Local potentials and smooth circle-bundle connection data

Let $(M,g,N,o)$ be the smooth, Hausdorff, second-countable, boundaryless oriented and time-oriented Lorentzian four-manifold of Oriented Lorentzian fields, observers and coordinate conventions. Its smooth real $k$-forms are sections of ${\bigwedge }^k{T}^{\ast }M$, written ${\mathrm{\Omega }}^k(M)$, and $d$ is the exterior derivative of Lorentzian exterior calculus and Hodge star. A smooth two-form $F$ is closed when $dF=0$ and exact when $F=dA$ for some smooth real one-form $A$ on the whole base.

By De rham cohomology, $H_{\mathrm{d}\mathrm{R}}^2(M)=Z^2(M)\mathrm{/}{B}^2(M)$, where $Z^2(M)$ is the vector space of closed smooth two-forms and $B^2(M)$ is the vector space of exact smooth two-forms. Thus, for closed $F$, the statement $[F]=0$ means exactly that $F=dA$ for a global smooth one-form $A$. This equivalence is the quotient definition; it does not assert that every closed form is exact.

Identify $\mathbb{C}$ with ${\mathbb{R}}^2$ by $z=x+iy$ and set $U(1)=S^1:=\{(x,y)\in {\mathbb{R}}^2:x^2+y^2=1\}$. Its topology is the subspace topology. Specify its smooth structure by the four charts

$V_{x,ϵ}=\{(x,y)\in S^1:ϵx>0\},(x,y)\mapsto y,y\mapsto (ϵ\sqrt{1-y^2},y),$

and

$V_{y,ϵ}=\{(x,y)\in S^1:ϵy>0\},(x,y)\mapsto x,x\mapsto (x,ϵ\sqrt{1-x^2}),$

for $ϵ\in \{-1,1\}$, with each coordinate range $(-1,1)$. They cover $S^1$; their transition maps are restrictions of the positive square-root branch and coordinate projections, hence are smooth by The positive square-root branch is smooth and $C^k$ Euclidean maps are closed under componentwise algebra and composition. In these coordinates multiplication is the restriction of $(x,y)(u,v)=(xu-yv,xv+yu)$ and inversion is $(x,y)\mapsto (x,-y)$, so both are smooth; they are also continuous for the stated topology. Thus this is a smooth Lie group and, in particular, a topological group as required by Principal g bundle and associated fiber bundle.

Let $E:\mathbb{R}\to U(1)$ be $E(t)=e^{it}=(\cos t,\sin t)$; the Cartesian formula is $\exp (x+iy)=e^x(\cos y+i\sin y)$, $\mathrm{\mid }\exp (x+iy)\mathrm{\mid }=e^x$, and $e^{i\pi }+1=0$. The trigonometric derivative identities of The derivatives of sine and cosine are cosine and minus sine, iterated, show that $E$ is smooth in the $C^{\mathrm{\infty }}$ sense of $C^k$ Euclidean maps and diffeomorphisms: every derivative of either coordinate is $\pm \sin$ or $\pm \cos$, and these functions are continuous by A function differentiable at $c$ is continuous at $c$. Surjectivity is $t\mapsto (\cos t,\sin t)$ is a bijection from $[0,2\pi )$ onto the real unit circle. If $E(t)\in V_{x,ϵ}$, the chart coordinate $\sin t$ has derivative $\cos t=x\mathrm{\ne }0$; if $E(t)\in V_{y,ϵ}$, the coordinate $\cos t$ has derivative $-\sin t=-y\mathrm{\ne }0$. The smooth inverse-function theorem Choice-free smooth inverse function theorem in Euclidean space therefore makes $E$ a diffeomorphism from some open interval about $t$ onto an open circle arc. Every smooth map from an open subset of $M$ to $U(1)$ consequently has, near each point, a smooth real phase lift obtained by composing with such a local inverse.

A normalized circle-connection datum on $M$ consists of a supplied countable open cover $(U_i{)}_{i\in I}$, smooth transition maps $g_{ij}:U_i\cap U_j\to U(1)$, smooth real one-forms $a_i\in {\mathrm{\Omega }}^1(U_i)$, and a fixed constant ${\kappa }_0>0$. Require $g_{ii}=1$ and $g_{ij}{g}_{jk}=g_{ik}$ on every triple intersection. Around each point of $U_i\cap U_j$, take any smooth real phase lift $f_{ij}$ with $g_{ij}=e^{i{f}_{ij}}$; require

on every such phase-lift neighborhood. If $f$ and $\tilde{f}$ are two lifts on a connected neighborhood, $\ker (\exp )=2\pi i\mathbb{Z}$, and $\exp z=\exp w$ exactly when $z-w\in 2\pi i\mathbb{Z}$ gives $f-\tilde{f}\in 2\pi \mathbb{Z}$ pointwise; continuity makes this difference constant, so $df=d\tilde{f}$. Thus the compatibility condition is independent of the chosen lift. The globally defined form $\lambda (g):=-i{g}^{-1}dg$ is real, and The complex exponential is entire and its complex derivative is itself together with The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$ gives $\lambda (g)=df$ on every phase-lift neighborhood. Hence the equivalent lift-free condition is $a_j=a_i+\lambda (g_{ij})$.

The right principal $U(1)$-bundle convention is as defined in Principal g bundle and associated fiber bundle: it has a continuous right action and locally trivial equivariant product charts, with free and transitive action on each fibre. The compatible transition functions are the gluing data for such a bundle, in the convention that a point with fibre coordinate $z_i$ in chart $i$ has coordinate $z_j=g_{ij}^{-1}{z}_i$ in chart $j$. No global real-valued angle on $U(1)$ is assumed.

The constant ${\kappa }_0$ is a chosen normalization with units of webers; it is not Planck's constant and is not a unit of electric charge. On a chart whose coordinates have length units, the normalized connection components $a_i$ have units ${\mathrm{m}}^{-1}$; the corresponding local potential $A_i:={\kappa }_0{a}_i$ has components $\mathrm{W}\mathrm{b}{\mathrm{m}}^{-1}$, and its curvature $F_i:=d{A}_i={\kappa }_{0}d{a}_i$ has components $\mathrm{W}\mathrm{b}{\mathrm{m}}^{-2}$. This definition imposes no flux or charge quantization.

The typed conventions formerly reserved on EMG-MG01 are authored earlier on this page: Finite-real supremum and infimum domain, Real metric and pseudometric convention, Metric nonnegativity from the three axioms, Explicit rational and real power regimes, Sequence-limit criterion with explicit choice use, Boundaryless domain for geometric completeness, and Set-sized objects and definable-class schemas. They fix the finite-real, topology, power, choice, geometric-domain and set-size interfaces. The finite chart formulas here make no countable selection; the explicit countable-choice hypothesis of the sequence criterion is spent only in that lemma's reverse implication. Authored carriers and local proofs do not themselves assert independent acceptance or repair an unrelated original supplier.