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Fixing the Ernst constant by integrable asymptotic twist

Statement

In a stationary vacuum end $R \times {∣ x ∣ > R}$ with $T = \partial_{t}$ , suppose $g_{00} = - 1 + 2 M_{0} / r + O_{6} (r^{- 2})$ , $g_{i j} = δ_{i j} + O_{6} (r^{- 1})$ and $g_{0 i} = O_{6} (r^{- 2})$ . If a smooth primitive $s$ of $η = 2 i_{T} (F + i ⋆_{g} F)$ is defined on a connected domain containing the end, where $F = \frac{1}{2} d T^{♭}$ , it can be shifted by exactly one complex constant so that $s = f + i ψ \to 1$ uniformly in angle on that end, with $f = - g (T, T)$ and $ψ = O (r^{- 2})$ . The normalized primitive is stationary and unique on its connected domain.

Facts & Assumptions

[F1]

Metric, orientation, Killing and Hodge conventions are Stationary Lorentzian data and tensor conventions and Complex self-dual forms and metric-dependent contraction; differentiated decay uses Differentiated asymptotic bounds and parameter uniformity.

[F2]

In vacuum $η$ is closed, its real part is $d f$ , and $F = \frac{1}{2} d T^{♭}$ . Vacuum Killing and Ernst differential identities

Proof

**Given:**The hypotheses in the Statement and the declared supplier interfaces.

Goal: Prove the stated conclusions.

1.1givenF1F2algebra

By [F2], $Re s - f$ is constant on the connected domain. Subtract this real constant. In end coordinates, stationarity gives $F_{i j} = \frac{1}{2} (\partial_{i} g_{0 j} - \partial_{j} g_{0 i}) = O (r^{- 3})$ and $F_{0 i} = - \frac{1}{2} \partial_{i} g_{00} = O (r^{- 2})$ . The inverse metric has bounded spatial components and $g^{0 i} = O (r^{- 2})$ . In $(⋆_{g} F)_{0 i} = \frac{1}{2} ϵ_{0 i}^{a b} F_{a b}$ the purely spatial contributions are $O (r^{- 3})$ and terms involving a time index acquire a factor $g^{0 j}$ and are $O (r^{- 4})$ . Hence $\partial_{i} Im s = 2 (⋆_{g} F)_{0 i} = O (r^{- 3})$ , uniformly in angle.

2.1givenF1F2step 1.1∎

Integrating this bound along each radial ray gives a finite limit $c (ω)$ and $∣ Im s (r ω) - c (ω) ∣ \leq C r^{- 2}$ . Join two directions by a great-circle arc of length at most $π r$ on the sphere of radius $r$ ; step 1.1 bounds the difference of their imaginary values by $C π r^{- 2}$ . Passing to the radial limits proves that all $c (ω)$ are one constant $c$ , and the radial bound is uniform. Subtract $i c$ to obtain $ψ = O (r^{- 2})$ and $s \to 1$ since $f \to 1$ . Antisymmetry gives $η (T) = 2 F (T, T) = 0$ , so $T (s) = 0$ . Two primitives differ by a constant on a connected domain; their equal limit at infinity forces that constant to vanish.

Depends on

Used by · 1

Closed Ernst form and existence of normalized potential

Dependency tree · 2 levels

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