Geometric electrovacuum equations and charge convention

Work with the Axiom of Choice, in particular countable choice. Let $M$ be a smooth Hausdorff second-countable four-manifold without boundary, $g$ a smooth Lorentzian metric of signature $(-,+,+,+)$, $N$ a smooth timelike field selecting the future direction, and $o$ an independently specified orientation. Let $F$ be a smooth real two-form. The metric-selected connection is the torsion-free metric-compatible connection $\mathrm{\nabla }$; use $R(X,Y)Z={\mathrm{\nabla }}_X{\mathrm{\nabla }}_{Y}Z-{\mathrm{\nabla }}_Y{\mathrm{\nabla }}_{X}Z-{\mathrm{\nabla }}_{[X,Y]}Z$ and its Ricci contraction ${\mathrm{Ric}}_{bd}=R^a{}_{bad}$. These conventions follow Oriented Lorentzian fields, observers and coordinate conventions.

The Hodge map ${\star }_g$ is determined by $g$ and $o$ through $\alpha \wedge {\star }_g\beta =⟨\alpha ,\beta {⟩}_g{\mathrm{v}\mathrm{o}\mathrm{l}}_g$, where the pairing is the metric-induced pairing on forms and ${\mathrm{v}\mathrm{o}\mathrm{l}}_g$ agrees with $o$; see Lorentzian exterior calculus and Hodge star. A source-free geometric electrovacuum datum is the tuple $(M,g,N,o,F)$ satisfying the equations in Geometric electrovac equations and flux normalization with zero cosmological constant:

$dF=0,d({\star }_{g}F)=0,{\mathrm{Ric}}_{ab}=2(F_{ac}{F}_b{}^c-\frac{1}{4}{g}_{ab}{F}_{cd}{F}^{cd})\mathrm{.}$

Indices are raised using $g^{-1}$. These are defining tensor equations, with no global potential, asymptotic flatness, horizon, or existence assertion included.

Let $S\subset M$ be a compact oriented smooth embedded two-manifold without boundary. It is spacelike when the restriction of $g$ to every tangent plane $T_{p}S$ is positive definite. Its orientation is specified independently of $o$. With $i:S\hookrightarrow M$ the inclusion, define the geometric electric and magnetic flux parameters by

$Q_E(S):=\frac{1}{4\pi }{\int }_S{i}^{\ast }({\star }_{g}F),Q_M(S):=\frac{1}{4\pi }{\int }_S{i}^{\ast }F\mathrm{.}$

The orientation and factor $4\pi$ are fixed before charges are compared; no SI constants are inserted into this normalization.

The cited Lorentzian and Hodge suppliers define the connection, curvature, exterior derivative and smooth Hodge map under the stated hypotheses. The equations therefore compare smooth tensors of matching types and select a class of tuples; they do not assert that a specified tuple satisfies them. The electrovacuum supplier fixes the displayed curvature and flux normalization. For the broader surfaces used here, the inclusion $i:S\hookrightarrow M$ is smooth, so Pullback of forms is smooth functorial and preserves wedges makes $i^{\ast }F$ and $i^{\ast }({\star }_{g}F)$ smooth two-forms on $S$. By Compact support of a differential form, the support of each is the closure in $S$ of its nonzero locus. Each support is therefore closed in compact $S$ and is compact by A closed subspace of a compact space is compact, and a finite union of compact subspaces is compact. Integration on an oriented embedded submanifold supplies their integrals and invariance under orientation-preserving reparametrization. Thus both displayed fluxes are well-defined real numbers. With $g$, $o$ and $F$ fixed, reversing the chosen orientation of $S$ leaves both pulled-back forms unchanged and negates each integral by Orientation reversal changes the integral sign, applied to these compactly supported top forms. Thus both flux signs reverse. The choice assumption includes the countable choice carried by the Lorentzian and Hodge suppliers.