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Invertibility, Lorentz signature and stationary norm

PDF p. 8: (2.4)–(2.5) and following display

Statement

Let $(M, g, N, T)$ be stationary Lorentzian data: $g$ is a smooth Lorentzian metric of signature $(-, +, +, +)$ , $N$ is a smooth timelike time-orientation field, and $T$ is Killing. On an open set $U$ let a normalized Ernst potential $σ = f + i ψ$ be supplied, where $f = - g (T, T)$ , and let a supplied pole $p \in U$ satisfy $T_{p} = 0$ . For $q \in [0, 1]$ put $D_{q} = ∣ q (1 + σ) + (1 - σ) ∣^{2}$ and $d_{q} = D_{q} (p)$ . On an open subset $V \subseteq U$ where $D_{q} > 0$ , suppose the smooth stationary Harrison one-form $W_{q}$ is supplied, including its normalization $W_{q} (T) = d_{q} - D_{q}$ . Define $A_{q} v = v + D_{q}^{- 1} W_{q} (v) T$ and $g_{q} = (D_{q} / 4) A_{q}^{*} g$ .

Then $d_{q} > 0$ , $\det A_{q} = d_{q} / D_{q} > 0$ , $A_{q}^{- 1} w = w - d_{q}^{- 1} W_{q} (w) T$ , and $g_{q}$ is Lorentzian. It is stationary, $L_{T} g_{q} = 0$ , and $g_{q} (T, T) = d_{q}^{2} g (T, T) / (4 D_{q})$ . The smooth field $A_{q}^{- 1} N$ defines a time orientation of $g_{q}$ for which $A_{q}$ carries future $g_{q}$ -timelike vectors to future $g$ -timelike vectors. If the supplied family is the single-potential family $W_{q} = (1 - q)^{2} W_{0} - 4 q (1 - q) T^{♭}$ , then $g_{1} = g$ .

Statement references

Stationary Lorentzian data and tensor conventions

Killing two-form and normalized Ernst notation

Stationary Harrison one-form and metric

Single-potential construction of the Harrison one-form family

Rank-one tangent changes, signature and Hodge transfer

The Lie derivative of a tensor field

Facts & Assumptions

[F1]

Stationary Lorentzian data provide the time-oriented Lorentz metric, its metric dual $T^{♭} = g (T, \cdot)$ , and a Killing field satisfying $L_{T} g = 0$ . Stationary Lorentzian data and tensor conventions

[F2]

The Ernst potential satisfies $d σ_{a} = 2 T^{b} F_{b a}$ , where $F$ is an antisymmetric complex two-form. Killing two-form and normalized Ernst notation

[F3]

The local one-form interface uses $D_{q} = ∣ q (1 + σ) + (1 - σ) ∣^{2}$ and $d_{q} = D_{q} (p)$ at a supplied zero $T_{p} = 0$ , on the domain $D_{q} > 0$ . These definitions require neither a maximal mass-one section nor a bifurcation surface. Stationary Harrison one-form and metric

[F4]

A supplied Harrison one-form is stationary and satisfies $W_{q} (T) = d_{q} - D_{q}$ ; its metric is the displayed pullback metric. Stationary Harrison one-form and metric

[F5]

For a Lorentz form, $D, d > 0$ and $W (T) = d - D$ , the rank-one map $A = I + D^{- 1} T \otimes W$ has determinant $d / D$ , inverse $I - d^{- 1} T \otimes W$ , and $(D / 4) A^{*} g$ has the original inertia. Rank-one tangent changes, signature and Hodge transfer

[F6]

The single-potential family is $W_{q} = (1 - q)^{2} W_{0} - 4 q (1 - q) T^{♭}$ . Single-potential construction of the Harrison one-form family

[F7]

For a smooth tensor field $B$ of any type, its Lie derivative along $T$ is $L_{T} B = {\frac{d}{d t} ∣}_{t = 0} Φ_{t}^{*} B$ , where $Φ_{t}$ is the local flow of $T$ . Pullback acts by the inverse differential on contravariant slots and by the differential on covariant slots. This applies to the mixed tensor $A_{q}$ as well as to $g_{q}$ . The Lie derivative of a tensor field

Proof

Given: The data, parameter, positive-denominator domain and supplied one-form in the Statement.

Goal: Establish the determinant, inverse, signature, stationarity, norm, time orientation and conditional endpoint identity.

1.1givenF1F3F4F5algebra

At the pole, $f (p) = 0$ . Expanding the squared modulus gives $d_{q} = (1 + q)^{2} + (1 - q)^{2} ψ (p)^{2} > 0$ for $q \in [0, 1]$ . At each point of $V$ , $D_{q} > 0$ and $W_{q} (T) = d_{q} - D_{q}$ , so the rank-one result applies with $D = D_{q}$ and $d = d_{q}$ . It gives the stated determinant and inverse and shows that $g_{q}$ has the Lorentzian inertia of $g$ . Smoothness follows from the smooth data and positive denominators.

1.2givenF1F2F3F4F7algebra

Antisymmetry in the Ernst equation gives $T (σ) = 2 T^{a} T^{b} F_{b a} = 0$ , hence $L_{T} D_{q} = 0$ . Also $L_{T} T = [T, T] = 0$ , while $L_{T} W_{q} = 0$ and $L_{T} g = 0$ . Therefore $L_{T} A_{q} = 0$ and differentiating the pullback expression yields $L_{T} g_{q} = 0$ .

2.1givenF1F4step 1.1algebra

The normalization gives $A_{q} T = (1 + D_{q}^{- 1} W_{q} (T)) T = (d_{q} / D_{q}) T$ . Substitution into the pullback metric gives $g_{q} (T, T) = d_{q}^{2} g (T, T) / (4 D_{q})$ . The smooth field $A_{q}^{- 1} N$ is $g_{q}$ -timelike because its squared norm is $(D_{q} / 4) g (N, N) < 0$ . For any $g_{q}$ -timelike $v$ , positivity of $D_{q} / 4$ gives $g_{q} (v, A_{q}^{- 1} N) = (D_{q} / 4) g (A_{q} v, N)$ , proving the asserted correspondence of future cones.

3.1givenF3F6step 1.1algebra∎

If the supplied one-forms are the single-potential family, its formula gives $W_{1} = 0$ ; the denominator definition gives $D_{1} = 4$ . Consequently $A_{1} = I$ and $g_{1} = g$ . This endpoint uses the specified family choice in addition to the equations for an individual Harrison one-form.

Source notes

The article passage attributes the original claims. The proof uses the stated definitions and rank-one lemma, and the endpoint identity uses the family specified in the article's equation (2.9).

Depends on

Used by · 16

Dependency tree · 2 levels

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Sources

The uniqueness of smooth stationary black holes · PDF pages 8–9, equations (2.4)–(2.5), following display, and equation (2.9)