Geometric electrovac equations and flux normalization

Let $(M,g,N,o)$ be a smooth oriented, time-oriented Lorentzian four-manifold of signature $(-,+,+,+)$; $o$ is the manifold orientation, $N$ is a smooth future timelike field, and the metric volume form is chosen to agree with $o$. Let $\mathrm{\nabla }$ be the metric-selected connection, let $\mathcal{F}$ be a smooth real two-form on $M$, and use the curvature convention $R(X,Y)Z={\mathrm{\nabla }}_X{\mathrm{\nabla }}_{Y}Z-{\mathrm{\nabla }}_Y{\mathrm{\nabla }}_{X}Z-{\mathrm{\nabla }}_{[X,Y]}Z$. Define $Q_{ab}(\mathcal{F}):={\mathcal{F}}_{ac}{\mathcal{F}}_b{}^c-\frac{1}{4}{g}_{ab}{\mathcal{F}}_{cd}{\mathcal{F}}^{cd},$ where both raised indices use $g^{-1}$. A geometric electrovac datum with zero cosmological constant is the tuple $(M,g,N,o,\mathcal{F})$ satisfying $d\mathcal{F}=0,d({\star }_g\mathcal{F})=0,{\mathrm{Ric}}_{ab}=2Q_{ab}(\mathcal{F})$ at every point of $M$. Here ${\star }_g$ is the Lorentzian Hodge map determined by $g$ and $o$; on two-forms ${\star }_g^2=-1$. No material current, matter tensor, global potential, asymptotic flatness, positive mass, horizon, or extension condition is part of this definition.

For flux integration, an oriented embedded smooth sphere means a compact smooth two-dimensional manifold $S$ without boundary, homeomorphic to the unit sphere $\{x\in {\mathbb{R}}^3:x\cdot x=1\}$, with a specified orientation in the sense of Oriented smooth manifolds and oriented charts and a smooth inclusion $i:S\hookrightarrow M$. Here smooth embedding means an injective immersion that is a homeomorphism onto its image with the subspace topology (Smooth embeddings); identify $S$ with that image. The sphere orientation is specified independently of the spacetime orientation; for coordinate spheres used below it is the stated outward orientation.

For a smooth two-form $\eta$ on a neighborhood of $i(S)$, use the pullback convention of The pullback of a differential form, $(i^{\ast }\eta {)}_p(v,w)={\eta }_{i(p)}(d{i}_{p}v,d{i}_{p}w),p\in S,v,w\in T_{p}S\mathrm{.}$ By Pullback of forms is smooth functorial and preserves wedges, this is a smooth top form on $S$. Its support is the closure in $S$ of its nonzero locus (Compact support of a differential form), hence is closed in compact $S$ and compact by A closed subspace of a compact space is compact, and a finite union of compact subspaces is compact. Thus $i^{\ast }\eta \in {\mathrm{\Omega }}_c^2(S)$.

Write ${\mathcal{I}}_S$ for Integral of a compactly supported top form: for a subordinate smooth partition $({\rho }_j)$ and connected charts ${\varphi }_j$, it sets ${\mathcal{I}}_S(\omega )={\sum }_j{I}_{{\varphi }_j}({\rho }_j\omega )$, where $I_{{\varphi }_j}$ is the signed chart integral of the compactly supported coefficient. Only finitely many terms are nonzero. The value is independent of charts, partition and refinement by Independence of atlas, partition and refinement. Define the surface notation by ${\int }_S\eta :={\mathcal{I}}_S(i^{\ast }\eta )\mathrm{.}$ This agrees with Integration on an oriented embedded submanifold; a reparametrization here means an orientation-preserving diffeomorphism $F:T\to S$ of oriented smooth two-manifolds (a smooth bijection with smooth inverse and orientation-preserving differential). For such $F$, the same value is ${\mathcal{I}}_T((iF{)}^{\ast }\eta )$. The integration definition and its independence theorem assume ${\mathrm{A}\mathrm{C}}_{\omega }$, already retained in the geometric background below. For $\eta =\mathcal{F}$ and $\eta ={\star }_g\mathcal{F}$, smoothness of the given field and Hodge map makes both pulled-back top forms admissible; their integrals are finite real numbers. Define the geometric electric and magnetic flux parameters by $q_e(S):=\frac{1}{4\pi }{\int }_S{\star }_g\mathcal{F},q_m(S):=\frac{1}{4\pi }{\int }_S\mathcal{F}\mathrm{.}$ When $M$ and the coordinates have length units, these parameters have units of length. A magnetic flux need not have a global potential; no flux quantization is included. The background Lorentzian geometry and Hodge conventions carry the countable-choice assumption ${\mathrm{A}\mathrm{C}}_{\omega }$ of their named suppliers; the algebraic identities in this definition require no additional choice principle.