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Regular Lorentzian metric from the normalized rank-one map

Statement

Assume the normalized family in Lifting a normalized stationary primitive to the Harrison parameter, with $D_{q} > 0$ and $D_{q} (p) > 0$ . Define $A_{q} v = v + D_{q}^{- 1} W_{q} (v) T$ and $g_{q} = (D_{q} / 4) A_{q}^{*} g$ . Then $A_{q}$ is orientation preserving and invertible, $g_{q}$ is Lorentzian and smooth through $f = 0$ , and $g_{q} = \frac{1}{4} (D_{q} g + T^{♭} \otimes W_{q} + W_{q} \otimes T^{♭} - (f / D_{q}) W_{q} \otimes W_{q})$ . Also $g_{1} = g$ and $g_{q} (T, T) = D_{q} (p)^{2} g (T, T) / (4 D_{q})$ ; in particular its sign is unchanged.

Facts & Assumptions

[F1]

Metric duals and $f$ use Stationary Lorentzian data and tensor conventions, and fiberwise algebraic pullbacks use Complex self-dual forms and metric-dependent contraction.

[F2]

The one-forms and their contraction normalization are supplied by Lifting a normalized stationary primitive to the Harrison parameter.

[F3]

The smooth denominator is $D_{q} = ∣ 1 + q - (1 - q) σ ∣^{2}$ for the smooth scalar $σ = f + i ψ$ and $q \in [0, 1]$ , as in Positive Harrison denominator and its parameter identity. In particular, evaluating its defining formula at $q = 1$ gives $D_{1} = ∣ 2 ∣^{2} = 4$ .

Proof

**Given:**The hypotheses in the Statement and the Facts below.

Goal: Establish the stated conclusions.

1.1givenF1F2

Write $D = D_{q}$ , $C = D_{q} (p)$ and $W = W_{q}$ . Since $D + W (T) = C$ , direct multiplication gives $(I + T \otimes W / D)^{- 1} = I - T \otimes W / C$ . If $T \neq 0$ , a basis starting with $T$ makes this rank-one map triangular with diagonal $(C / D, 1, 1, 1)$ ; if $T = 0$ , it is the identity and $C = D$ . Thus its determinant is $C / D > 0$ . Invertible algebraic pullback and positive rescaling preserve Lorentz signature.

2.1givenF1F2F3algebrastep 1.1∎

Expanding $g (A_{q} u, A_{q} v)$ gives the stated tensor formula because $g (T, T) = - f$ . Its only denominator is the positive $D_{q}$ , so it is smooth through $f = 0$ . The identity $W_{1} = 0$ from [F2] and the evaluation $D_{1} = 4$ in [F3] give $g_{1} = g$ . Finally $A_{q} T = (C / D) T$ , hence the asserted contraction and sign identity. All coefficients are smooth in $q$ and spacetime wherever $D_{q} > 0$ .

Source notes

This conditional calculation follows §3.2 (3.11), (3.20)–(3.23) of the cited independent source. No existence of the stationary primitive or Einstein–Maxwell solution is inferred from this calculation alone.

Depends on

Used by · 2

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

External dependencies

Alexandru F. Radu, Rigidity of stationary black holes with small ADM angular momentum · Section 3.2, equations (3.11), (3.12), (3.20)–(3.23) and the paragraph immediately following (3.23).

The normalized rank-one factorization has determinant D_q(p)/D_q, the stated inverse and Lorentz signature; its tensor expansion is regular at f=0, its stationary-field contraction preserves sign, and W_1=0 with D_1=4 recovers the original metric.

Hypotheses: The source uses its smooth stationary vacuum data and normalized parameter one-form with W_q(T)=D_q(p)-D_q and positive D_q and D_q(p). Here only the smooth conditional algebraic data supplied by the declared local dependencies are used.

Reading limits: Consulted the cited HTML equations and adjacent explanations only. This reference attributes the existing rank-one calculation; no field equations, primitive existence or global extension result is imported. The inverse, determinant, expansion and contractions are explicitly calculated in the local proof.

Sources

Alexandru F. Radu, Rigidity of stationary black holes with small ADM angular momentum · Section 3.2, equations (3.11), (3.12), (3.20)–(3.23) and the paragraph immediately following (3.23).