Nonlinear electromagnetic constitutive model with a $C^2$ Lagrangian

Let $(M,g,N,o)$ be a smooth, Hausdorff, second-countable, boundaryless, oriented and time-oriented Lorentzian four-manifold with signature $(-,+,+,+)$, independent orientation $o$, and the torsion-free metric-compatible connection $\mathrm{\nabla }$ selected by $g$. Fix a smooth principal $U(1)$ bundle sector and a smooth connection $\mathcal{A}$ in that sector. In the inherited physical-curvature normalization, its curvature is a smooth real two-form $F\in {\mathrm{\Omega }}^2(M)$ with $dF=0$. On a supplied countable trivializing cover $(U_i)$, write its compatible local physical-potential representatives as $A_i\in {\mathrm{\Omega }}^1(U_i)$, so $F{\mathrm{\mid }}_{U_i}=d{A}_i$ and, in the normalization of Local potentials and smooth circle-bundle connection data, $A_i={\kappa }_0{a}_i$ with $A_j-A_i={\kappa }_0\lambda (g_{ij})$ on overlaps. No global potential is required.

A permitted variation is specified by a smooth compactly supported global real one-form $a\in {\mathrm{\Omega }}^1(M)$, with the same component units as each $A_i$ if units are attached. For a dimensionless parameter $ϵ\in \mathbb{R}$, define $A_{i,ϵ}:=A_i+ϵa{\mathrm{\mid }}_{U_i}$. Because $a$ is global, these local representatives retain the same transition functions and define a smooth connection ${\mathcal{A}}_{ϵ}$ in the same bundle sector. Equivalently, the normalized local representatives shift as $a_{i,ϵ}=a_i+ϵ{\kappa }_0^{-1}a{\mathrm{\mid }}_{U_i}$. Thus $F_{ϵ}{\mathrm{\mid }}_{U_i}=d{A}_{i,ϵ}=F{\mathrm{\mid }}_{U_i}+ϵda$, so globally $F_{ϵ}=F+ϵda$. The shorthand ${\mathcal{A}}_{ϵ}=\mathcal{A}+ϵa$ denotes this affine shift of local physical-potential representatives; it does not assert direct addition of a base one-form to a principal-bundle connection one-form on its total space.

Raise indices with $g^{-1}$ and let ${\star }_g$ be the Hodge map determined by $g$ and $o$. Define the scalar invariants $X:=-\frac{1}{4}{F}_{ab}{F}^{ab},Y:=-\frac{1}{4}{F}_{ab}({\star }_{g}F{)}^{ab}\mathrm{.}$ Let $L:{\mathbb{R}}^2\to \mathbb{R}$ be $C^2$, and write $L_X,L_Y$ for its first partial derivatives evaluated at $(X,Y)$. Define the contravariant antisymmetric tensor $H^{ab}:=L_X{F}^{ab}+L_Y({\star }_{g}F{)}^{ab}\mathrm{.}$ The source-free local action density is $L(X,Y){\mathrm{v}\mathrm{o}\mathrm{l}}_g$. For a continuous compactly supported top form, use the finite chart-partition integral ${\mathcal{I}}_M$ of the continuous-integral definition linked below; it agrees with the usual manifold integral on smooth forms. This convention retains Countable Choice, meaning that every countable family of nonempty sets has a choice function, as inherited from the smooth integral. Its relative action difference under a compact variation is ${\mathcal{I}}_M([L(X(F_{ϵ}),Y(F_{ϵ}))-L(X(F),Y(F))]{\mathrm{v}\mathrm{o}\mathrm{l}}_g)$; this difference has compact support even when the unvaried integral over $M$ is not finite. The connection is stationary when the derivative of this relative action at $ϵ=0$ vanishes for every permitted $a$.

In the mathematical normalization, $L=X$ is the Maxwell quadratic term. If units are attached, the components of $F$ have a fixed unit $\mathfrak{F}$, $X,Y,L$ have unit ${\mathfrak{F}}^2$, and coefficients $\alpha ,\beta$ in $L=X+\alpha X^2+\beta Y^2$ have unit ${\mathfrak{F}}^{-2}$, so $\alpha X$ and $\beta Y$ are dimensionless. This normalization is not an SI action prefactor. The coefficients are arbitrary parameters, not QED coefficients. This definition supplies no quantum origin, source coupling, characteristic-cone condition, hyperbolicity, causality, or solution-existence claim.

Oriented Lorentzian fields, observers and coordinate conventions

Local potentials and smooth circle-bundle connection data

Circle gluing, curvature and connection variations

Lorentzian exterior calculus and Hodge star

Continuous compact-support integration for the constitutive action

A source-free nonlinear constitutive equation is a model distinct from linear Maxwell unless $L=X$ (up to an overall constant). Positivity and causality of its characteristic cone require a separate principal-symbol analysis. If an Euler–Heisenberg interpretation is later supplied, its coefficients and thresholds must come from a stated QFT calculation and approximation regime; none is assigned here.