Selected classical field topology and optional circle normalization

Let $(M,g,N,o)$ be the smooth, Hausdorff, second-countable, oriented and time-oriented Lorentzian four-manifold of Einstein–Maxwell primitive quantities and SI conventions, with signature $(-,+,+,+)$ and no boundary. Write ${\mathrm{\Omega }}^k(M)$ for the smooth real sections of ${\mathrm{\Lambda }}^k{T}^{\ast }M$. The classical electromagnetic field type is a smooth real two-form $F\in {\mathrm{\Omega }}^2(M)$. In length-coordinate charts its components have SI unit tesla; its integral over an oriented smooth two-surface has unit weber. The exterior derivative is denoted $d$. A field is closed when $dF=0$. Closedness is a field condition, not the same condition as having a global potential.

For an open set $U\subseteq M$, a local potential for $F$ on $U$ is a smooth real one-form $A_U\in {\mathrm{\Omega }}^1(U)$ satisfying $d{A}_U=F{\mathrm{\mid }}_U$. It is a global potential when $U=M$. In a length-coordinate chart the components of $A_U$ have unit tesla metre. This classical real-field description chooses $F$ itself and requires no bundle or global $A$.

An optional circle-normalized connection sector adds the mathematical data of a smooth right principal $U(1)$ bundle $\pi :P\to M$ and a connection, using the local-potential and circle-data conventions of Local potentials and smooth circle-bundle connection data. Here $U(1)=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }=1\}$ with multiplication. Concretely, supply a countable open cover $\{U_i\}$ and smooth transition maps $g_{ij}:U_i\cap U_j\to U(1)$ satisfying $g_{ii}=1$ and $g_{ij}{g}_{jk}=g_{ik}$. On overlap pieces with smooth real lifts $g_{ij}=e^{i{f}_{ij}}$, the normalized local connection one-forms $a_i$ satisfy $a_j=a_i+d{f}_{ij}$; in local fibre angles the convention is ${\theta }_j={\theta }_i-f_{ij}$, so $d{\theta }_i+a_i$ patches. The local curvature forms $d{a}_i$ agree. Fix an independent constant ${\kappa }_0>0$ with unit weber, and set $A_i={\kappa }_0{a}_i$ and $F{\mathrm{\mid }}_{U_i}={\kappa }_{0}d{a}_i$. Thus $a_i$ has component unit ${\mathrm{m}}^{-1}$ in length coordinates, while $A_i$ has component unit tesla metre. Choosing $P$ and ${\kappa }_0$ is extra model data; no claim is made that every smooth closed real $F$ is the curvature of such a fixed-normalization connection.

The selected topological sector is the fixed pair consisting of the bundle $P$ and the normalization ${\kappa }_0$; its connection is data on that fixed bundle. Changing $P$ or ${\kappa }_0$ selects a different sector.

In a trivialization, a gauge transformation is a single-valued smooth map $h:U\to U(1)$; its local real lifts may differ by integer multiples of $2\pi$. A connection on a trivial bundle is gauge trivial when one such global map takes its normalized connection form to zero. For a closed piecewise-$C^1$ loop $\gamma$ lying in a trivialization, horizontal transport is defined by $\dot{\theta }=-a(\dot{\gamma })$; the endpoint fibre rotation is its holonomy. When $a$ is one global one-form along the loop this is $\exp (-i{\int }_{\gamma }a)$. These definitions introduce no quantum state or quantum phase. The constant ${\kappa }_0$ is not identified with Planck's constant or a unit of electric charge.

Let $(M,g,N,o)$ be a smooth, oriented and time-oriented Lorentzian four-manifold with signature $(-,+,+,+)$, and let $F\in {\mathrm{\Omega }}^2(M)$ 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 $dF=0$ 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 $V\subseteq {\mathbb{R}}^4$ star-shaped about a point $p$ (that is, $p+t(x-p)\in V$ for every $x\in V$ and $0\le t\le 1$), then $F{\mathrm{\mid }}_U$ 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

$A_{\nu }(x)={\int }_0^{1}t(x-p{)}^{\mu }{F}_{\mu \nu }(p+t(x-p))dt\mathrm{.}$

For an explicit global example, take $M_{\ast }={\mathbb{R}}_{x^0}\times ({\mathbb{R}}^3\setminus \{0\})$ with Minkowski metric $-(d{x}^0{)}^2+d{x}^2+d{y}^2+d{z}^2$, orientation $d{x}^0\wedge dx\wedge dy\wedge dz$, and $x^0,x,y,z$ measured in metres. Let $r>0$ be spatial radius, let $\theta ,\varphi$ be dimensionless spherical angles, and for $b\in \mathbb{R}$ with unit weber set

$F_b=b\sin \theta d\theta \wedge d\varphi \mathrm{.}$

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 $U_N=S_r^2\setminus \{S\}$, $U_S=S_r^2\setminus \{N\}$, and $O=U_N\cap U_S$. The corresponding open patches in $M_{\ast }$ 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 $\varphi$ is defined modulo $2\pi$ on $O$, while $d\varphi$ is a global one-form there. The form $F_b$ is smooth and closed on $M_{\ast }$, with outward sphere flux ${\int }_{S_r^2}{F}_b=4\pi b$. For $b\mathrm{\ne }0$ it has no global smooth one-form potential on $M_{\ast }$. It has local north/south potentials

$A_N=b(1-\cos \theta )d\varphi ,A_S=-b(1+\cos \theta )d\varphi \mathrm{.}$

For the fixed circle normalization ${\kappa }_0>0$ of Selected classical field topology and optional circle normalization, these patch as the stated normalized $U(1)$ connection exactly when $2b\mathrm{/}{\kappa }_0\in \mathbb{Z}$: on the overlap, $A_S-A_N=-2bd\varphi$ and the transition is $g_{NS}(\varphi )=\exp (-2ib\varphi \mathrm{/}{\kappa }_0)$. 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 $M_0={\mathbb{R}}_{x^0}\times S^1\times {\mathbb{R}}^2$. Write $w\in S^1\subset \mathbb{C}$ for the circle coordinate and $\nu =-i{w}^{-1}dw$ for its global angular one-form, normalized by ${\int }_{S^1}\nu =2\pi$. For dimensionless $a\in \mathbb{R}$, the global one-form $A={\kappa }_{0}a\nu$ has $F=dA=0$, while its normalized connection has holonomy $e^{-2\pi ia}$ around the positive circle. It is gauge trivial for integer $a$ and is not gauge trivial for noninteger $a$. On an angle chart $w=e^{i\varphi }$, $\nu =d\varphi$. 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.

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.