General MOTS first variation before electrovac specialization

Let $S$ be a smooth spacelike codimension-two surface in a smooth time-oriented Lorentzian four-manifold, with null-expansion conventions as in Null expansions, screen quotients and focal data. Choose smooth future null normals $l,k$ with $g(l,k)=-2$, and assume the outgoing expansion ${\theta }_l=0$ identically. For a smooth normal field $v$ with $g(l,v)=1$, write $v=-k\mathrm{/}2+Vl$, $u=k\mathrm{/}2+Vl$, $s_A=-g(k,{\mathrm{\nabla }}_{A}l)\mathrm{/}2$, and $Y=V\mathrm{\mid }{\chi }_l{\mathrm{\mid }}^2+G(l,u)$, where $G=\mathrm{Ric}-{\mathrm{Scal}}_{g}g\mathrm{/}2$ and $\mathrm{\mid }{\chi }_l{\mathrm{\mid }}^2$ uses the positive surface metric. For each smooth real function $\varphi$ under consideration, supply a smooth map $\mathrm{\Phi }:S\times I\to M$ on a common open interval $I\ni 0$, with ${\mathrm{\Phi }}_0$ the inclusion, ${\mathrm{\partial }}_{\sigma }\mathrm{\Phi }{\mathrm{\mid }}_0=\varphi v$, and each ${\mathrm{\Phi }}_{\sigma }$ a spacelike immersion. Supply also a smooth field $l_{\sigma }(p)\in T_{{\mathrm{\Phi }}_{\sigma }(p)}M$ of future nonzero null normals to these slices, with $l_0=l$. Let ${\theta }_{\sigma }$ denote their null expansion pulled back to $S$; equivalently, it is the ${\mathrm{\Phi }}_{\sigma }^{\ast }g$-trace of $(X,Y)\mapsto g({\mathrm{\nabla }}_X^{{\mathrm{\Phi }}_{\sigma }}{l}_{\sigma },d{\mathrm{\Phi }}_{\sigma }(Y))$, where ${\mathrm{\nabla }}^{{\mathrm{\Phi }}_{\sigma }}$ is the pullback connection. Define ${\delta }_{\varphi v}{\theta }_l={{\mathrm{\partial }}_{\sigma }{\theta }_{\sigma }\mid }_0$. At the stated MOTS this derivative depends only on $\varphi v$ and the base normal $l$, not on the other choices of the supplied family and changing null gauge. On a noncompact $S$, existence of a common interval is an explicit requirement; the same formula applies locally on any supplied surface patch.

For such a supplied variation, ${\delta }_{\varphi v}{\theta }_l=L_v\varphi$ with $L_v\varphi =-{\mathrm{\Delta }}_S\varphi +2s\cdot D\varphi +(R_S\mathrm{/}2-Y-\mathrm{\mid }s{\mathrm{\mid }}^2+{\mathrm{div}}_{S}s)\varphi \mathrm{.}$ Here $R_S$ is the intrinsic scalar curvature, so $R_S\mathrm{/}2$ is the positive-round-sphere Gauss curvature in two dimensions. Use the sign agreement in Agreement of expansion and normal-connection conventions. In the outward hypersurface gauge Surface measure, flux densities and null normal gauge, take $l=L$, $k=2\underset{‾}{L}$, $v=\nu$; then $s=\zeta$ and $Y=\mathrm{\mid }{\chi }_L{\mathrm{\mid }}^2\mathrm{/}2+G(L,n)$ by Outward unit variation in the normalized null basis and General curvature coefficient in the outward hypersurface gauge. Thus at a geometric electrovac surface, if additionally ${\chi }_L=0$ and $\mathrm{Ric}(L,L)=0$, the zeroth coefficient is $K_S+\mathrm{div}\zeta -\mathrm{\mid }\zeta {\mathrm{\mid }}^2-e^2-b^2$, by Electrovac contraction in the normalized normal plane. The null Ricci condition is supplied by Zero null optics supplies the missing null Ricci condition when its actual open null-congruence hypotheses hold. Mere MOTS status does not justify this last specialization.

The general first-variation formula, including the curvature coefficient Y, is imported from Andersson–Mars–Simon, §3 Lemma 3.1 and Lemma 3.2, equations (2)–(7), printed pp. 7–8. The general immersed-surface family, its varying null normal and the MOTS independence assertion are imported from the preceding variation-family paragraph in §3, printed p. 7, and Lemma 3.1. The supplied family and its common parameter interval are explicit hypotheses in the Statement. No local derivation of the Appendix A covariant variation calculation is claimed.

The five local lemmas separate the algebraic null normalization, second-form and normal-connection signs, coefficient conversion, electrovac contraction, and null-hypersurface condition required by the application. For the outward hypersurface specialization only, the normal-graph supplier provides the actual family and its smooth null gauge, on its stated admissible domain. It is an instance of the general immersed-surface variation defined above; no such family for an arbitrary normal $v$ is obtained from that supplier. The article-specific horizon hypotheses must be supplied by the consumer; this input makes no claim that an arbitrary MOTS has zero null second form or zero null Ricci contraction.