Horizon expansion variation and Einstein–Maxwell curvature

PDF p. 26: (4.4) and following sentence

Adopt the regular stationary vacuum model with original exterior section maximal, mass normalized to one, and stationary field $T$ nontrivial on the bifurcation sphere $S$. Fix $q\in (0,1]$. Use one smooth Harrison construction on the exterior and an open neighborhood $O$ of $S$ common to the electrovacuum and horizon-preservation results, with the same normalized Ernst potential, pole and stationary primitive, in the invariant horizon gauge required by the latter. Denote its metric and electromagnetic two-form by $g_q,F_q$.

Suppose a smooth spacelike maximal section ${\mathrm{\Sigma }}_q$ with boundary $S$ and the boundary data of the horizon notation are supplied: smooth metric and two-form extensions on an open neighborhood $U_S$ of $S$, a smooth spacelike section extension across $S$, an outward unit normal, and an admissible normal-graph tube. Require that on an open neighborhood $O_S\subseteq U_S\cap O$ of $S$ the supplied metric and two-form extensions coincide with these same Harrison fields, with matching orientations and time orientations. Use the section extension and normal-graph tube near $S$ inside $O_S$.

Let $n_q$ be the future unit normal to the extended section and ${\nu }_q$ the outward unit normal to $S$ in it. Set $L_q=n_q+{\nu }_q$ and ${\underset{‾}{L}}_q=(n_q-{\nu }_q)\mathrm{/}2$, so that $g_q(L_q,{\underset{‾}{L}}_q)=-1$. With ${\mathrm{\nabla }}^{(q)}$ the Levi–Civita connection of $g_q$, define the normal connection one-form by ${\zeta }_q(X)=-g_q({\underset{‾}{L}}_q,{\mathrm{\nabla }}_X^{(q)}{L}_q)$ for $X$ tangent to $S$. The oriented area form of ${\gamma }_q=g_q{\mathrm{\mid }}_{TS}$ is $d{\mu }_q$; define $e_q,b_q$ by $({\star }_{g_q}{F}_q){\mathrm{\mid }}_{TS}=e_{q}d{\mu }_q$ and $F_q{\mathrm{\mid }}_{TS}=b_{q}d{\mu }_q$.

Let $\mathrm{̸}{\mathrm{\nabla }}^{(q)}$, $\mathrm{̸}{\mathrm{div}}^{(q)}$ and $\mathrm{̸}{\mathrm{\Delta }}^{(q)}$ be the intrinsic connection, divergence and divergence-of-gradient Laplacian of ${\gamma }_q$; divergence of a one-form means divergence of its metric-dual vector. Let $K_q$ be its Gauss curvature with the positive-round-sphere convention. The outgoing expansion is the ${\gamma }_q$-trace of $(X,Y)\mapsto g_q({\mathrm{\nabla }}_X^{(q)}{L}_q,Y)$. Pulling back the outgoing expansion of the normal graph with its future null normal $n_q+{\nu }_{q,u}$ defines ${\mathrm{\Theta }}_q(u)$; write ${\mathcal{L}}_q\varphi ={\frac{d}{dϵ}\mid }_0{\mathrm{\Theta }}_q(ϵ\varphi )$.

Then $S$ has zero outgoing expansion and, for every smooth function $\varphi$ on $S$, ${\mathcal{L}}_q\varphi =-\mathrm{̸}{\mathrm{\Delta }}^{(q)}\varphi +2{\zeta }_q\cdot \mathrm{̸}{\mathrm{\nabla }}^{(q)}\varphi +(K_q+\mathrm{̸}{\mathrm{div}}^{(q)}{\zeta }_q-\mathrm{\mid }{\zeta }_q{\mathrm{\mid }}^2-e_q^2-b_q^2)\varphi \mathrm{.}$ Moreover ${\mathrm{Ric}}_{g_q}(L_q,{\underset{‾}{L}}_q)=e_q^2+b_q^2$. The dot product and norm are taken with ${\gamma }_q$. The conclusions are conditional on the supplied section and graph data.

The original-section maximality premise is imposed on the actual physical exterior section $C={\mathrm{\Sigma }}_0\cap \mathcal{E}$ for the original metric $g$, so $H_g{\mathrm{\mid }}_C=0$. Where this result uses the original reference section ${\mathrm{\Sigma }}_1$, it denotes the closure $B=C\cup S$ of this actual section. Maximality of a separately labelled chart-side region or another supplied section is not substituted for this premise. The spherical identification and the physical stationary end chart remain independent. Regular non-degenerate stationary vacuum spacetime Spacelike sections, maximality and lapse–shift data

Stationary vacuum model and asymptotic normalization

Regular non-degenerate stationary vacuum spacetime

Spacelike sections, maximality and lapse–shift data

Mass-one convention and Harrison denominator

Stationary Lorentzian data and tensor conventions

Horizon null frame, flux densities and stability operator notation

Surface measure, flux densities and null normal gauge

Geometric electrovacuum equations and charge convention

Proposition 2.1: Einstein–Maxwell equations and regularity

Preservation of bifurcate horizons and surface gravity

Normal boost action and invariant two-forms at a fixed surface

Electrovac contraction in the normalized normal plane

General MOTS first variation before electrovac specialization

The Axiom of Choice

The article reference attributes equation (4.4) and the subsequent mixed-null Ricci identity. The deduction above uses the completed suppliers and does not use the article's assertion as a premise. The boundary compatibility is an explicit condition on the supplied data. The notation definition alone permits more general extensions and does not establish their electrovacuum equations or Killing symmetry. The proof here applies both geometric results only to the common Harrison fields specified in the Statement; no existence of compatible section or graph data is asserted.

These premises specify the generated conditional scope and are not attributed to the original article. The substantive local horizon conclusions are retained. The immutable original extraction is unchanged; in particular, original Theorem 1.1 retains its unresolved source-assessment status.