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A common positive Harrison domain across the compact sphere

Statement

Let $S$ be a compact embedded smooth two-dimensional submanifold of stationary Lorentzian data, with positive-definite induced metric, with $T$ tangent to $S$ , and with a supplied zero $p \in S$ of $T$ . On an open neighborhood $U$ of $S$ , suppose $σ = f + i ψ$ with $ψ : U \to R$ , and suppose a smooth stationary one-form $W_{0}$ is given (stationary means $L_{T} W_{0} = 0$ ), with $f = - g (T, T)$ , $F = \frac{1}{2} d T^{♭}$ , $d W_{0} = 4 ((f - 1) F + ψ ⋆_{g} F)$ and $W_{0} (T) = D_{0} (p) - D_{0}$ , where $D_{0} = ∣ 1 - σ ∣^{2}$ . Set $D_{q} = ∣ 1 + q - (1 - q) σ ∣^{2}$ and $W_{q} = (1 - q)^{2} W_{0} - 4 q (1 - q) T^{♭}$ . There is a common open neighborhood $O \subseteq U$ of $S$ with $D_{q} > 0$ and $D_{q} + W_{q} (T) = d_{q} > 0$ for every $q \in [0, 1]$ , where $d_{q} = D_{q} (p)$ . On $O$ , the polynomial curl and contraction identities are $d W_{q} = 4 (1 - q) (((1 - q) f - (1 + q)) F + (1 - q) ψ ⋆_{g} F)$ and $W_{q} (T) = d_{q} - D_{q}$ .

Facts & Assumptions

[F1]

The metric, forms and ZFC convention are Stationary Lorentzian data and tensor conventions, Complex self-dual forms and metric-dependent contraction and The Axiom of Choice. Stationarity of the one-form $W_{0}$ means $L_{T} W_{0} = 0$ , with the Lie derivative of a differential form defined in The Lie derivative of a differential form.

[F2]

The denominator calculation is recorded in Positive Harrison denominator and its parameter identity. Its needed algebra and neighborhood construction are repeated below; the supplier's additional smoothness and stationarity assumptions on $σ$ are not assumed here or used to invoke its full statement.

[F3]

The polynomial one-form calculation is recorded in Lifting a normalized stationary primitive to the Harrison parameter. Its contraction and curl calculations are repeated below directly from the assumed identities for $W_{0}$ , without differentiating $ψ$ or invoking the supplier under unverified scalar hypotheses.

[F4]

The surface is an embedded two-dimensional smooth submanifold with its inherited topology, as in Embedded submanifolds and slice charts. Positive-definiteness of its induced metric is the explicit spacelike assumption in the Statement.

[F5]

For real $a, b$ , $∣ a + i b ∣^{2} = a^{2} + b^{2}$ , with the real and imaginary coordinates and modulus of Real and imaginary parts, complex conjugation, and modulus.

[F6]

Smooth scalar maps include continuity in Cʳ and smooth maps between smooth manifolds. The open-neighborhood criterion for continuity is Continuity of a map of topological spaces at a point and globally.

[F7]

The exterior derivative is defined by The exterior derivative by the invariant vector-field formula. Its defining sums are linear in the form, so $d (a α + b β) = a d α + b d β$ for spacetime-constant real $a, b$ and smooth forms $α, β$ .

Proof

Given: The hypotheses in the Statement and the mathematical conventions in [F1]–[F7].

Goal: Prove the common positivity and the polynomial identities without additional hypotheses on the scalar $ψ$ .

1.1givenF1F2F4F6

By the induced-metric hypothesis and tangency, $g (T, T) \geq 0$ on $S$ , so $f \leq 0$ there. In coordinates $f = - g_{a b} T^{a} T^{b}$ is a finite sum of products of smooth functions, hence is smooth and continuous. Define $O = {x \in U : f (x) < \frac{1}{2}}$ . The open-neighborhood criterion in [F6], applied to $(- \infty, \frac{1}{2})$ , makes $O$ open in $U$ and hence in the ambient manifold. It contains $S$ and is independent of $q$ . This gives the neighborhood construction in [F2] using only the regularity of $f$ .

2.1givenF1F2F5algebrastep 1.1

For $x \in O$ and $q \in [0, 1]$ , put $a_{q} = 1 + q - (1 - q) f = 2 q + (1 - q) (1 - f)$ . Since $1 - f > \frac{1}{2}$ , $a_{q} \geq \frac{1}{2}$ (also at $q = 1$ ). By [F5], $D_{q} = a_{q}^{2} + (1 - q)^{2} ψ^{2} \geq \frac{1}{4} > 0$ . At $p$ , $T (p) = 0$ gives $f (p) = 0$ , so $d_{q} = (1 + q)^{2} + (1 - q)^{2} ψ (p)^{2} \geq 1$ . Neither estimate requires continuity or stationarity of $ψ$ .

3.1givenF1F2F3F5algebrastep 2.1

Repeating the pointwise calculation of [F2], subtract $(1 - q)^{2} D_{0}$ from $D_{q}$ using [F5]. The $ψ^{2}$ terms cancel, leaving $(1 + q - (1 - q) f)^{2} - (1 - q)^{2} (1 - f)^{2} = 4 q (1 - (1 - q) f)$ . Thus $D_{q} = (1 - q)^{2} D_{0} + 4 q (1 - (1 - q) f)$ , and $d_{q} = (1 - q)^{2} D_{0} (p) + 4 q$ . For the contraction calculation of [F3], $T^{♭} (T) = - f$ gives $W_{q} (T) = (1 - q)^{2} (D_{0} (p) - D_{0}) + 4 q (1 - q) f = d_{q} - D_{q}$ . Consequently $D_{q} + W_{q} (T) = d_{q} \geq 1$ on the same $O$ . These are pointwise algebraic identities with no scalar regularity premise.

4.1givenF1F3F7algebrastep 3.1∎

For the curl calculation of [F3], hold $q$ fixed and use [F7] and $d T^{♭} = 2 F$ to obtain $d W_{q} = 4 (1 - q)^{2} ((f - 1) F + ψ ⋆_{g} F) - 8 q (1 - q) F = 4 (1 - q) (((1 - q) f - (1 + q)) F + (1 - q) ψ ⋆_{g} F)$ . No derivative of $ψ$ is taken. The defining expression for $W_{q}$ is polynomial in $q$ with smooth spacetime coefficients. Also $D_{0} = D_{0} (p) - W_{0} (T)$ is smooth by the assumed contraction identity, and the formula in step 3.1 makes $D_{q}$ smooth jointly in spacetime and $q$ . The curl's displayed right-hand side equals the smooth form $d W_{q}$ by the assumed curl identity and the calculation just given. Hence these are smooth local identities on all of $O$ , including either side of any horizon lying in $O$ ; no restriction to an exterior is used.

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