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Boost dependence of expansion variation and the MOTS gauge

Statement

Assume countable choice ${A C}_{ω}$ . Let $S$ and its canonical graph gauge be as in the linear-expansion variation lemma. Let $u (t, p)$ be a jointly smooth admissible graph path on a common open parameter interval containing $0$ , with $u (0, p) = 0$ . A marginally outer trapped surface (MOTS) here means $θ_{L_{0}} = 0$ identically on $S$ for the chosen outward future null ray. For any jointly smooth real $f (t, p)$ with $f (0, p) = 0$ , replace the normals along $F_{u (t)}$ by ${\hat{L}}_{t} = e^{f (t, p)} L_{u (t)}$ and ${\hat{\underline{L}}}_{t} = e^{- f (t, p)} {\underline{L}}_{u (t)}$ . If $\dot{u} (0) = ϕ$ , their pulled-back expansion obeys ${\partial_{t} {\hat{Θ}}_{t} ∣}_{0} = L ϕ + \dot{f} (0, \cdot) θ_{L_{0}} .$ Hence at a MOTS this variation is unchanged by every such smooth boost fixing the base normal. If the base scale changes by a smooth $f_{0} (p)$ , the variation at a MOTS is instead multiplied by $e^{f_{0}}$ ; absolute equality is not asserted for different base scales. A smooth reparametrization of the graph by diffeomorphisms $η_{t} : S \to S$ with $η_{0} = i d$ adds $d θ_{L_{0}} ({\dot{η}}_{0})$ , which is also zero at a MOTS.

Facts & Assumptions

[F1]

Canonical outward normals and null gauge on a graph fixes the base normal $L_{0} = n + ν$ and the graph gauge.

[F2]

Under the geometric and graph-domain hypotheses inherited in the Statement and countable choice in [F6], Linear and path-independent normal-graph variation of expansion supplies the canonical variation $L ϕ$ for the stated jointly smooth admissible graph path with $u (0, p) = 0$ .

[F3]

Null normal pairs and screen geometry supplies the expansion trace and its boost scaling; The chain rule for differentials of smooth maps supplies differentiation of smooth composites.

[F4]

Affine connection on a smooth manifold supplies the connection product rule $\nabla_{X} (a Y) = X (a) Y + a \nabla_{X} Y$ . The along-surface derivative convention in [F3] permits applying this rule to local ambient extensions.

[F5]

Diffeomorphisms and local diffeomorphisms of manifolds defines the diffeomorphisms $η_{t} : S \to S$ used for reparametrization. The assumed smooth family means $(t, p) \mapsto η_{t} (p)$ is jointly smooth.

[F6]

Adopt countable choice ${A C}_{ω}$ in the sense of The Axiom of Countable Choice (AC_ω). It meets the inherited graph-construction requirements of [F2] and the smooth tangent-bundle and time-orientation conventions of [F3]. No stronger choice principle is assumed.

Proof

Given: The geometric and graph-domain data of the Statement, countable choice in [F6], the jointly smooth admissible graph path with $u (0, p) = 0$ , and the smooth boost with specified base scale.

Goal: Establish the precise extent of gauge independence.

1.1F1F2F3F4F6givenalgebra

The path hypotheses and [F6] meet the requirements of [F2] and [F3]. For any graph-tangent $X, Y$ , the product rule in [F4], applied through the along-surface convention in [F3], yields $g (\nabla_{X} (e^{f} L), Y) = e^{f} g (\nabla_{X} L, Y) + e^{f} X (f) g (L, Y)$ . The last term is zero because $L$ is normal. Taking the surface trace gives $\hat{Θ} = e^{f} Θ$ ; reciprocal scaling of the other normal preserves the pairing $- 1$ . Differentiating and using [F2] gives $\partial_{t} \hat{Θ} ∣_{0} = e^{f_{0}} (L ϕ + {\dot{f}}_{0} θ_{L_{0}})$ . Setting $f_{0} = 0$ gives the displayed formula; setting $θ_{L_{0}} = 0$ gives both base-scale assertions. Without the MOTS condition, the additional term generally remains.

2.1F2F3F5step 1.1algebra∎

Pullback by the smooth diffeomorphism family in [F5] replaces the scalar by $Θ_{t} (η_{t} (p))$ . The chain rule differentiates this to $\partial_{t} Θ_{t} ∣_{0} + d θ_{L_{0}} ({\dot{η}}_{0})$ . A function identically zero has zero differential, so the second term vanishes at a MOTS. These computations establish gauge and parameterization independence only with the stated base-scale and smoothness hypotheses.

Depends on

Used by · 1

Surface measure, flux densities and null normal gauge

Dependency tree · 2 levels

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