Fixed-endpoint curved charge variation

Let $(M,g,N)$ be a smooth, time-oriented Lorentzian four-manifold of signature $(-,+,+,+)$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model). Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ for the canonical tangent-bundle smooth structure, as required for the metric-selected Levi–Civita connection $\mathrm{\nabla }$ of Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields. Let $O\subseteq M$ be open and let $A$ be a $C^2$ covector field on $O$; define the $C^1$ two-form $F=dA$ by $F_{ab}={\mathrm{\partial }}_a{A}_b-{\mathrm{\partial }}_b{A}_a$ in each chart. Fix constants $c>0$, $m>0$ and $q\in \mathbb{R}$. Let $z:[a,b]\to O$ be a $C^2$ future-directed timelike curve, where $a<b$ are proper-time values in seconds and $U=dz\mathrm{/}d\tau$ satisfies $g(U,U)=-c^2$. For every $C^2$ fixed-endpoint variation $z_{ϵ}$ through timelike curves in $O$, with variation field $J={\mathrm{\partial }}_{ϵ}{z}_{ϵ}{\mathrm{\mid }}_{ϵ=0}$ and $J(a)=J(b)=0$, define

$S[z_{ϵ}]=-mc{\int }_a^b\sqrt{-g({\mathrm{\partial }}_{\tau }{z}_{ϵ},{\mathrm{\partial }}_{\tau }{z}_{ϵ})}d\tau +q{\int }_a^{b}A({\mathrm{\partial }}_{\tau }{z}_{ϵ})d\tau \mathrm{.}$

Then the first variation is

$\delta S={\int }_a^b[-mg(J,{\mathrm{\nabla }}_{U}U)+qF(J,U)]d\tau \mathrm{.}$

Consequently, $z$ is stationary under all local chart-supported fixed-endpoint variations if and only if

$m({\mathrm{\nabla }}_{U}U{)}_a=q{F}_{ab}{U}^b,$

equivalently $m{\mathrm{\nabla }}_{U}U=q{F}^{\mathrm{♯}}(U)$, where $F^{\mathrm{♯}}(v)$ is defined by $g(F^{\mathrm{♯}}(v),w)=F(w,v)$. The statement is local to a potential domain; no global potential is assumed.

Let $(M,g,N)$ be a smooth, time-oriented Lorentzian four-manifold of signature $(-,+,+,+)$; assume ${\mathrm{A}\mathrm{C}}_{\omega }$ for its canonical tangent-bundle smooth structure, so the metric-selected Levi–Civita connection $\mathrm{\nabla }$ exists by Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields. Let $O\subseteq M$ be open and let $F$ be a $C^1$ alternating covariant two-tensor field on $O$ (a two-form); define $F^{\mathrm{♯}}$ by $g(F^{\mathrm{♯}}(v),w)=F(w,v)$. Fix $m,c>0$, $q\in \mathbb{R}$, an event $x_0\in O$, and $U_0\in T_{x_0}M$ with $g(U_0,U_0)=-c^2$ future directed. Then the initial-value problem

$\dot{x}=U,{\mathrm{\nabla }}_{U}U=(q\mathrm{/}m)F^{\mathrm{♯}}(U),(x({\tau }_0),U({\tau }_0))=(x_0,U_0)$

has a unique local $C^2$ solution on an open proper-time interval containing ${\tau }_0$. Any two solutions with these initial data agree on every connected part of their common domain containing ${\tau }_0$. Along the solution, $g(U,U)=-c^2$ and $U$ remains future directed. The result is local; it asserts no completeness or global existence for arbitrary external fields.

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 $F$.

Use the smooth, time-oriented Lorentzian background $(M,g,N)$, signature $(-,+,+,+)$, and the distinct chart, observer, congruence, tetrad and worldline meanings of Particle spacetime, observers and worldlines. The metric and time orientation are prescribed data, not dynamical variables. A charged spinning test body in this definition consists of the following model quantities along a massive worldline $z:I\to M$:

• $I\subseteq \mathbb{R}$ is an open interval whose parameter is proper time $\tau$ in seconds; $z$ is $C^2$, future-directed and timelike, and $U:=dz\mathrm{/}d\tau$ obeys $g(U,U)=-c^2$. The SI conversion constant is $c>0$ metres per second; in a flat specialization use the affine Minkowski model and the proper-time decomposition of Oriented time-oriented affine Minkowski space (the spacetime primitives) and Norms of four-velocity and four-acceleration, and the observer rest-space decomposition.
• The constant rest-mass parameter $m>0$ has unit kilogram, and the constant charge parameter $q\in \mathbb{R}$ has unit coulomb. They are primitive model parameters; this definition does not derive them from the background field or from another framework.
• The kinetic four-momentum is the derived vector $P:=mU$ along $z$, with unit kilogram metre per second. It is distinct from the potential-dependent covector defined in Gauge boundary term and kinetic/canonical momentum.
• The spin or polarization vector is a $C^1$ tangent vector field $s(\tau )\in T_{z(\tau )}M$ constrained by $g(s,U)=0$, so it lies in the positive-definite instantaneous rest space $U^{\perp }$. Its unit is angular momentum, kilogram metre squared per second. This is a classical model vector; no quantization or quantum spin state is defined here.
• The gyromagnetic ratio $g_s\in \mathbb{R}$ is a dimensionless primitive species parameter. In the instantaneous rest space the model defines the magnetic-moment vector by $\mu =(g_{s}q\mathrm{/}2m)s$, with unit ampere metre squared. No value $g_s=2$ is assumed, and this constitutive relation is not deduced from Lorentz covariance or classical electromagnetism.
• The prescribed external electromagnetic field is a $C^1$ alternating covariant two-tensor $F$ on an open region of $M$, independent of the unknown curve. Define $F^{\mathrm{♯}}:TM\to TM$ by $g(F^{\mathrm{♯}}(v),w)=F(w,v)$. In the flat specialization, a named inertial chart with $x^0=ct$ and $x^i$ in metres uses $F_{0i}=-E_i\mathrm{/}c$ and $F_{ij}={ϵ}_{ijk}{B}_k$, matching The electromagnetic two-form, its observer decomposition and the four-current. On the curved background, $F$ is only a two-form; at an event its components in a chosen orthonormal tetrad use the same algebraic convention. Such a tetrad is a basis field, not a coordinate chart or an observer congruence, as distinguished in Particle spacetime, observers and worldlines. The components in an orthonormal tetrad have tesla units. This is an explicit curved-background model input. The cited EM supplier is a flat Minkowski definition and does not derive a curved field or curved Maxwell system.
• On any open potential domain where a $C^2$ covector field $A$ exists, $F=dA$ with components $F_{ab}={\mathrm{\partial }}_a{A}_b-{\mathrm{\partial }}_b{A}_a$ and $A$ has tesla metre components in the stated coordinates. No global potential is assumed, and no Maxwell equation or constitutive law is imposed here.

This is a test-body model on a fixed background: self-gravity, gravitational backreaction, self-field evaluation, radiation reaction, collisions, electric dipole moments, field-gradient translational forces and higher multipoles are outside this definition. Units follow the framework-local SI conventions of Framework-local SI units, the vacuum constants and the exact/measured status of their values and do not identify quantities across frameworks.

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.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ for the canonical tangent-bundle smooth structure, and let $\mathrm{\nabla }$ be the metric-selected Levi–Civita connection of Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields. Let $U$ and the prescribed external two-form $F$ be as in Charged and spinning test-body quantities, and let the translational motion obey the adopted action law External-field charged-particle action. For a $C^1$ spin or polarization vector $s$ along the worldline with $g(s,U)=0$, define $G:=g_s\mathrm{/}2$ and $K:=F(s,U)=g(s,F^{\mathrm{♯}}(U))$. Adopt the transport law

${\mathrm{\nabla }}_{U}s=\frac{q}{m}[G{F}^{\mathrm{♯}}(s)-(G-1)\frac{U}{c^2}K]\mathrm{.}$

Here ${\mathrm{\nabla }}_{U}s$ denotes the covariant derivative of the section $s$ along the specified curve, rather than differentiation of an unspecified ambient extension. More generally, for any $C^1$ vector field $V$ along this $C^2$ worldline $z$, define

$(D_{\tau }V{)}^a:=\frac{d{V}^a}{d\tau }+{\mathrm{\Gamma }}^a{}_{bc}(z(\tau ))U^b{V}^c,{\mathrm{\nabla }}_{U}s:=D_{\tau }s\mathrm{.}$

The components of $V$ are $C^1$ functions of $\tau$ in each smooth chart, and the displayed derivative is continuous. This is the pullback-connection derivative on $z^{\ast }TM$. To verify that the component definition is independent of the chart, let $y=y(x)$ and $J^{\alpha }{}_a:=\mathrm{\partial }{y}^{\alpha }\mathrm{/}\mathrm{\partial }{x}^a$. The connection transformation law gives

${\mathrm{\Gamma }}^{\mathrm{\prime }\alpha }{}_{\beta \gamma }{J}^{\beta }{}_b{J}^{\gamma }{}_c=J^{\alpha }{}_a{\mathrm{\Gamma }}^a{}_{bc}-{\mathrm{\partial }}_b{J}^{\alpha }{}_c\mathrm{.}$

Since $V^{\mathrm{\prime }\alpha }=J^{\alpha }{}_a{V}^a$, the ordinary chain and product rules give ${\dot{V}}^{\mathrm{\prime }\alpha }=J^{\alpha }{}_a{\dot{V}}^a+({\mathrm{\partial }}_b{J}^{\alpha }{}_c)U^b{V}^c$. The two transition-derivative terms cancel, leaving $(D_{\tau }V{)}^{\mathrm{\prime }\alpha }=J^{\alpha }{}_a(D_{\tau }V{)}^a$. Thus no smoothness of the curve beyond $C^2$, or of the section beyond $C^1$, is required. Metric compatibility also gives $\frac{d}{d\tau }g(V,W)=g(D_{\tau }V,W)+g(V,D_{\tau }W)$ for $C^1$ sections $V,W$: insert the component formulas and ${\mathrm{\partial }}_b{g}_{ac}={\mathrm{\Gamma }}^d{}_{ba}{g}_{dc}+{\mathrm{\Gamma }}^d{}_{bc}{g}_{ad}$. These conventions specify the mathematical derivative in the adopted transport law; they add no further physical motion assumption.

This is an ideal classical external-field dipole model with constant $q,m,g_s$. Its flat homogeneous-field specialization is the BMT polarization law in the stated RPM convention. The formula on a curved background, and its use for a spatially varying prescribed field, are separately adopted model assumptions; neither follows from the flat BMT source or from general covariance alone.

The model excludes an electric dipole moment, field-gradient or Stern–Gerlach translational forces, spin backreaction, radiation reaction, collisions and higher multipoles. The parameter $g_s$ is not fixed to $2$ and no measured value is asserted. Using this classical vector to represent a real quantum particle's polarization is a correspondence assumption; the postulate does not construct quantum spin or claim empirical validation.

Assume the charged test-body quantities and local potential domain of Charged and spinning test-body quantities, the flat-space sign convention of The electromagnetic two-form, its observer decomposition and the four-current, and the adopted fixed-endpoint action postulate External-field charged-particle action, and assume ${\mathrm{A}\mathrm{C}}_{\omega }$ for the canonical tangent-bundle smooth structure. For a finite $C^2$ future-timelike curve segment $z:[a,b]\to O$ in that domain, where $a<b$ are proper-time endpoints, and the local $C^2$ fixed-endpoint variations of the postulate, choose the orientation-preserving parameter $\lambda =\tau$; the segment is stationary for the adopted action if and only if

$m{\mathrm{\nabla }}_{U}U=q{F}^{\mathrm{♯}}(U),({\mathrm{\nabla }}_{U}U{)}_a=\frac{q}{m}{F}_{ab}{U}^b\mathrm{.}$

This is a model deduction: it establishes the equation equivalent to stationarity of the postulated functional, not the physical truth of the postulate.