Gauge boundary term and kinetic/canonical momentum

Let $M$ be a smooth Hausdorff second-countable four-manifold without boundary and $O\subseteq M$ open. Adopt ${\mathrm{A}\mathrm{C}}_{\omega }$ of The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) for the tangent-bundle structure used in Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model. On $O$, let $g$ be a $C^2$ symmetric nondegenerate covariant two-tensor of signature $(-,+,+,+)$ and $N$ a $C^1$ tangent vector field with $g(N,N)<0$. Here $C^k$ means that all chart components have continuous derivatives through order $k$. A tangent vector $v$ is future timelike when $g(v,v)<0$ and $g(v,N)<0$, as in that supplier; these definitions apply at the stated finite regularity. Let $A$ be a $C^2$ one-form on $O$ and $F=dA$, where $(dA{)}_{ab}={\mathrm{\partial }}_a{A}_b-{\mathrm{\partial }}_b{A}_a$. Let $\chi \in C^3(O,\mathbb{R})$ and set $A^{\chi }:=A+d\chi$. For $a<b$, any $C^2$ future-timelike curve segment $z:[a,b]\to O$ parameterized by proper time $\tau$ so $g(z^{\mathrm{\prime }},z^{\mathrm{\prime }})=-c^2$ for $c>0$, define its four-velocity by $U:=dz\mathrm{/}d\tau =z^{\mathrm{\prime }}$, and $U^{\mathrm{♭}}:=g(U,\cdot )$. All constants in this mathematical lemma are real scaling parameters, without an asserted physical unit. For constants $m>0$ and $q\in \mathbb{R}$ and the functional

$S_A[z]=-mc{\int }_a^b\sqrt{-g(z^{\mathrm{\prime }},z^{\mathrm{\prime }})}d\tau +q{\int }_a^{b}A(z^{\mathrm{\prime }})d\tau ,$

the field strength is unchanged, $d{A}^{\chi }=dA$, and

$S_{A^{\chi }}[z]-S_A[z]=q(\chi (z(b))-\chi (z(a)))\mathrm{.}$

The same action formula and boundary identity apply to any $C^2$ future-timelike curve on $[a,b]$ with an arbitrary parameter: replace $\tau$ by that parameter and use its velocity in the integrands. For ${ϵ}_0>0$, a fixed-endpoint variation is a jointly $C^2$ map $Z:(-{ϵ}_0,{ϵ}_0)\times [a,b]\to O$, extending to a neighbourhood of the closed interval, with $Z(0,\cdot )=z$, fixed endpoint values, and future-timelike parameter velocities; its members need not retain proper-time normalization. Stationarity means vanishing of the action derivative at $ϵ=0$ for every such variation. Thus the difference is constant over every fixed-endpoint variation, so stationarity and the kinetic covector $p:=m{U}^{\mathrm{♭}}$ are gauge invariant. If the local covector ${\pi }_A:=p+qA$ is defined, then ${\pi }_{A^{\chi }}={\pi }_A+qd\chi$. These are local statements on $O$; they assert no global exactness of $F$.