationary for the displayed action. The potential and metric are fixed inputs, $m>0$ and $q$ are constant, and this premise asserts neither a global potential nor global trajectory existence. The admissible parameter changes are the $C^2$ orientation-preserving diffeomorphisms specified above.