General spacelike two-surfaces, null gauges and fluxes

Assume Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $(M,g,N)$ be a smooth oriented, time-oriented Lorentzian four-manifold with signature $(-,+,+,+)$ and a smooth metric connection $\mathrm{\nabla }$, meaning any $g$-compatible affine connection; torsion freedom is not assumed. Let $i:S\hookrightarrow M$ be a nonempty compact oriented smooth embedded two-manifold without boundary with positive-definite induced metric $q=i^{\ast }g$. No containing spacelike hypersurface is required. Supply a global ordered pair of future null normal fields $\mathrm{\ell },k$ with $g(\mathrm{\ell },k)=-1$; the choice of $\mathrm{\ell }$ labels the outgoing direction. The library convention $g(\mathrm{\ell },k)=-2$ is converted by halving its second normal.

For tangent fields $u,v$ on $S$, identify them with their images under $di$ in metric pairings and interpret ${\mathrm{\nabla }}_u\mathrm{\ell }$ as $(i^{\ast }\mathrm{\nabla }{)}_u\mathrm{\ell }$ in the pullback tangent bundle $i^{\ast }TM$. Put ${\chi }_{\mathrm{\ell }}(u,v)=g({\mathrm{\nabla }}_u\mathrm{\ell },v)$, ${\theta }_{(\mathrm{\ell })}={\mathrm{tr}}_q{\chi }_{\mathrm{\ell }}$ and $\mathrm{\Omega }(u)=-g(k,{\mathrm{\nabla }}_u\mathrm{\ell })$. A marginal outer trapped surface (MOTS) means ${\theta }_{(\mathrm{\ell })}=0$ everywhere; no sign for the other null expansion is added. Write $d{\mu }_q$ for the positive area density and $A={\int }_{S}d{\mu }_q$. For a smooth real two-form $F$, use the chosen orientations to define $F{\mathrm{\mid }}_{TS}=bd{A}_q$ and $({\star }_{g}F){\mathrm{\mid }}_{TS}=ed{A}_q$, where $d{A}_q$ is the positive oriented area form. Set $Q_M=(4\pi {)}^{-1}{\int }_{S}F=(4\pi {)}^{-1}{\int }_{S}bd{\mu }_q$ and $Q_E=(4\pi {)}^{-1}{\int }_S{\star }_{g}F=(4\pi {)}^{-1}{\int }_{S}ed{\mu }_q$.

Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model

Null normal pairs and screen geometry

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

Riemannian volume form on an oriented manifold

Integration on an oriented embedded submanifold

Lorentzian exterior calculus and Hodge star

Affine connection on a smooth manifold

Pullback connection

Pullback connection is well defined and functorial

Integral of a compactly supported smooth density

Orientation-free density integration and its properties

The Axiom of Countable Choice (AC_ω)

Given: The objects and hypotheses in the Definition, including Countable Choice.

Goal: Establish the assertions and well-definedness stated there.

1.1 Multiplying the library second null normal by $1\mathrm{/}2$ changes its pairing with the first from $-2$ to $-1$ and preserves its future null direction. These are the supplied smooth normal fields; no global pair is inferred from local existence. [given, A1, F1]

1.2 The area form $d{A}_q$ is nowhere zero and smooth. Both pulled-back two-forms are smooth, since $F$, the Hodge star and $i$ are smooth. Thus each is uniquely a smooth scalar multiple of $d{A}_q$: in a positive surface chart, divide its coefficient by $\sqrt{\det q}>0$. The ratios agree on overlaps by the top-form transformation law, giving global smooth $b,e$. [A1, F2]

2.1 Apply the pullback construction to the connection on $TM$ and the smooth embedding $i$. In any pulled-back frame $e_a$, write $\mathrm{\ell }={\mathrm{\ell }}^a{e}_a$ and let ${\omega }^a{}_b$ be the ambient connection matrix. Then $(i^{\ast }\mathrm{\nabla }{)}_u\mathrm{\ell }$ has coefficients $u({\mathrm{\ell }}^a)+{\omega }^a{}_b(di(u)){\mathrm{\ell }}^b$, which are smooth and depend only on the section along $S$. Frame independence is supplied by the pullback theorem. If a local ambient extension of $\mathrm{\ell }$ is used, the section identity gives the same formula; two such extensions agree after pullback, so their tangential derivatives agree. The connection's function-linearity in $u$, smooth metric pairing and invertibility of the positive matrix $q$ now make ${\chi }_{\mathrm{\ell }}$, its $q$-trace and $\mathrm{\Omega }$ well-defined smooth tensors or functions. In the metric-selected case, ${\chi }_{\mathrm{\ell }}$ and ${\theta }_{(\mathrm{\ell })}$ coincide with the null-screen definitions; $\mathrm{\Omega }$ is the independent local definition above. The general argument uses no torsion restriction. [A1, F1, F3, step 1.1]

2.2 Compactness of $S$ makes $d{\mu }_q$, $bd{\mu }_q$, $ed{\mu }_q$ and the pulled-back top forms compactly supported on $S$. Choose a common chart partition in positive surface charts. The localized coefficients of $bd{A}_q$ and $bd{\mu }_q$ are the same functions ${\rho }_{j}b\sqrt{\det q}$; their respective form and density integrals therefore have identical finite chart sums. Replacing $b$ by $e$ gives the second equality. The submanifold integral convention then proves both stated flux identities, with finite integrals. [A1, F2, F4, step 1.2]

3.1 The area density is nonzero and nonnegative because $S$ is nonempty and $q$ is positive definite. Strict positivity of density integration gives $A>0$. With ambient orientation fixed, reversing the surface orientation negates $d{A}_q$ and both oriented flux integrals, while leaving $d{\mu }_q$ and $A$ unchanged; equivalently $b,e$ both change sign. Thus both charges change sign and their squares are unchanged. [F2, F4, step 1.2, step 2.2] ∎