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Inverse Harrison relation and Killing-scalar identity

PDF p. 28: (5.2) and contraction display

Statement

Adopt the stationary vacuum model and regular non-degenerate stationary construction, with original exterior section spacelike and maximal, ADM mass one, and stationary Killing field $T$ not identically zero on the bifurcation sphere $S$ . Use the supplied normalized Ernst potential $σ = f + i ψ$ , where $f = - g (T, T)$ and $ψ$ is real, and the single regular Harrison family with its invariant one-form $W_{0}$ and pole $p \in S$ satisfying $T_{p} = 0$ . Let $F_{a b} = \nabla_{a} T_{b}$ , $F = F + i ⋆_{g} F$ , $D_{0} = ∣ 1 - σ ∣^{2}$ , $d_{0} = D_{0} (p)$ , and $A_{0} v = v + D_{0}^{- 1} W_{0} (v) T$ . The pullback $A_{0}^{*}$ acts on each covariant slot. Let $g_{0}$ and $F_{0}$ be the limiting Harrison metric and real electromagnetic two-form, and set $H_{0} = F_{0} + i ⋆_{g_{0}} F_{0}$ .

On $S$ the inverse Harrison relation and metric relation are

$F = - \frac{1 - σ}{1 - \overline{σ}} (A_{0}^{- 1})^{*} H_{0}, g_{0} = \frac{∣ 1 - σ ∣^{2}}{4} A_{0}^{*} g .$

With both indices on each form raised by its own metric, and with complex bilinear contraction without conjugation or a factor of one half,

$F_{a b} F_{g}^{a b} = {(\frac{1 - σ}{1 - \overline{σ}})}^{2} \frac{∣ 1 - σ ∣^{4}}{16} (H_{0})_{a b} (H_{0})_{g_{0}}^{a b} = - \frac{1}{4} (1 - σ)^{4} .$

Statement references

Stationary vacuum model and asymptotic normalization

Regular non-degenerate stationary vacuum spacetime

Spacelike sections, maximality and lapse–shift data

Mass-one convention and Harrison denominator

Killing two-form and normalized Ernst notation

Complex self-dual forms and metric-dependent contraction

Stationary Harrison one-form and metric

Harrison electromagnetic form

Invariant Ernst data and smooth K-invariant gauge

Facts & Assumptions

[A1]

The model, regular construction, maximal section and mass-one convention are as specified in the statement. Stationary vacuum model and asymptotic normalization Regular non-degenerate stationary vacuum spacetime Spacelike sections, maximality and lapse–shift data Mass-one convention and Harrison denominator

[F1]

The normalized Ernst and complex-form conventions define $F$ , $F$ , the metric Hodge stars, fiberwise pullback and the full complex bilinear contraction used above. Killing two-form and normalized Ernst notation Complex self-dual forms and metric-dependent contraction

[F2]

The invariant construction supplies the indicated smooth potential, pole and one-form with $W_{0} (T) = d_{0} - D_{0}$ . The denominator result gives $D_{0} > 0$ on a neighborhood of $S$ . Invariant Ernst data and smooth K-invariant gauge Positivity of the Harrison denominator and horizon bound

[F3]

The Harrison definitions give $g_{0} = (D_{0} / 4) A_{0}^{*} g$ and, at $q = 0$ , $H_{0} = - ((1 - \overline{σ}) / (1 - σ)) A_{0}^{*} F$ . The latter uses $H_{0} = F_{0} + i ⋆_{g_{0}} F_{0}$ . Stationary Harrison one-form and metric Harrison electromagnetic form

[F4]

For the supplied pole and positive denominator, $d_{0} > 0$ and $A_{0}$ is orientation preserving and invertible. Its metric transfer gives $(A_{0}^{*} U)_{a b} (A_{0}^{*} U)_{g_{0}}^{a b} = (4 / D_{0})^{2} U_{a b} U_{g}^{a b}$ for any complex two-form $U$ . Invertibility, Lorentz signature and stationary norm Rank-one tangent changes, signature and Hodge transfer

[F5]

At each point of $S$ , in the oriented $g_{0}$ -orthonormal frame of the limiting normal-form result, $(F_{0})_{01} = - 1$ , $(F_{0})_{23} = 0$ , $(⋆_{g_{0}} F_{0})_{01} = 0$ , $(⋆_{g_{0}} F_{0})_{23} = 1$ , and the mixed normal–tangential components of $F_{0}$ vanish. Boost invariance and limiting Maxwell normal form

Proof

Given: The model, construction and conventions in the Statement.

Goal: Prove the inverse relation and the displayed contraction identity on $S$ .

1.1givenF2F3F4algebra

By positivity of $D_{0}$ , both $1 - σ$ and $1 - \overline{σ}$ are nonzero. The map $A_{0}$ is invertible, so applying $(A_{0}^{- 1})^{*}$ to the electromagnetic formula in [F3] and multiplying by the reciprocal scalar gives the inverse relation. The metric relation is the first formula in [F3].

1.2F1F5algebra

In the frame of [F5], $H_{0, 01} = - 1$ and $H_{0, 23} = i$ , while all mixed components vanish: the Hodge star maps the normal and tangent area forms to each other in this oriented orthonormal frame. Raising with $g_{0} = diag (- 1, 1, 1, 1)$ in that frame and summing both orders of each antisymmetric index pair gives $(H_{0})_{a b} (H_{0})_{g_{0}}^{a b} = 2 ((- 1) (1) + i^{2}) = - 4$ .

2.1F3F4step 1.1step 1.2algebra∎

The transfer formula in [F4] with $U = F$ gives $(A_{0}^{*} F)_{g_{0}}^{2} = (4 / D_{0})^{2} F_{g}^{2}$ . Substitution of the $q = 0$ electromagnetic formula yields the first displayed contraction equality. Finally $D_{0} = (1 - σ) (1 - \overline{σ})$ and step 1.2 give $F_{g}^{2} = - \frac{1}{4} (1 - σ)^{4}$ . This proves the identity at every point of $S$ .

Source notes

The article passage attributes the original relations. The argument above uses the stated construction and normal-form suppliers, rather than the article's assertion as a premise.

Depends on

Used by · 2

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Sources

The uniqueness of smooth stationary black holes · PDF p. 28, equation (5.2) and the following contraction display