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Stationary transfer of timelike curves

PDF p. 15: Proof of Lemma 3.2; (3.8); PDF p. 16: Proof of Lemma 3.2; (3.8)

Statement

Assume the regular stationary vacuum model, with the actual exterior section $C = Σ_{0} \cap E$ maximal, mass normalized to one, and $T ∣_{S} \equiv̸ 0$ . Let $E$ be the open exterior and write its stationary product as $E ≅ R \times C$ , with flow $ϕ_{s}^{T} (t, x) = (t + s, x)$ . Supply the smooth normalized Ernst potential $σ = f + i ψ$ and, for a fixed $q \in [0, 1]$ , a smooth stationary Harrison one-form $W_{q}$ on $E$ , with the pole $p$ at which $T = 0$ and the associated $D_{q}$ , $d_{q}$ , $A_{q}$ and $g_{q}$ as in Stationary Harrison one-form and metric. Orient $g_{q}$ so that $A_{q}$ sends its future timelike vectors to future $g$ -timelike vectors.

For any smooth future $g$ -timelike curve $γ : [0, 1] \to E$ , put $α_{q} (λ) = - d_{q}^{- 1} \int_{0}^{λ} (W_{q})_{γ (μ)} (γ^{'} (μ)) d μ$ and $γ_{q} (λ) = ϕ_{α_{q} (λ)}^{T} (γ (λ))$ . Then $γ_{q} (0) = γ (0)$ and, if $γ (λ) = (t (λ), x (λ))$ in the product, $γ_{q} (λ) = (t (λ) + α_{q} (λ), x (λ))$ . Moreover $γ_{q}^{'} = d ϕ_{α_{q}}^{T} (A_{q}^{- 1} γ^{'})$ and $(g_{q})_{γ_{q}} (γ_{q}^{'}, γ_{q}^{'}) = \frac{D_{q} (γ)}{4} g_{γ} (γ^{'}, γ^{'}) < 0$ . Thus $γ_{q}$ is future $g_{q}$ -timelike.

Facts & Assumptions

[A1]

The regular stationary vacuum assumptions and maximality of the actual exterior section $C = Σ_{0} \cap E$ are those in Stationary vacuum model and asymptotic normalization, Regular non-degenerate stationary vacuum spacetime and Spacelike sections, maximality and lapse–shift data. Work in ZFC, including countable choice as used by the calculus supplier. The Axiom of Choice

[F1]

Under maximality of the actual exterior section $C$ , the stationary product result identifies $E$ with $R \times C$ by the complete flow of $T$ . In particular all real flow times preserve $E$ , and $f < 1$ throughout $E$ . Remark 1.3: lapse bounds and stationary product

[F2]

The normalized Ernst data give $f = - g (T, T)$ and $σ = f + i ψ$ ; the normalization defines $D_{q} = ∣ q (1 + σ) + (1 - σ) ∣^{2}$ and $d_{q} = D_{q} (p)$ . Stationary Lorentzian data and tensor conventions Killing two-form and normalized Ernst notation Mass-one convention and Harrison denominator

[F3]

The supplied one-form is stationary and satisfies $W_{q} (T) = d_{q} - D_{q}$ . The associated map is $A_{q} v = v + D_{q}^{- 1} W_{q} (v) T$ , and $g_{q} = (D_{q} / 4) A_{q}^{*} g$ . Stationary Harrison one-form and metric

[F4]

On any open subset where $D_{q} > 0$ , $d_{q} > 0$ , $A_{q}^{- 1} v = v - d_{q}^{- 1} W_{q} (v) T$ , $g_{q}$ is stationary and Lorentzian, and the stated transported time orientation is well defined. The stationary flow preserves $g$ , $D_{q}$ and $A_{q}$ . Invertibility, Lorentz signature and stationary norm

[F5]

The chain rule applies to smooth maps of manifolds. The fundamental theorem of calculus differentiates the integral of a continuous real-valued function, including one-sided endpoint derivatives. The chain rule for differentials of smooth maps Fundamental theorem of calculus for Banach-valued continuous curves

Proof

Given: The model, supplied Harrison data and future $g$ -timelike curve in the Statement.

Goal: Establish the corrected curve's initial point, spatial path, tangent, norm and future direction.

1.1givenF1F2F3F4algebra

The assumed maximality is $H = 0$ on the actual exterior section $C = Σ_{0} \cap E$ , so it matches the conditional stationary-product and norm conclusions of [F1]. No identification with the specified outer side of the hypersurface chart is required. By [F1], $f < 1$ throughout $E$ . Put $B_{q} = q (1 + σ) + (1 - σ)$ . By [F2], $Re B_{q} = 2 q + (1 - q) (1 - f)$ . For $0 \leq q < 1$ the second term is positive and the first is nonnegative; for $q = 1$ the real part is $2$ . Thus $D_{q} = ∣ B_{q} ∣^{2} \geq (Re B_{q})^{2} > 0$ throughout $E$ . The supplied Ernst data at the pole and on the exterior, together with the supplied Harrison one-form, now meet the positive-domain hypotheses of [F4] on the exterior. No Ernst-potential domain containing the whole bifurcation sphere is used.

2.1givenA1F1F3F4F5step 1.1

The integrand $λ \mapsto W_{q} (γ^{'} (λ))$ is smooth, and $d_{q} > 0$ . By the fundamental theorem of calculus, $α_{q} (0) = 0$ and $α_{q}^{'} = - d_{q}^{- 1} W_{q} (γ^{'})$ , with one-sided derivatives at the endpoints. The complete product flow therefore defines $γ_{q}$ throughout $[0, 1]$ ; its product coordinates give the claimed initial point and spatial path.

3.1givenF1F4F5step 2.1algebra

Apply the chain rule to $(s, z) \mapsto ϕ_{s}^{T} (z)$ and $(α_{q}, γ)$ . Since $\partial_{s} ϕ_{s}^{T} (z) = T_{ϕ_{s}^{T} (z)} = d ϕ_{s}^{T} (T_{z})$ , we obtain $γ_{q}^{'} = d ϕ_{α_{q}}^{T} (γ^{'} + α_{q}^{'} T) = d ϕ_{α_{q}}^{T} (A_{q}^{- 1} γ^{'})$ .

4.1givenF3F4step 1.1step 3.1algebra

The stationary flow preserves $g_{q}$ , since it preserves $g$ , $D_{q}$ and $A_{q}$ . Hence the pullback formula and $A_{q} A_{q}^{- 1} = I$ give $(g_{q})_{γ_{q}} (γ_{q}^{'}, γ_{q}^{'}) = (g_{q})_{γ} (A_{q}^{- 1} γ^{'}, A_{q}^{- 1} γ^{'}) = \frac{D_{q} (γ)}{4} g_{γ} (γ^{'}, γ^{'}) < 0$ . Positivity follows from step 1.1 and timelikeness from the given curve.

5.1givenF4step 3.1step 4.1∎

At each point of $γ$ , the transported orientation makes $A_{q}^{- 1} γ^{'}$ future $g_{q}$ -timelike. The flow is a continuous path of $g_{q}$ -isometries starting at the identity. Applied to a future timelike vector, its image remains timelike and continuously remains in the same time-cone component; hence each flow map preserves the chosen future cone. Step 3.1 therefore makes $γ_{q}^{'}$ future directed. This proves all the assertions.

Source notes

The cited article passage attributes the transfer and equation (3.8). The argument above uses the completed product and Harrison suppliers and the stated calculus facts; it does not use the article claim as a premise.

The actual-exterior maximality wording repairs the generated supplier contract. It uses the current conditional product result and does not identify the independent hypersurface and physical end charts or attribute this revised scope to the article.

Depends on

Used by · 4

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Sources

The uniqueness of smooth stationary black holes · PDF pages 15–16, proof of Lemma 3.2, equation (3.8)