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Local potentials and explicit global obstruction in the model

Physical scope

The smooth classical field branch uses only the adopted homogeneous Maxwell law dF=0: local potentials exist on star-shaped coordinate charts; on the punctured spatial domain a nonzero sphere flux obstructs a global potential; a fixed U(1) normalization restricts the explicit north/south connection patching, while a separate flat example distinguishes zero curvature from gauge triviality. No coupled Einstein solution, physical magnetic monopole, or quantum charge quantization is asserted.

Statement

Let $(M, g, N, o)$ be a smooth, oriented and time-oriented Lorentzian four-manifold with signature $(-, +, +, +)$ , and let $F \in Ω^{2} (M)$ be a smooth real field with the SI conventions of Selected classical field topology and optional circle normalization. Assume the smooth branch of the adopted curved Maxwell postulate Smooth curved vacuum Maxwell constitutive adoption applies on $M$ , with no singular source in its domain. Only its homogeneous clause $d F = 0$ is used below; no Einstein equation or coupled solution is asserted.

If $U \subseteq M$ is a nonempty star-shaped coordinate domain with chart image $V \subseteq R^{4}$ star-shaped about a point $p$ (that is, $p + t (x - p) \in V$ for every $x \in V$ and $0 \leq t \leq 1$ ), then $F ∣_{U}$ has a smooth local potential. Use a length-coordinate chart for the following formula; its coordinates have unit metre. The components can be taken to be

$A_{ν} (x) = \int_{0}^{1} t (x - p)^{μ} F_{μ ν} (p + t (x - p)) d t .$

For an explicit global example, take $M_{*} = R_{x^{0}} \times (R^{3} ∖ {0})$ with Minkowski metric $- (d x^{0})^{2} + d x^{2} + d y^{2} + d z^{2}$ , orientation $d x^{0} \land d x \land d y \land d z$ , and $x^{0}, x, y, z$ measured in metres. Let $r > 0$ be spatial radius, let $θ, ϕ$ be dimensionless spherical angles, and for $b \in R$ with unit weber set

$F_{b} = b \sin θ d θ \land d ϕ .$

For each fixed $(x^{0}, r)$ , let $S_{r}^{2}$ have its outward orientation, let $N, S$ be its north and south poles, and set $U_{N} = S_{r}^{2} ∖ {S}$ , $U_{S} = S_{r}^{2} ∖ {N}$ , and $O = U_{N} \cap U_{S}$ . The corresponding open patches in $M_{*}$ are ${\tilde{U}}_{N} = {r + z > 0}$ and ${\tilde{U}}_{S} = {r - z > 0}$ , whose restrictions to each $S_{r}^{2}$ are $U_{N}$ and $U_{S}$ . The angular coordinate $ϕ$ is defined modulo $2 π$ on $O$ , while $d ϕ$ is a global one-form there. The form $F_{b}$ is smooth and closed on $M_{*}$ , with outward sphere flux $\int_{S_{r}^{2}} F_{b} = 4 π b$ . For $b \neq 0$ it has no global smooth one-form potential on $M_{*}$ . It has local north/south potentials

$A_{N} = b (1 - \cos θ) d ϕ, A_{S} = - b (1 + \cos θ) d ϕ .$

For the fixed circle normalization $κ_{0} > 0$ of Selected classical field topology and optional circle normalization, these patch as the stated normalized $U (1)$ connection exactly when $2 b / κ_{0} \in Z$ : on the overlap, $A_{S} - A_{N} = - 2 b d ϕ$ and the transition is $g_{N S} (ϕ) = \exp (- 2 i b ϕ / κ_{0})$ . This integrality is a restriction of the selected connection sector; the classical real-field description permits every real $b$ .

There is also a flat-connection example on the smooth base $M_{0} = R_{x^{0}} \times S^{1} \times R^{2}$ . Write $w \in S^{1} \subset C$ for the circle coordinate and $ν = - i w^{- 1} d w$ for its global angular one-form, normalized by $\int_{S^{1}} ν = 2 π$ . For dimensionless $a \in R$ , the global one-form $A = κ_{0} a ν$ has $F = d A = 0$ , while its normalized connection has holonomy $e^{- 2 π i a}$ around the positive circle. It is gauge trivial for integer $a$ and is not gauge trivial for noninteger $a$ . On an angle chart $w = e^{i ϕ}$ , $ν = d ϕ$ . This is a classical connection statement on a smooth base, not a claim about a quantum phase or an Einstein solution. The punctured point in the sphere example is outside the field domain; no physical magnetic monopole or quantum electric-charge quantization is asserted.

Facts & Assumptions

Given: The smooth Lorentzian field model above, a smooth real two-form, and the positive fixed normalization $κ_{0}$ only when the optional circle sector is selected.

[A1]

The current draft Smooth curved vacuum Maxwell constitutive adoption adopts the smooth SI field law on an open subset $U$ of a smooth oriented, time-oriented Lorentzian four-manifold: for smooth real $F \in Ω^{2} (U)$ and smooth four-current $j$ , $d F = 0$ and $\nabla_{a}^{g} F^{a b} = - μ_{0} j^{b}$ . It excludes magnetic sources, material excitation laws, singular point or sheet sources, distributional products, and global-potential assumptions. This is a physical postulate, not a theorem; only $d F = 0$ is used here. Its batch-1 contract contains author formulation, scope, sources and non-derivation review; this is not independent acceptance, and current external prerequisite closure remains an owner obligation.

[A2]

The current draft Radial primitives for closed one- and two-forms states that for an open star-shaped $D \subseteq R^{4}$ and smooth closed real two-form $F = \frac{1}{2} F_{a b} d x^{a} \land d x^{b}$ , the displayed radial integral defines a smooth $A = A_{b} d x^{b}$ with $d A = F$ . Its complete local proof differentiates the integral and checks every coefficient. The typed foundational dependencies are now authored earlier on emg-mg02; this is local proof completion, not independent acceptance. Independent acceptance and current run readiness remain separate owner duties.

[A3]

For the punctured-Minkowski specialization equip $M_{*}$ with the future time orientation selected by $N_{*} : = \partial_{x^{0}}$ ; its displayed metric gives $g (N_{*}, N_{*}) = - 1$ , and its orientation is the displayed product orientation. This is the canonical Lorentzian tuple of the supplier. The current draft Sphere winding, nonexact curvature and flat holonomy states the smooth Cartesian punctured-space form, its unit-sphere flux and Stokes obstruction, the smooth north/south potentials and fixed- $κ_{0}$ transition criterion, and the flat example $A = κ_{0} a ν$ with $ν = - i w^{- 1} d w$ and holonomy $e^{- 2 π i a}$ . On an angle chart $w = e^{i ϕ}$ , its global angular one-form is $ν = d ϕ$ . Its complete local proof gives an ordered tetrahedral finite-cycle Stokes argument and gauge-period calculation. The gluing supplier now uses the actual local set-size definition and constructs the normalized bundle and curvature. The exact interfaces match; no independent acceptance is asserted.

[F1]

Selected classical field topology and optional circle normalization defines a closed classical field by $d F = 0$ , a local potential by $d A_{U} = F ∣_{U}$ , and the optional normalized connection by $A_{i} = κ_{0} a_{i}$ , $F ∣_{U_{i}} = κ_{0} d a_{i}$ , and $a_{j} = a_{i} + d f_{i j}$ when $g_{i j} = e^{i f_{i j}}$ . It assigns $F$ flux unit Wb and $κ_{0}$ unit Wb; the choice is not a quantum charge postulate.

[F2]

Stokes theorem for smooth singular chains states that for a smooth manifold $X$ , integer $k \geq 1$ , finite smooth real singular $k$ -chain $c$ , and smooth $(k - 1)$ -form $ω$ , $\int_{c} d ω = \int_{\partial c} ω$ .

Proof

1.1A2construct

Fix a nonempty star-shaped chart $U$ and its center $p$ . Define $A_{ν}$ by the displayed radial integral. The integrand is smooth in $(x, t)$ on $V \times [0, 1]$ ; on every relatively compact subdomain its $x$ -derivatives are continuous and uniformly bounded on the compact integration interval, so differentiation under the integral gives a smooth one-form.

1.2A3algebra

Equip the explicit $M_{*}$ witness with $N_{*} : = \partial_{x^{0}}$ as in [A3]. This vector field is smooth, globally defined and timelike since $g (N_{*}, N_{*}) = - 1$ ; select its future cone. The displayed metric and orientation, this time orientation, and the standard product smooth structure give exactly the tuple required by [A3]. On $M_{*}$ the Cartesian expression $F_{b} = b (x d y \land d z + y d z \land d x + z d x \land d y) / r^{3}$ is smooth because $r > 0$ . Direct differentiation gives $d F_{b} = b (\partial_{x} (x / r^{3}) + \partial_{y} (y / r^{3}) + \partial_{z} (z / r^{3})) d x \land d y \land d z = b (3 / r^{3} - 3 r^{2} / r^{5}) d x \land d y \land d z = 0$ . In spherical coordinates, the outward-oriented sphere integral is $\int_{S_{r}^{2}} F_{b} = b \int_{0}^{π} \sin θ d θ \int_{0}^{2 π} d ϕ = 4 π b$ .

1.3A3F1algebra

The north and south potentials in [A3] are smooth on ${\tilde{U}}_{N}$ and ${\tilde{U}}_{S}$ respectively, differentiate to $F_{b}$ , and differ by $A_{S} - A_{N} = - 2 b d ϕ$ on their overlap. Dividing by $κ_{0}$ gives the normalized connection transition lift $f_{N S} = - 2 b ϕ / κ_{0}$ , so $g_{N S} = e^{i f_{N S}}$ returns to its initial value after $ϕ \mapsto ϕ + 2 π$ exactly when $e^{- 4 π i b / κ_{0}} = 1$ . The map $x \mapsto e^{i x}$ has kernel $2 π Z$ , hence this is equivalent to $2 b / κ_{0} \in Z$ . If $n = 2 b / κ_{0} \in Z$ , then $g_{N S} = e^{- i n ϕ}$ is smooth and single-valued on the overlap and $a_{S} = a_{N} + d (- n ϕ)$ is the declared gluing rule; conversely any fixed-sector transition must be single-valued.

2.1A1A2step 1.1algebra

By antisymmetry of $F$ and $d F = 0$ from [A1], differentiation of the radial formula in step 1.1 gives $\partial_{ρ} A_{ν} - \partial_{ν} A_{ρ} = \int_{0}^{1} (2 t F_{ρ ν} + t^{2} (x - p)^{μ} \partial_{μ} F_{ρ ν}) (p + t (x - p)) d t = \int_{0}^{1} \partial_{t} [t^{2} F_{ρ ν} (p + t (x - p))] d t = F_{ρ ν} (x)$ . Hence $d A_{U} = F ∣_{U}$ .

2.2A3F2step 1.2algebra

For $b \neq 0$ , step 1.2 identifies the explicit field and its canonical Lorentzian tuple with [A3]. That supplier proves that $F_{b}$ has no global smooth one-form potential, using its ordered tetrahedral smooth singular two-cycle $c$ with $\partial c = 0$ and $\int_{c} F_{b} = 4 π b$ . Its proof establishes the radial projection as an orientation-preserving one-sheeted parametrization and justifies this integral by its finite-parametrization integration supplier. If a global $A$ satisfied $d A = F_{b}$ , finite-chain Stokes [F2] would give $4 π b = \int_{c} d A = \int_{\partial c} A = 0$ , a contradiction. Thus the stated obstruction follows with the exact supplier hypotheses and integration bridge.

3.1A3F1algebraconstruct∎

On $M_{0}$ , write $w = e^{i ϕ}$ on an angle chart. The supplier form $ν = - i w^{- 1} d w$ then equals the global angular form $d ϕ$ , since $d w = i e^{i ϕ} d ϕ$ ; its integral around the positive circle is $2 π$ . Thus $A = κ_{0} a ν$ is global and $d A = 0$ . The horizontal-lift equation over one positive turn is $\dot{θ} = - a \dot{ϕ}$ , so the endpoint fibre rotation is $e^{- 2 π i a}$ . A single-valued gauge map along the circle has a real lift on $[0, 2 π]$ whose endpoint difference is in $2 π Z$ , so it preserves this rotation. Therefore noninteger $a$ is not gauge trivial. If $a \in Z$ , $h (w) = w^{- a}$ is a well-defined $U (1)$ map and its local lift has derivative $- a d ϕ$ , which removes the connection.

Remarks

The source texts motivate the local/global distinction and exhibit patch potentials, but the proof uses the current conditional local radial and sphere-flux supplier interfaces above. The classical sphere field is smooth only on the punctured domain; no source at the removed point is included. The finite radial integral, explicit two-patch transition, flux integral, and circle holonomy calculation make no arbitrary-family choice. The typed sequence criterion states exactly its countable-choice hypothesis; the finite calculations here do not spend it. Independent supplier acceptance and source status remain owner review duties; the audited imported boundaryless-convention remark is not a missing prerequisite.

Local potentials and explicit global obstruction in the model

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources