Connection sectors, holonomy and presymplectic boundaries

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ for the smooth tangent-bundle and principal-connection convention. Let $(M,g,N,o)$ be the smooth, Hausdorff, second-countable, oriented and time-oriented Lorentzian four-manifold used for normalized circle-connection data. Fix a smooth right principal $U(1)$ bundle $P\to M$, one countable trivializing cover $(U_i)$ with transition functions $g_{ij}$ in the convention $z_j=g_{ij}^{-1}{z}_i$, and one normalization constant ${\kappa }_0>0$ measured in webers. Let $(a_i)$ and $(a_i^{\mathrm{\prime }})$ be two smooth normalized connection representatives for this same atlas, each satisfying $a_j=a_i+\lambda (g_{ij})$ and $a_j^{\mathrm{\prime }}=a_i^{\mathrm{\prime }}+\lambda (g_{ij})$, where $\lambda (g):=-i{g}^{-1}dg$. In length-coordinate charts, the components of $a_i$ and $a_i^{\mathrm{\prime }}$ have units ${\mathrm{m}}^{-1}$; the physical potentials $A_i={\kappa }_0{a}_i$ and $A_i^{\mathrm{\prime }}={\kappa }_0{a}_i^{\mathrm{\prime }}$ have components $\mathrm{W}\mathrm{b}{\mathrm{m}}^{-1}$.

Then ${\alpha }_i:=a_i^{\mathrm{\prime }}-a_i$ is the restriction of a unique smooth global real one-form $\alpha$ on $M$. The normalized curvatures $f{\mathrm{\mid }}_{U_i}=d{a}_i$ and $f^{\mathrm{\prime }}{\mathrm{\mid }}_{U_i}=d{a}_i^{\mathrm{\prime }}$ are global, and $f^{\mathrm{\prime }}-f=d\alpha$. The physical curvatures $F={\kappa }_{0}f$ and $F^{\mathrm{\prime }}={\kappa }_0{f}^{\mathrm{\prime }}$ satisfy $F^{\mathrm{\prime }}-F={\kappa }_{0}d\alpha$; in length-coordinate charts their components have units $\mathrm{W}\mathrm{b}{\mathrm{m}}^{-2}$. Consequently ${\int }_S{\iota }^{\ast }{F}^{\mathrm{\prime }}={\int }_S{\iota }^{\ast }F$ for every compact oriented smooth surface $S$ without boundary and smooth embedding $\iota :S\hookrightarrow M$. The representatives on a fixed bundle and atlas form an affine space over ${\mathrm{\Omega }}^1(M)$.

For a smooth local gauge map ${\mathrm{\Phi }}_h(x,z)=(x,h(x)z)$, the adopted pushforward action on horizontal distributions changes the local representative by $a_i\mapsto a_i-\lambda (h)$; this sign is part of the stated convention.

Local potentials and smooth circle-bundle connection data

Circle gluing, curvature and connection variations

Gauge transformations act on principal connections and carry curvature

The finite-chart diagram records the affine translation of a connection representative and the resulting curvature translation. It summarizes step 1.2 and step 2.1; the proof does not rely on the diagram.

\mathcal A(P) \arrow[r,"T_\alpha"] \arrow[d,"F_{\mathrm{phys}}"'] &
\mathcal A(P) \arrow[d,"F_{\mathrm{phys}}"] \\
\Omega^2(M) \arrow[r,"+\kappa_0\,d\alpha"'] & \Omega^2(M)

Here $\mathcal{A}(P)$ is the affine set of normalized connections on the fixed bundle and atlas, $T_{\alpha }$ adds the global one-form $\alpha$ to every local representative, and $F_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ sends a connection to its global physical curvature. Step 2.1 proves that $F_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(T_{\alpha }a)=F_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(a)+{\kappa }_{0}d\alpha$, which is exactly the commutativity asserted by the square. The proposition does not classify bundle sectors or claim that every flat difference is gauge trivial.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ as inherited from the smooth principal-connection convention. Fix an oriented, time-oriented smooth Lorentzian four-manifold $(M,g,N,o)$ and a fixed normalized $U(1)$ connection sector as in the fixed-topology affine-sector result. Its connection variations are global smooth real one-forms $\alpha ,\beta \in {\mathrm{\Omega }}^1(M)$, with components of unit ${\mathrm{m}}^{-1}$ in length-coordinate charts. Length-coordinate charts use metre-valued coordinates, with $x^0=c{t}_{\mathrm{s}\mathrm{e}\mathrm{c}}$ when time is recorded in seconds; $c$ is a coordinate unit-conversion scale in $\mathrm{m}{\mathrm{s}}^{-1}$, with no numerical measured value or physical role asserted. Impose the fixed-background Maxwell operator equations $d{\star }_{g}d\alpha =0$ and $d{\star }_{g}d\beta =0$, where ${\star }_g$ is the oriented Lorentzian Hodge map of signature $(-,+,+,+)$. Define the real three-form

${\omega }_g(\alpha ,\beta ):=\alpha \wedge {\star }_{g}d\beta -\beta \wedge {\star }_{g}d\alpha \mathrm{.}$

Let ${\iota }_{\mathrm{\Sigma }}:\mathrm{\Sigma }\hookrightarrow M$ be an oriented smooth embedding of a possibly noncompact three-dimensional manifold, and require the induced metric $\gamma :={\iota }_{\mathrm{\Sigma }}^{\ast }g$ to be positive definite. Its oriented Riemannian volume form is the nowhere-vanishing three-form ${\mathrm{v}\mathrm{o}\mathrm{l}}_{\gamma }$ whose expression in each positively oriented chart $y=(y^1,y^2,y^3)$ is $\sqrt{\det ({\gamma }_{ij})}d{y}^1\wedge d{y}^2\wedge d{y}^3$. Let ${\mu }_{\gamma }$ be the positive smooth density measure with coordinate expression $d{\mu }_{\gamma }=\sqrt{\det ({\gamma }_{ij})}{d}^{3}y$. The density-measure integration supplier listed below gives the chart-independent measure and integral; for a real $L^1({\mu }_{\gamma })$ function its signed integral is the finite difference of the integrals of its positive and negative parts, and its chart formula is absolutely convergent.

There is a unique smooth real function $u_{\alpha ,\beta }$ on $\mathrm{\Sigma }$ such that ${\iota }_{\mathrm{\Sigma }}^{\ast }{\omega }_g(\alpha ,\beta )=u_{\alpha ,\beta }{\mathrm{v}\mathrm{o}\mathrm{l}}_{\gamma }$. Define the pairing ${\mathrm{\Omega }}_{\mathrm{\Sigma }}(\alpha ,\beta )$ precisely when $u_{\alpha ,\beta }\in L^1(\mathrm{\Sigma },{\mu }_{\gamma })$, meaning ${\int }_{\mathrm{\Sigma }}\mathrm{\mid }{u}_{\alpha ,\beta }\mathrm{\mid }d{\mu }_{\gamma }<\mathrm{\infty }$, and then set ${\mathrm{\Omega }}_{\mathrm{\Sigma }}(\alpha ,\beta ):={\int }_{\mathrm{\Sigma }}{u}_{\alpha ,\beta }d{\mu }_{\gamma }$. This absolute-integrability condition is the convergence convention on noncompact $\mathrm{\Sigma }$ and makes the value independent of any exhaustion or ordering. In length coordinates on $\mathrm{\Sigma }$, the coefficients of $d\alpha$ and $d\beta$ have units ${\mathrm{m}}^{-2}$, those of ${\omega }_g$ have units ${\mathrm{m}}^{-3}$, $u_{\alpha ,\beta }$ has units ${\mathrm{m}}^{-3}$, and $d{\mu }_{\gamma }$ has volume units ${\mathrm{m}}^3$; thus the normalized integral is dimensionless. Replacing normalized variations by physical-potential variations ${\kappa }_0\alpha$ and ${\kappa }_0\beta$ multiplies the current by ${\kappa }_0^2$; an action prefactor is not fixed here.

This is a mathematical fixed-metric Maxwell-block definition. The equations are imposed on the normalized connection variations. The item does not identify ${\kappa }_{0}d\alpha$ with the geometric-normalized electrovac field of the coupled Einstein–Maxwell equations, prescribe a conversion between those normalizations, include metric variations, or claim to define the full coupled covariant phase-space current.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

Fixed-topology affine sector of normalized circle connections

Local potentials and smooth circle-bundle connection data

Variation of the Lorentzian Hodge map and the linearized Maxwell equation

Lorentzian exterior calculus and Hodge star

Oriented Lorentzian fields, observers and coordinate conventions

Measurable integration extends smooth density integration

The fixed-metric restriction is essential: the Hodge-variation result gives $\delta ({\star }_g\mathcal{F})={\star }_{g}f$ when the metric perturbation is zero. For nonzero metric perturbations the terms involving $h$ in the Hodge variation remain, and the full Einstein–Maxwell current needs additional gravitational and mixed contributions. The current formula is bilinear and antisymmetric by the symmetry of the Lorentzian two-form pairing.

Let $g$ be a fixed oriented Lorentzian metric of signature $(-,+,+,+)$ on a smooth four-manifold, and let $\alpha ,\beta \in {\mathrm{\Omega }}^1(M)$ be smooth normalized connection variations satisfying $d{\star }_{g}d\alpha =d{\star }_{g}d\beta =0$. For ${\omega }_g(\alpha ,\beta )=\alpha \wedge {\star }_{g}d\beta -\beta \wedge {\star }_{g}d\alpha$, one has $d{\omega }_g(\alpha ,\beta )=0$. For every smooth real function $\chi$ on $M$, replacing $\alpha$ by $\alpha +d\chi$ changes the current by $d(\chi {\star }_{g}d\beta )$. Hence, for every finite smooth singular three-chain $c$, the integrated change is ${\int }_{\mathrm{\partial }c}\chi {\star }_{g}d\beta$; it vanishes when $c$ is a cycle.

For a noncompact oriented spacelike hypersurface ${\iota }_{\mathrm{\Sigma }}:\mathrm{\Sigma }\hookrightarrow M$, suppose both ${\mathrm{\Omega }}_{\mathrm{\Sigma }}(\alpha ,\beta )$ and ${\mathrm{\Omega }}_{\mathrm{\Sigma }}(\alpha +d\chi ,\beta )$ are defined by the absolute-integrability convention of the preceding presymplectic-current definition. Write ${\iota }_{\mathrm{\Sigma }}^{\ast }({\omega }_g(\alpha +d\chi ,\beta )-{\omega }_g(\alpha ,\beta ))=u{\mathrm{v}\mathrm{o}\mathrm{l}}_{\gamma }$. For every compact Borel exhaustion $K_n\subseteq K_{n+1}$ with ${\bigcup }_n{K}_n=\mathrm{\Sigma }$, one has ${\int }_{K_n}ud{\mu }_{\gamma }\to {\int }_{\mathrm{\Sigma }}ud{\mu }_{\gamma }$. If the exhaustion has finite smooth singular three-chains $c_n$ supported in $K_n$ satisfying ${\int }_{c_n}\tau ={\int }_{K_n}{u}_{\tau }d{\mu }_{\gamma }$ for every smooth three-form $\tau =u_{\tau }{\mathrm{v}\mathrm{o}\mathrm{l}}_{\gamma }$ on $\mathrm{\Sigma }$, using the induced metric and density fixed in that definition, then

${\mathrm{\Omega }}_{\mathrm{\Sigma }}(\alpha +d\chi ,\beta )-{\mathrm{\Omega }}_{\mathrm{\Sigma }}(\alpha ,\beta )={\lim }_{n\to \mathrm{\infty }}{\int }_{\mathrm{\partial }{c}_n}\chi {\star }_{g}d\beta \mathrm{.}$

The limit exists and is independent of the compact exhaustion and its smooth-chain representatives. For a fixed $\beta$, the pairing shift vanishes exactly when this limiting boundary flux is zero. Given any linear subspace $\mathcal{V}$ of smooth real one-form solutions $\beta$ of $d{\star }_{g}d\beta =0$ for which both pairings are defined, call the exact gauge direction $d\chi$ degenerate on $\mathcal{V}$ when its pairing shift vanishes for every $\beta \in \mathcal{V}$. Thus degeneracy on $\mathcal{V}$ holds exactly when this limit vanishes for every $\beta \in \mathcal{V}$.

Fixed-background Maxwell presymplectic current

Lorentzian exterior calculus and Hodge star

Stokes theorem for smooth singular chains

The calculation treats an exact gauge variation with a globally defined real parameter $\chi$. A circle-valued large gauge transformation need not admit such a global phase; this identity alone therefore does not declare every flat connection difference to be presymplectically degenerate.

Let $V$ be a finite-dimensional real vector space with a nondegenerate alternating bilinear form $\omega :V\times V\to \mathbb{R}$. For a linear subspace $W\subseteq V$, define $W^{\omega }:=\{v\in V:\omega (v,w)=0\text{ for every }w\in W\}$, and assume $W^{\omega }\subseteq W$ (so $W$ is coisotropic). Then the quotient vector space $W\mathrm{/}{W}^{\omega }$, whose elements are cosets $[u]:=u+W^{\omega }$, carries the well-defined symplectic form $\bar{\omega }([u],[v])=\omega (u,v)$. This proposition concerns only finite-dimensional constrained mode models; it does not assert that an infinite-dimensional Einstein–Maxwell solution space has a smooth symplectic quotient.

Isotropic, coisotropic, symplectic, and Lagrangian subspaces

Symplectic reduction of a coisotropic vector subspace

For $W=V$ the quotient is $V$ itself; for a Lagrangian constraint subspace $W=W^{\omega }$ the quotient is the zero symplectic vector space. These are finite-dimensional boundary cases of the stated result, not evidence for an infinite-dimensional solution-space reduction.