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Cartan normalization at the bifurcation pole

PDF p. 8: §2.1, paragraph after (2.2)

Statement

Adopt the stationary vacuum model and regular stationary vacuum spacetime data, with bifurcation sphere $S$ , $T ∣_{S} \equiv̸ 0$ , and maximal exterior section $Σ_{0} \cap E$ . The local Hawking-field result gives a point $p \in S$ with $T_{p} = 0$ . Let $U$ be a connected open set containing $p$ on which a smooth normalized Ernst potential $σ = f + i ψ$ is supplied, where $f = - g (T, T)$ and $F_{a b} = \nabla_{a} T_{b}$ . Fix $q \in [0, 1]$ and put $D_{q} = ∣ q (1 + σ) + (1 - σ) ∣^{2}$ and $d_{q} = D_{q} (p)$ .

Suppose a smooth real one-form $W_{q}$ on $U$ is stationary, meaning $L_{T} W_{q} = 0$ , and satisfies $d W_{q} = 4 (1 - q) (((1 - q) f - (1 + q)) F + (1 - q) ψ ⋆_{g} F)$ . Then $d (W_{q} (T) + D_{q}) = L_{T} W_{q} - ι_{T} d W_{q} + d D_{q} = 0$ on $U$ , and consequently $W_{q} (T) + D_{q} = d_{q}$ throughout $U$ . In particular, smoothness at the pole forces $W_{q} (T) = d_{q} - D_{q}$ ; the conclusion is conditional on the supplied one-form and does not assert its existence.

Statement references

Stationary vacuum model and asymptotic normalization

Regular non-degenerate stationary vacuum spacetime

Spacelike sections, maximality and lapse–shift data

Local Hawking and axial fields, boost and pole

Stationary Lorentzian data and tensor conventions

Killing two-form and normalized Ernst notation

Interior product of a form by a vector field

Cartan's magic formula

A smooth function with zero differential is constant on each connected component

Facts & Assumptions

[A1]

Adopt the stationary vacuum model and the regular spacetime hypotheses, including the stated maximal exterior section and $T ∣_{S} \equiv̸ 0$ . A maximal spacelike section has vanishing mean curvature. Stationary vacuum model and asymptotic normalization Regular non-degenerate stationary vacuum spacetime Spacelike sections, maximality and lapse–shift data

[F1]

Under these hypotheses the local Hawking-field result supplies a pole $p \in S$ at which $T_{p} = 0$ . Local Hawking and axial fields, boost and pole

[F2]

The stationary data define the smooth metric, Killing field and Levi–Civita connection. The Ernst potential convention gives $d f = 2 ι_{T} F$ and $d ψ = 2 ι_{T} ⋆_{g} F$ . Stationary Lorentzian data and tensor conventions Killing two-form and normalized Ernst notation

[F3]

By the inline definitions in the Statement, the denominator and its pole value are $D_{q} = ∣ q (1 + σ) + (1 - σ) ∣^{2}$ and $d_{q} = D_{q} (p)$ .

[F4]

Cartan’s identity for a smooth one-form is $L_{T} W = d (ι_{T} W) + ι_{T} d W$ . Cartan's magic formula

[F5]

A smooth real-valued function whose differential vanishes is constant on each connected component of its manifold. A smooth function with zero differential is constant on each connected component

[F6]

For a smooth one-form $W$ and smooth two-form $B$ , the interior product inserts $T$ into the first slot: $ι_{T} W = W (T)$ and $(ι_{T} B) (V) = B (T, V)$ . Interior product of a form by a vector field

Proof

Given: The model and data, connected set $U$ , potential, parameter and one-form in the Statement.

Goal: Prove the differential identity and the forced value of the contraction.

1.1givenF2F3F6algebra

Write $a = (1 - q) f - (1 + q)$ . The denominator is $D_{q} = a^{2} + (1 - q)^{2} ψ^{2}$ , so $d D_{q} = 2 (1 - q) (a d f + (1 - q) ψ d ψ)$ . Using the two Ernst identities and the assumed equation for $d W_{q}$ gives $ι_{T} d W_{q} = 2 (1 - q) (a d f + (1 - q) ψ d ψ) = d D_{q}$ .

2.1givenA1F1F3F4F5F6step 1.1∎

Cartan’s identity and stationarity give $d (W_{q} (T) + D_{q}) = L_{T} W_{q} - ι_{T} d W_{q} + d D_{q} = 0$ . By [F5], this smooth real-valued function is constant on the connected manifold $U$ . At the supplied pole, smoothness makes $W_{q}$ an ordinary covector and $T_{p} = 0$ , so $W_{q} (T) (p) = 0$ . Its constant value is therefore $D_{q} (p) = d_{q}$ , proving the stated normalization.

Source notes

The article citation attributes the original normalization claim. The argument above uses the explicitly supplied potential equation, the one-form equation, Cartan’s identity and the zero-differential constancy proposition.

Depends on

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Sources

The uniqueness of smooth stationary black holes · PDF page 8, §2.1, equations (2.1)–(2.2) and the following paragraph