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Positive Harrison denominator and its parameter identity

Statement

On an open domain $U$ in stationary Lorentzian data, let $σ = f + i ψ$ be a smooth complex scalar, where $f = - g (T, T)$ and $ψ : U \to R$ is real-valued and smooth. Assume stationarity: both $f$ and $ψ$ are constant along each local integral curve of $T$ contained in $U$ . Let $0 \leq q \leq 1$ and set $D_{q} = ∣ 1 + q - (1 - q) σ ∣^{2}$ . If $f < 1$ on a region $E \subseteq U$ , then $D_{q} > 0$ there for every $q$ ; in particular this applies when $E$ is an exterior region. If $S \subseteq U$ is a compact embedded surface with $f \leq 0$ on $S$ , there is an open neighborhood $O \subseteq U$ of $S$ , independent of $q$ , with $D_{q} > 0$ on $O$ . Moreover $D_{q} = (1 - q)^{2} D_{0} + 4 q (1 - (1 - q) f)$ , and at a point $p \in U$ with $T (p) = 0$ , $D_{q} (p) = (1 - q)^{2} D_{0} (p) + 4 q > 0$ .

Facts & Assumptions

[F1]

The smooth metric, Killing field, local flow and real scalar $f = - g (T, T)$ use Stationary Lorentzian data and tensor conventions. The local integral curves in the stationarity assumption are the trajectories of that local flow.

[F2]

A scalar value $z = a + i b$ has real coordinates $a, b$ and squared modulus $∣ z ∣^{2} = a^{2} + b^{2}$ . These scalar operations, including real and imaginary parts, use Real and imaginary parts, complex conjugation, and modulus.

[F3]

An embedded surface means an embedded two-dimensional smooth submanifold, with the subspace topology inherited from the ambient manifold, as in Embedded submanifolds and slice charts.

[F4]

Smooth real scalar functions on $U$ are smooth maps to $R$ ; the smooth-map definition includes continuity on $U$ . Cʳ and smooth maps between smooth manifolds

[F5]

For a continuous map and an open neighborhood $V$ of its value at a point, there is an open neighborhood of the point mapped into $V$ . Continuity of a map of topological spaces at a point and globally

Proof

Given: The hypotheses in the Statement and the mathematical interfaces in [F1]–[F5].

Goal: Establish both positivity claims and the parameter identity, including positivity at a zero of $T$ .

1.1givenF1F2algebra

Write $a_{q} = Re (1 + q - (1 - q) σ) = 1 + q - (1 - q) f = 2 q + (1 - q) (1 - f)$ . At any point with $f < 1$ , this is positive: at $q = 0$ it is $1 - f > 0$ , and at $q > 0$ its first summand is positive and its second is nonnegative. Since $ψ$ is real-valued, $D_{q} = a_{q}^{2} + (1 - q)^{2} ψ^{2} \geq a_{q}^{2} > 0$ . This proves positivity on $E$ .

1.2givenF2algebra

Using the squared-modulus formula, subtract $(1 - q)^{2} D_{0}$ from $D_{q}$ . The $ψ^{2}$ terms cancel, leaving $(1 + q - (1 - q) f)^{2} - (1 - q)^{2} (1 - f)^{2}$ . Its factorization is $2 q \cdot 2 (1 - (1 - q) f) = 4 q (1 - (1 - q) f)$ , which proves the stated identity for every $q$ .

2.1givenF3F4F5step 1.1

Set $O = {x \in U : f (x) < 1}$ . Continuity of $f$ shows that $O$ is open: for each $x \in O$ , apply [F5] to the open real interval $(- \infty, 1)$ to obtain an open neighborhood of $x$ in $U$ contained in $O$ . Because $U$ is open in the ambient manifold, these neighborhoods are also ambient open. The inequality $f \leq 0$ on $S$ gives $S \subseteq O$ . This $O$ is independent of $q$ , and step 1.1 gives $D_{q} > 0$ throughout $O$ for every $q \in [0, 1]$ . In particular the asserted neighborhood exists for the compact embedded surface with its inherited topology; compactness is not needed by this explicit construction.

3.1givenF1F2algebrastep 1.2∎

If $T (p) = 0$ , then $f (p) = 0$ . Step 1.2 gives $D_{q} (p) = (1 - q)^{2} D_{0} (p) + 4 q$ , while [F2] gives $D_{0} (p) = 1 + ψ (p)^{2}$ and $D_{q} (p) = (1 + q)^{2} + (1 - q)^{2} ψ (p)^{2} \geq 1$ . This establishes positivity also at both parameter endpoints and completes all the claims.

Source notes

The denominator and parameter formulas are attributed to Radu, Section 3.1 (3.5), (3.7), and Section 3.2 (3.17). The proof here is a local calculation; the explicit neighborhood in step 2.1 replaces the source's compactness argument. No existence of a stationary primitive or Einstein–Maxwell solution is inferred from this calculation.

Depends on

Used by · 4

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

External dependencies

Alexandru F. Radu, Rigidity of stationary black holes with small ADM angular momentum · Section 3.1, equations (3.2), (3.5), (3.7); Section 3.2, equation (3.17) and the positivity paragraph after (3.19).

The cited passages define the denominator, give its real-part bound and parameter identity, and obtain a common positive neighborhood of the bifurcation sphere.

Hypotheses: The paper uses normalized stationary vacuum Ernst data with real norm and twist potentials, f<1 on the exterior and f<=0 on the bifurcation sphere. Only the scalar formulas are adapted here.

Reading limits: Consulted the named HTML passages. The conditional algebra and explicit sublevel neighborhood are proved locally below; no vacuum equation, global primitive existence, or other conclusion of the paper is imported.

Sources

Alexandru F. Radu, Rigidity of stationary black holes with small ADM angular momentum · Section 3.1, equations (3.2), (3.5), (3.7); Section 3.2, equation (3.17) and the positivity paragraph after (3.19).