Horizon null frame, flux densities and stability operator notation

PDF p. 26: Proof of Theorem 4.2, definitions before (4.4)

Assume the axiom of choice, used through its countable-choice consequence in the normal-graph construction. Fix $q\in (0,1]$ and supplied Harrison data giving an oriented, time-oriented Lorentzian metric $g_q$ of signature $(-,+,+,+)$ and a smooth real electromagnetic two-form $F_q$ on the Harrison open set $V$. Let ${\mathrm{\Sigma }}_q$ be a supplied smooth spacelike maximal section with boundary $S$, as in the stationary maximal-graph setting: spacelike means that the induced metric is positive definite, and maximal means that the trace of its second fundamental form vanishes. As additional boundary data, supply an open spacetime neighborhood $U_S$ of $S$ and smooth extensions of both $g_q$ and $F_q$ to $U_S$: the extended metric is oriented, time-oriented and Lorentzian of the same signature, the extended two-form is real, and both agree with the Harrison fields on $U_S\cap V$, with matching orientations and time orientations. Write $g_q,F_q$ for these specified fields on $V\cup U_S$. Assume $S$ is a smooth oriented closed surface, and supply within $U_S$ a smooth spacelike extension of the section, a smooth outward unit normal ${\nu }_q$ to $S$ in that extension, and an admissible outward normal-graph tube. The unit normal $n_q$ to the extended section is future-directed. The orientation of $S$ fixes its area form $d{\mu }_q$; the spacetime orientation fixes the Hodge star ${\star }_{g_q}$.

Set $L_q=n_q+{\nu }_q$ and ${\underset{‾}{L}}_q=(n_q-{\nu }_q)\mathrm{/}2$, so $g_q(L_q,{\underset{‾}{L}}_q)=-1$. Both are future null normals. For tangent $X$ define the normal-connection one-form ${\zeta }_q(X)=-g_q({\underset{‾}{L}}_q,{\mathrm{\nabla }}_X^{(q)}{L}_q)$, where ${\mathrm{\nabla }}^{(q)}$ is the Levi–Civita connection of $g_q$; its components in a tangent frame $e_A$ are $({\zeta }_q{)}_A={\zeta }_q(e_A)$. Define the electric and magnetic flux densities $e_q,b_q$ by $({\star }_{g_q}{F}_q){\mathrm{\mid }}_{TS}=e_{q}d{\mu }_q$ and $F_q{\mathrm{\mid }}_{TS}=b_{q}d{\mu }_q$.

Write ${\gamma }_q=g_q{\mathrm{\mid }}_{TS}$. Let $\mathrm{̸}{\mathrm{\nabla }}^{(q)}$ be its Levi–Civita connection, $\mathrm{̸}{\mathrm{div}}^{(q)}$ the trace of the intrinsic covariant derivative, and $\mathrm{̸}{\mathrm{\Delta }}^{(q)}=\mathrm{̸}{\mathrm{div}}^{(q)}{\mathrm{grad}}_{{\gamma }_q}$ the scalar Laplacian. Divergence of a one-form means divergence of its ${\gamma }_q$-dual vector field. Let $K_q={\gamma }_q(R^{{\gamma }_q}(e_1,e_2)e_2,e_1)$ for a ${\gamma }_q$-orthonormal frame, with $R(X,Y)Z=\mathrm{̸}{\mathrm{\nabla }}_X\mathrm{̸}{\mathrm{\nabla }}_{Y}Z-\mathrm{̸}{\mathrm{\nabla }}_Y\mathrm{̸}{\mathrm{\nabla }}_{X}Z-\mathrm{̸}{\mathrm{\nabla }}_{[X,Y]}Z$.

For an admissible normal graph $S_u$ over $S$, use its outward unit normal ${\nu }_{q,u}$ in the extended section and its future outgoing null normal $L_{q,u}=n_q+{\nu }_{q,u}$. Its outgoing expansion is the trace, in the induced surface metric, of $(X,Y)\mapsto g_q({\mathrm{\nabla }}_X^{(q)}{L}_{q,u},Y)$. Pull this expansion back to $S$ along the graph map and denote it by ${\mathrm{\Theta }}_q(u)$. For smooth $\varphi$ define ${\mathcal{L}}_q\varphi ={\frac{d}{dϵ}\mid }_0{\mathrm{\Theta }}_q(ϵ\varphi )$: this is the first variation under displacement $\varphi {\nu }_q$ in the specified null gauge. If the initial outgoing expansion vanishes, $S$ is a marginally outer trapped surface, abbreviated MOTS, and ${\mathcal{L}}_q$ is its outward stability operator in this section. No coefficient formula or sign of this operator is asserted here.

For the separate comparison-sphere convention, additionally supply the stationary product end with coordinates $(t,x)\in \mathbb{R}\times \{x\in {\mathbb{R}}^3:\mathrm{\mid }x\mathrm{\mid }>a\}$, $a>0$, and $T={\mathrm{\partial }}_t$, using the same original end as the stationary graph data. For $R_0>a$, set $r=\mathrm{\mid }x\mathrm{\mid }$ and $C_q=\{t=0,r=R_0{q}^{-2\mathrm{/}3}\}$, as in the coordinate comparison-sphere definition, and require $C_q\subset V$. Assume $C_q$ is spacelike for $g_q$ and supply a smooth future outward null normal ${\mathrm{\ell }}_q$ along it satisfying $g_q(T,{\mathrm{\ell }}_q)<0$. Use the positive constant $d_q=D_q(p)$ from the stationary graph's Harrison normalization, and define ${\widehat{L}}_q=\frac{d_q}{-4g_q(T,{\mathrm{\ell }}_q)}{\mathrm{\ell }}_q,g_q(T,{\widehat{L}}_q)=-d_q\mathrm{/}4.$ This is the unique positive rescaling of the supplied ${\mathrm{\ell }}_q$ with that normalization. The horizon convention $L_q=n_q+{\nu }_q$ and the comparison convention ${\widehat{L}}_q$ are separate; the article uses the symbol $L_q$ in both settings.

Harrison electromagnetic form

Stationary maximal graphs and normalized graph heights

Surface measure, flux densities and null normal gauge

Null expansions, screen quotients and focal data

Complex self-dual forms and metric-dependent contraction

The Axiom of Choice

Comparison spheres and their stationary cylinders

Mass-one convention and Harrison denominator

Given: The supplied positive-parameter data, compatible smooth boundary extensions, orientations, normals and admissible normal-graph tube in the Definition; for the separate comparison convention, its additional end, sphere and null-normal data.

Goal: Justify the displayed notation and the existence of the stated linearization under those hypotheses.

1.1 The unit normals satisfy $g_q(n_q,n_q)=-1$, $g_q({\nu }_q,{\nu }_q)=1$ and $g_q(n_q,{\nu }_q)=0$. Consequently $L_q$ and ${\underset{‾}{L}}_q$ are null, have pairing $-1$, and are future-directed. The supplied extensions place a smooth Lorentzian metric and a smooth real two-form on the same open neighborhood $U_S$ of $S$, agreeing with the Harrison fields wherever they overlap $V$. Apply the surface-data supplier to the spacelike extension of the section in this neighborhood: it makes the normal connection one-form well defined. The smooth Hodge star there and the nonvanishing oriented area form give each restricted two-form a unique smooth scalar coefficient, defining $e_q$ and $b_q$ for these supplied fields. [given, F1, F2, F4, algebra]

2.1 The surface-data supplier gives the intrinsic connection, divergence, Laplacian and frame-independent Gauss curvature with the displayed conventions. The null-screen definition identifies the expansion used in the graph construction, and the surface-data supplier establishes existence and linearity of its normal-graph derivative in the stated gauge, with countable choice for that construction. The MOTS and stability-operator names apply under the explicitly stated vanishing condition. [given, A1, F2, F3, step 1.1]

3.1 For the separate comparison data, the coordinate supplier places $C_q$ in the supplied end, and the explicit inclusion $C_q\subset V$ makes $g_q(T,{\mathrm{\ell }}_q)$ defined. Since $d_q>0$ and $g_q(T,{\mathrm{\ell }}_q)<0$, the scalar $s_q=d_q\mathrm{/}(-4g_q(T,{\mathrm{\ell }}_q))$ is smooth and strictly positive on $C_q$. Thus ${\widehat{L}}_q=s_q{\mathrm{\ell }}_q$ remains future outward null and normal to $C_q$, and linearity gives $g_q(T,{\widehat{L}}_q)=-d_q\mathrm{/}4$. If another positive rescaling $s{\mathrm{\ell }}_q$ has this pairing, division by the nonzero $g_q(T,{\mathrm{\ell }}_q)$ forces $s=s_q$. This verifies precisely the separate normalization on the supplied outgoing ray. [given, F5, algebra] ∎

The article citation attributes the notation and the separate comparison-sphere normalization. The boundary extensions and the comparison sphere's spacelikeness and negative-pairing null normal are explicit supplied data; their existence is not asserted here. The justification uses the stated suppliers and local algebra, and does not use the article's variation formula as a premise.