Surface measure, flux densities and null normal gauge

Work under ${\mathrm{A}\mathrm{C}}_{\omega }$ for the normal-graph construction and chart-partition integration. Let $(M,g)$ be a smooth oriented, time-oriented Lorentzian four-manifold with signature $(-,+,+,+)$, let $\mathrm{\Sigma }$ be a smooth spacelike hypersurface, and let $S\subset \mathrm{int}\mathrm{\Sigma }$ be a smooth oriented embedded spacelike two-surface without boundary with a supplied smooth outward unit normal $\nu$ in $h=g{\mathrm{\mid }}_{T\mathrm{\Sigma }}$. The future unit normal of $\mathrm{\Sigma }$ is $n$. The induced metric on $S$ is $q$, and $d{\mu }_q$ denotes its positive oriented area form (its associated positive density is used in unsigned area integrals). Define $L=n+\nu ,\underset{‾}{L}=(n-\nu )\mathrm{/}2,g(L,\underset{‾}{L})=-1.$ For $U\in TS$ set $\zeta (U)=-g(\underset{‾}{L},{\mathrm{\nabla }}_{U}L)$, where $\mathrm{\nabla }$ is the spacetime metric connection. For any supplied smooth real two-form $F$ on $M$, its magnetic and electric flux densities are the unique smooth functions $b,e$ on $S$ specified by $F{\mathrm{\mid }}_{TS}=bd{\mu }_q,({\star }_{g}F){\mathrm{\mid }}_{TS}=ed{\mu }_q\mathrm{.}$ Let $V_q$ be the Radon Borel measure determined by the positive Riemannian density $\mathrm{\mid }d{\mu }_q\mathrm{\mid }$. Here integrable means absolutely integrable for $V_q$: ${\int }_S\mathrm{\mid }f\mathrm{\mid }d{V}_q<\mathrm{\infty }$. When $b$ and $e$, respectively, are integrable, define their magnetic and electric fluxes by ${\int }_{S}bd{V}_q$ and ${\int }_{S}ed{V}_q$. For compact $S$ this holds automatically, and these fluxes equal the oriented top-form integrals ${\int }_{S}F{\mathrm{\mid }}_{TS}$ and ${\int }_S({\star }_{g}F){\mathrm{\mid }}_{TS}$. For noncompact $S$, the same notation for an absolutely integrable top form is interpreted by the stated density integral. The orientation of $M$ fixes ${\star }_g$, and the supplied orientation of $S$ fixes the signs of $b,e$.

All surface derivatives use the Levi–Civita connection $D^q$ of $q$: $q({\mathrm{grad}}_{q}f,X)=df(X)$, ${\mathrm{div}}_{q}X=\mathrm{tr}(Y\mapsto D_Y^{q}X)$, and ${\mathrm{\Delta }}_{q}f={\mathrm{div}}_q{\mathrm{grad}}_{q}f$. On one-forms, divergence means divergence of the $q$-dual vector field. The Gauss curvature is $K_q=q(R^q(e_1,e_2)e_2,e_1)$ for a $q$-orthonormal frame, with $R^q(X,Y)Z=D_X^q{D}_Y^{q}Z-D_Y^q{D}_X^{q}Z-D_{[X,Y]}^{q}Z$; this is the positive-round-sphere convention.

Fix an admissible outward normal-graph tube $T(p,r)={\exp }_p^h(r{\nu }_p)$ as in the graph-domain lemma; on compact $S$ such a uniform tube exists. For every admissible $u$, let $F_u(p)=T(p,u(p))$, let ${\nu }_u$ be the outward graph unit normal and set $L_u=n{\mathrm{\mid }}_{F_u(S)}+{\nu }_u$. Let $\mathrm{\Theta }(u)=F_u^{\ast }({\theta }_{L_u})$, where ${\theta }_{L_u}={\mathrm{tr}}_{q_u}g(\mathrm{\nabla }{L}_u,\cdot )$ is the outgoing expansion and $q_u=F_u^{\ast }h$. Define the normal-graph expansion linearization by $\mathcal{L}\varphi ={\frac{d}{dϵ}\mid }_0\mathrm{\Theta }(ϵ\varphi )\mathrm{.}$ For closed $S$ (compact without boundary) this acts on every smooth function. On a noncompact $S$ a global variation needs an actual common admissible parameter interval; compactly supported directions in a supplied tube suffice, while the local differential expression is defined on all smooth germs. When ${\theta }_L=0$ identically, $S$ is a MOTS and $\mathcal{L}$ in this fixed gauge is called its outward stability operator in $\mathrm{\Sigma }$. It is a linear differential operator of order at most two. Smooth boosts fixing the base $L$ do not change this variation at a MOTS; changing its base positive scale multiplies it by that scale. The explicit curvature and matter coefficient formula is a separate first-variation result, not part of this definition.

The orthogonality and normalization in [F1] give the displayed null pairing, as also calculated in [F4]. Covariant derivatives in directions tangent to $S$ are independent of any ambient extensions by [F1], so $\zeta$ is a smooth one-form. By [F2], every smooth two-form on the oriented two-dimensional $S$ is a unique scalar multiple of $d{\mu }_q$: in a positive chart the scalar is its coefficient divided by $\sqrt{\det q}$. The nonzero denominator proves existence, uniqueness and smoothness of $b,e$; the top-form transformation law makes these ratios agree on overlaps. The positive density $\mathrm{\mid }d{\mu }_q\mathrm{\mid }$ gives $V_q$ by [F7], and smooth $b,e$ are Borel functions. Absolute integrability in [F7] defines each finite signed flux even on a noncompact $S$. If $S$ is compact, every smooth function on $S$ has compact support, so the smooth compact-support comparison in [F7] proves its integrability. In a positive chart, the coefficient of $fd{\mu }_q$ and that of the signed density $f\mathrm{\mid }d{\mu }_q\mathrm{\mid }$ are both $f\sqrt{\det q}$. Their partition-weighted chart integrals therefore coincide by [F6] and [F7]. Taking $f=b,e$ proves the two compact oriented-flux equalities; on noncompact $S$, the $L^1$ chart formula in [F7] gives the declared absolutely convergent extension. Countable choice is used in the inherited chart-partition integration constructions of [F6] and [F7]. The intrinsic connection and tensorial contractions in [F3] define the displayed gradient, divergence and Laplacian independently of coordinates. The curvature definition is antisymmetric in its first two slots. Applying metric compatibility to the commutator $D_X^q{D}_Y^q-D_Y^q{D}_X^q-D_{[X,Y]}^q$ acting on $q(Z,W)$ gives $q(R^q(X,Y)Z,W)+q(Z,R^q(X,Y)W)=0$, proving antisymmetry in the final two slots. Under a change of orthonormal frame, these two antisymmetries make the displayed value multiply by the square of the frame determinant, which is one in dimension two. Thus $K_q$ is well defined even under an orientation-reversing frame change. Finally [F4] gives actual admissible graph domains and a smooth fixed gauge, and [F5] proves well-definedness, linearity, path independence and the stated restrictions on boost invariance of the derivative. No Einstein equation, compactness without explicit mention, geodesic completeness or arbitrary parameter-dependent null normalization is presumed.