Stable MOTS area–charge inequality

Assume Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ for the surface, flux and deformation constructions below. In the smooth four-dimensional classical Einstein–Maxwell model, adopt geometric units and explicitly assume the field equation $G_{ab}+\mathrm{\Lambda }{g}_{ab}=8\pi (T_{ab}^{\mathrm{E}\mathrm{M}}+{\tau }_{ab})$, where $\mathrm{\Lambda }\ge 0$ and $T_{ab}^{\mathrm{E}\mathrm{M}}=(4\pi {)}^{-1}(F_{ac}{F}_b{}^c-\frac{1}{4}{g}_{ab}{F}_{cd}{F}^{cd})$. Here $G=\mathrm{Ric}-\frac{1}{2}Rg$, with the metric-selected (torsion-free, metric-compatible) connection ${\mathrm{\nabla }}^g$, and the smooth non-electromagnetic stress tensor $\tau$ satisfies the dominant energy condition: $-{\tau }^a{}_b{v}^b$ is future causal or zero for each future timelike $v$. Charged matter is allowed; source-free Maxwell equations, axial symmetry and asymptotic flatness are not assumptions of this result.

Let $S$ be a nonempty connected closed orientable spacelike two-surface with supplied future null normals $\mathrm{\ell },k$, $g(\mathrm{\ell },k)=-1$, and outgoing expansion ${\theta }_{(\mathrm{\ell })}=0$, using the general spacelike-surface data with its connection fixed to the same ${\mathrm{\nabla }}^g$ used for $G$. Use this connection for the outgoing expansion and for all deformations and expansion variations below. Suppose it is spacetime stably outermost in the exact sense of the spacetime-stably-outermost definition: some smooth $\gamma \ge 0$, $\psi >0$ give $X=\gamma \mathrm{\ell }-\psi k$ with ${\delta }_X{\theta }_{(\mathrm{\ell })}\ge 0$. The derivative is the general spacetime derivative supplied by the expansion-variation lemma, with boost invariance supplied by the boost-variation lemma; no containing spacelike slice is required. Then, with $A$, $e,b$ and the flux normalization of the general spacelike-surface data,

$A\ge 4\pi (Q_E(S{)}^2+Q_M(S{)}^2)\mathrm{.}$

The stronger source bound retained for applications is ${\int }_S(e^2+b^2)d{\mu }_q\le 4\pi (1-g_S)\le 4\pi$, where $g_S$ is the genus of the connected orientable surface (the number of handles). If this integral is positive, $g_S=0$. For a disconnected closed surface apply these assertions to each connected component satisfying the hypotheses; in particular $A_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}\ge 4\pi {\sum }_j(Q_{E,j}^2+Q_{M,j}^2)$. This does not assert a bound by the square of the sum of component charges.

For a surface lying in a spacelike section, a supplied smooth positive supersolution $f$ of its outward graph operator, $\mathcal{L}f\ge 0$, verifies the spacetime-stability hypothesis by the slice bridge in the spacetime-stably-outermost definition: $\mathrm{\ell }=n+\nu$, $k=(n-\nu )\mathrm{/}2$ and $X=f\nu =(f\mathrm{/}2)\mathrm{\ell }-fk$. Nonnegativity of a principal eigenvalue suffices only together with an applicable theorem furnishing its smooth positive eigenfunction. No such spectral theorem is silently assumed here.

General spacelike two-surfaces, null gauges and fluxes

Normal deformation families and an outgoing null gauge

Outgoing expansion variation at a general MOTS

MOTS boosts preserve the spacetime stability sign

Spacetime stably outermost condition and the slice bridge

The Axiom of Countable Choice (AC_ω)

Pointwise Maxwell and matter energy-condition vocabulary

Metric compatibility, torsion freedom and the metric-selected connection class

The area–charge theorem and the stronger squared-density bound are imported from Dain–Jaramillo–Reiris Theorem 2.1 and equation (31), not claimed as locally reconstructed proofs. The surface expansion and its variation use the same Levi–Civita connection ${\mathrm{\nabla }}^g$ as the curvature in the field equation, matching the opening convention of source §3. The source's null normals satisfy $g(\mathrm{\ell },k)=-1$, its MOTS condition is ${\theta }_{(\mathrm{\ell })}=0$, and its Definitions 3.1–3.2 require a positive stability weight in an outgoing non-timelike normal direction. The field equation has the sign $G+\mathrm{\Lambda }g=8\pi (T^{\mathrm{E}\mathrm{M}}+\tau )$ and the factor $1\mathrm{/}(4\pi )$ in $T^{\mathrm{E}\mathrm{M}}$. Under the explicitly assumed Countable Choice, the local helpers supply the general null variation and its exact relation to the slice graph derivative, so the import does not restrict the source's surface to a spacelike section. Connectedness is explicit for the genus and single-surface inequality; componentwise application preserves the valid disconnected scope. The field-law and energy hypotheses must be established in each application; this import neither establishes them nor proves spectral stability of a particular surface.