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Boost invariance and limiting Maxwell normal form

PDF p. 28: Proof of Proposition 5.1, first paragraph

Statement

Adopt the stationary vacuum model and regular non-degenerate stationary construction, with original exterior section spacelike and maximal, ADM mass one, and stationary field $T$ not identically zero on the bifurcation sphere $S$ . Use the single regular Harrison family and invariant primitive supplied in this construction. Let $g_{0}$ and $F_{0}$ be its limiting metric and real electromagnetic two-form, and let $K$ be the local Hawking field.

The local flow of $K$ fixes $S$ , acts by a nontrivial boost on the $g_{0}$ -normal plane to $S$ , and preserves $F_{0}$ . Consequently the mixed normal–tangential components of $F_{0}$ vanish. At each point choose a $g_{0}$ -orthonormal frame $(n = e_{0}, ν = e_{1}, e_{2}, e_{3})$ in the spacetime orientation, where $n$ is future timelike, $ν$ is the outward spacelike normal, and $(e_{2}, e_{3})$ is positively oriented for the outward end-sphere orientation of $S$ . Then $(F_{0})_{23} = (⋆_{g_{0}} F_{0})_{01} = 0$ and $(⋆_{g_{0}} F_{0})_{23} = - (F_{0})_{01} = 1$ ; all other independent components of $F_{0}$ vanish. The frame label $e_{0}$ is distinct from the electric flux density $e_{0} = 1$ .

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

Local Hawking and axial fields, boost and pole

Invariant Ernst data and smooth K-invariant gauge

Proposition 2.1: Einstein–Maxwell equations and regularity

Corollary 4.1: area and pointwise limiting flux rigidity

Surface measure, flux densities and null normal gauge

Lorentzian exterior calculus and Hodge star

Facts & Assumptions

[A1]

The stated model and regularity hypotheses include the oriented stationary geometry, the bifurcation sphere, the maximal original exterior section, and the mass-one normalization. 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 local Hawking field $K$ is Killing for the original metric, commutes with $T$ , vanishes on $S$ , and has a positive-rate normal boost there. Its flow fixes $S$ pointwise and its differential is the identity on $T S$ and a nontrivial boost on the original normal plane. The local-field result also supplies a pole $p \in S$ with $T (p) = 0$ . Local Hawking and axial fields, boost and pole Normal boost action and invariant two-forms at a fixed surface

[F2]

The invariant-gauge result supplies a smooth normalized Ernst potential $σ$ and a single smooth primitive $W_{0}$ near $S$ , both invariant under $K$ . The regular Harrison result supplies the limiting Lorentzian metric and real two-form on a neighborhood of $S$ . Here $σ = f + i ψ$ , $f = - g (T, T)$ and $ψ$ is real; $W_{0}$ is stationary, $d W_{0} = 4 ((f - 1) F + ψ ⋆_{g} F)$ and $W_{0} (T) = D_{0} (p) - D_{0}$ for the pole in [F1], with $D_{0} = ∣ 1 - σ ∣^{2}$ and $F = \frac{1}{2} d T^{♭}$ . For this same normalized potential, $d σ = 2 ι_{T} (F + i ⋆_{g} F)$ with $F = \frac{1}{2} d T^{♭}$ . Closed Ernst form and existence of normalized potential Invariant Ernst data and smooth K-invariant gauge Proposition 2.1: Einstein–Maxwell equations and regularity

[F3]

Put $D_{0} = ∣ 1 - σ ∣^{2}$ and $A_{0} v = v + D_{0}^{- 1} W_{0} (v) T$ . The Harrison definitions give $g_{0} = (D_{0} / 4) A_{0}^{*} g$ and express the complex electromagnetic form as a scalar depending on $σ$ times $A_{0}^{*} (F + i ⋆_{g} F)$ , where $F = \frac{1}{2} d T^{♭}$ ; $F_{0}$ is its real part. These formulas apply where $D_{0} > 0$ . Stationary Harrison one-form and metric Harrison electromagnetic form Complex self-dual forms and metric-dependent contraction

[F4]

The limiting flux result gives $b_{0} = 0$ and $e_{0} = 1$ pointwise. The limiting-flux interface uses the Riemannian induced metric $γ_{0} = g_{0} ∣_{T S}$ on $S$ . By definition, $F_{0} ∣_{T S} = b_{0} d μ_{0}$ and $(⋆_{g_{0}} F_{0}) ∣_{T S} = e_{0} d μ_{0}$ , where $d μ_{0}$ is the area form for the stated orientation. Corollary 4.1: area and pointwise limiting flux rigidity Surface measure, flux densities and null normal gauge

[F5]

For either oriented Lorentzian metric of signature $(-, +, +, +)$ , the Hodge star is uniquely characterized by $α \land ⋆ β = ⟨ α, β ⟩ v o l$ . In a positively oriented orthonormal coframe, $⋆ (e^{0} \land e^{1}) = - e^{2} \land e^{3}$ and $⋆ (e^{2} \land e^{3}) = e^{0} \land e^{1}$ . Lorentzian exterior calculus and Hodge star

[F6]

For a compact embedded spacelike surface with tangent stationary field $T$ and a zero $p$ , the potential and stationary primitive identities in [F2] imply a common open neighborhood of that surface on which $D_{q} = ∣ 1 + q - (1 - q) σ ∣^{2} > 0$ for all $q \in [0, 1]$ ; in particular $D_{0} > 0$ there. A common positive Harrison domain across the compact sphere

Proof

Given: The model, construction, orientations and frame in the Statement.

Goal: Establish the limiting boost invariance and the displayed components.

1.1A1F1F2F3F6algebra

By [A1], $S$ is an embedded compact sphere inside the original spacelike section, so its induced $g$ -metric is positive definite. At $s \in S$ , $T_{s} S$ lies in both horizon tangent spaces; transversality makes their intersection two-dimensional, hence equal to $T_{s} S$ . Since $T$ is tangent to both branches, it is tangent to $S$ . The pole in [F1], smooth potential and stationary primitive identities in [F2] therefore meet every hypothesis of [F6]. Applied on their common neighborhood, [F6] gives an open $O$ containing $S$ with $D_{0} > 0$ . Restrict to $O$ when using [F3].

2.1F1F2F3F5step 1.1

The local flow of $K$ preserves $g$ and $T$ by [F1]. It therefore preserves $T^{♭}$ and $F = \frac{1}{2} d T^{♭}$ . Being an isometry connected to the identity, it preserves the original metric, orientation, exterior pairing and metric volume form. Pulling back the characterizing identity in [F5] and using uniqueness gives $Φ_{t}^{*} (⋆_{g} F) = ⋆_{g} (Φ_{t}^{*} F)$ , so it also preserves $⋆_{g} F$ . By [F2] it also preserves $σ$ and $W_{0}$ . Hence it preserves $D_{0}$ , $A_{0}$ , $g_{0}$ , and the complex Harrison electromagnetic form in [F3], and thus preserves its real part $F_{0}$ . These are local assertions on $O$ wherever the local flow is defined.

3.1F1F4step 2.1algebra

Fix $p \in S$ . The flow fixes $p$ and acts identically on $T_{p} S$ . Its differential preserves $g_{0}$ by step 2.1, so it preserves the $g_{0}$ -orthogonal normal plane $N_{p}$ . Since $T_{p} S$ is spacelike for $g_{0}$ by [F4], the quotient $T_{p} M / T_{p} S$ is naturally isomorphic to $N_{p}$ through orthogonal projection. On that quotient the differential has eigenvalues $e^{κ t}$ and $e^{- κ t}$ with $κ > 0$ : this follows from the original normal boost in [F1]. Consequently its action on $N_{p}$ is a nontrivial Lorentz boost with those eigenvalues. If $v_{\pm}$ are corresponding null eigenvectors and $X \in T_{p} S$ , invariance gives $F_{0} (v_{\pm}, X) = e^{\pm κ t} F_{0} (v_{\pm}, X)$ . A sufficiently small nonzero $t$ forces both values to vanish. The eigenvectors span $N_{p}$ , proving all mixed normal–tangential components vanish.

4.1F4F5step 3.1algebra∎

In the positive tangent frame, $d μ_{0} (e_{2}, e_{3}) = 1$ . Thus [F4] gives $(F_{0})_{23} = 0$ and $(⋆_{g_{0}} F_{0})_{23} = 1$ . In the stated oriented orthonormal Lorentz frame, the two basis identities in [F5] give $(⋆_{g_{0}} F_{0})_{23} = - (F_{0})_{01}$ and $(⋆_{g_{0}} F_{0})_{01} = (F_{0})_{23}$ . Hence $(F_{0})_{01} = - 1$ and $(⋆_{g_{0}} F_{0})_{01} = 0$ . Antisymmetry and step 3.1 account for every remaining independent component of $F_{0}$ .

Source notes

The article reference attributes the original normal-form claim. Its printed display uses $⋆_{g_{q}}$ while describing a $g_{0}$ -orthonormal frame; the limiting calculation above uses $⋆_{g_{0}}$ throughout. The proof uses the declared suppliers and does not use the article's assertion as a premise.

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Sources

The uniqueness of smooth stationary black holes · PDF p. 28, proof of Proposition 5.1, first paragraph