Spacetime stably outermost condition and the slice bridge

Assume Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ for the supplied general-surface and spacetime-deformation constructions and, in the slice setting below, the normal-graph construction.

For a closed MOTS with supplied global future null pair $g(\mathrm{\ell },k)=-1$, say that it is spacetime stably outermost when there are smooth functions $\bar{\gamma }\ge 0$ and $\psi >0$ such that, with $x=\bar{\gamma }\mathrm{\ell }-k$ and $X=\psi x=\gamma \mathrm{\ell }-\psi k$ where $\gamma =\psi \bar{\gamma }$, one has ${\delta }_X{\theta }_{(\mathrm{\ell })}\ge 0$ pointwise. The derivative is the general spacetime deformation derivative, not a derivative restricted by definition to a spacelike slice. Stability with respect to a specified normal direction $v$ means the same condition with $X=\psi v$ for some positive $\psi$.

When $S$ is also in a spacelike hypersurface with future unit normal $n$ and outward unit normal $\nu$ within it, use $\mathrm{\ell }=n+\nu$, $k=(n-\nu )\mathrm{/}2$. Its normal-graph operator $\mathcal{L}$ is the derivative defined in the slice supplier. If a smooth positive function $f$ satisfies $\mathcal{L}f\ge 0$, then this supplies the spacetime-stability witness $X=f\nu =(f\mathrm{/}2)\mathrm{\ell }-fk$, namely $\psi =f$ and $\bar{\gamma }=1\mathrm{/}2$. A principal-eigenvalue argument can supply such an $f$ only after its own positive-eigenfunction theorem and hypotheses have been supplied.

General spacelike two-surfaces, null gauges and fluxes

Outgoing expansion variation at a general MOTS

MOTS boosts preserve the spacetime stability sign

Surface measure, flux densities and null normal gauge

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 Under [A1], the quantified functions and the well-defined derivative of [F1] give an unambiguous predicate. The equivalence of the two displayed coefficient descriptions follows from $\gamma =\psi \bar{\gamma }$ and $\psi >0$. The normal velocity is spacelike or null by the calculation in [F2], while [F2] also makes the predicate independent of a positive rescaling of the chosen pair. [A1, F1, F2]

2.1 Under [A1], the slice supplier gives actual admissible normal-graph deformations. In the slice gauge, direct substitution gives $\mathrm{\ell }\mathrm{/}2-k=\nu$. The graph with initial velocity $f\nu$ is one of the spacetime deformation families in [F1], and its outgoing normal has precisely the same base $\mathrm{\ell }=n+\nu$. Independence of the family and of its extension gauge at a MOTS in [F1] identifies its derivative with ${\delta }_{f\nu }{\theta }_{(\mathrm{\ell })}$. Thus $\mathcal{L}f\ge 0$ supplies exactly the stated witness. This proves the bridge from an actual positive supersolution; it makes no unsupported spectral-existence assertion. [A1, F1, F3, step 1.1] ∎