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Lifting a normalized stationary primitive to the Harrison parameter

Statement

Work on an open domain $U$ containing $p$ , with $ψ : U \to R$ smooth and real-valued, $f = - g (T, T)$ , and both $f$ and $ψ$ constant along every local integral curve of $T$ contained in $U$ . Assume the stationary Lorentzian and denominator data specified in [F1]–[F2], a real two-form $F = \frac{1}{2} d T^{♭}$ , and a smooth stationary one-form $W_{0}$ (meaning $L_{T} W_{0} = 0$ ) with $d W_{0} = 4 ((f - 1) F + ψ ⋆_{g} F)$ and $W_{0} (T) = D_{0} (p) - D_{0}$ , where $T (p) = 0$ . Define $W_{q} = (1 - q)^{2} W_{0} - 4 q (1 - q) T^{♭}$ for $q \in [0, 1]$ . Then $W_{q} (T) = D_{q} (p) - D_{q}$ , $d W_{q} = 4 (1 - q) (((1 - q) f - (1 + q)) F + (1 - q) ψ ⋆_{g} F)$ , $W_{1} = 0$ , and $W_{q}$ is smooth in spacetime and $q$ .

For the decay assertion, take the spatial components in one fixed Cartesian exterior chart $x = (x^{1}, x^{2}, x^{3})$ with $r = ∣ x ∣ > R_{0} > 0$ , and let $T_{i}$ denote the spatial components of $T^{♭}$ . Interpret differentiated estimates and parameter uniformity by [F3]. If $(W_{0})_{i} = O_{5} (r^{- 3})$ and $T_{i} = O_{5} (r^{- 2})$ in this same chart, then $(W_{q})_{i} = O_{5} (r^{- 2})$ uniformly for $q \in [0, 1]$ .

Statement references

Stationary Lorentzian data and tensor conventions

Complex self-dual forms and metric-dependent contraction

Positive Harrison denominator and its parameter identity

Differentiated asymptotic bounds and parameter uniformity

The exterior derivative by the invariant vector-field formula

The Lie derivative of a differential form

Facts & Assumptions

[F1]

Stationarity of $(g, T)$ and $T^{♭}$ use Stationary Lorentzian data and tensor conventions, and the real two-form and star use Complex self-dual forms and metric-dependent contraction. A smooth one-form $ω$ is called stationary when $L_{T} ω = 0$ , with the Lie derivative of a differential form defined in The Lie derivative of a differential form.

[F2]

On the open domain $U$ containing $p$ specified in the Statement, $σ = f + i ψ$ is smooth, $f = - g (T, T)$ , and $ψ : U \to R$ is smooth and real-valued. Both $f$ and $ψ$ are constant along every local integral curve of $T$ contained in $U$ . Thus the scalar $D_{q} = ∣ 1 + q - (1 - q) σ ∣^{2}$ obeys the polynomial identity of Positive Harrison denominator and its parameter identity on $U$ for $q \in [0, 1]$ .

[F3]

The fixed Cartesian exterior chart, multi-index derivatives, componentwise $O_{m} (r^{- b})$ bounds and parameter-uniform estimates are those of Differentiated asymptotic bounds and parameter uniformity. Uniformity requires a common radius and derivative constants for every parameter value. Only this fixed-chart notation is used; no existence or coordinate invariance of an end is asserted.

[F4]

The operator $d$ is the exterior derivative of The exterior derivative by the invariant vector-field formula. For a smooth real one-form $ω$ and smooth vector fields $X, Y$ , its formula is $(d ω) (X, Y) = X (ω (Y)) - Y (ω (X)) - ω ([X, Y])$ .

Proof

Given: The hypotheses in the Statement and the interfaces in [F1]–[F4].

Goal: Establish the polynomial identities, smoothness and conditional uniform decay.

1.1givenF1F2algebra

Since $T^{♭} (T) = - f$ and $T (p) = 0$ gives $f (p) = 0$ , contraction yields $W_{q} (T) = (1 - q)^{2} (D_{0} (p) - D_{0}) + 4 q (1 - q) f$ . Subtract the polynomial identity $D_{q} = (1 - q)^{2} D_{0} + 4 q (1 - (1 - q) f)$ at the variable point from its value at $p$ to obtain $D_{q} (p) - D_{q}$ .

1.2givenF1F4algebra

Expanding the three terms of [F4] for $a α + b η$ , with real constants $a, b$ and smooth real one-forms $α, η$ , gives $d (a α + b η) = a d α + b d η$ : each derivative and evaluation is real-linear in the one-form. With $q$ fixed, the coefficients $(1 - q)^{2}$ and $- 4 q (1 - q)$ are spacetime constants, so this linearity and the given $d T^{♭} = 2 F$ give $d W_{q} = 4 (1 - q)^{2} ((f - 1) F + ψ ⋆_{g} F) - 8 q (1 - q) F$ . The coefficient of $F$ simplifies to $4 (1 - q) ((1 - q) f - (1 + q))$ , giving the asserted curl. The defining formula for $W_{q}$ is polynomial in $q$ with smooth stationary coefficients and gives $W_{1} = 0$ .

2.1givenF1F3algebra∎

In the chart of the decay assertion, let $β$ be any multi-index with $∣ β ∣ \leq 5$ . By [F3] the two input estimates give a common radius $R > \max {R_{0}, 1}$ and constants $A_{β}, B_{β} \geq 0$ with $∣ \partial^{β} (W_{0})_{i} ∣ \leq A_{β} r^{- 3 - ∣ β ∣}$ and $∣ \partial^{β} T_{i} ∣ \leq B_{β} r^{- 2 - ∣ β ∣}$ for every spatial component and $r \geq R$ . These inputs have no $q$ dependence. Since $0 \leq (1 - q)^{2} \leq 1$ and $0 \leq 4 q (1 - q) \leq 1$ on $[0, 1]$ , differentiating the defining component formula gives $∣ \partial^{β} (W_{q})_{i} ∣ \leq A_{β} r^{- 3 - ∣ β ∣} + B_{β} r^{- 2 - ∣ β ∣} \leq (A_{β} + B_{β}) r^{- 2 - ∣ β ∣}$ . The same radius and constants work for all $q \in [0, 1]$ , precisely the required uniform $O_{5} (r^{- 2})$ estimate.

Source notes

The polynomial construction and normalization are attributed to Radu, Section 3.2 (3.12)–(3.18); the fixed-chart differentiated notation is stated in Section 1.1 after (1.1). The proof here is a local conditional calculation, including the explicit uniform derivative bound. Neither existence of the stationary primitive nor the Einstein–Maxwell equations are imported or established here.

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Used by · 4

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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 1.1, differentiated notation after (1.1); Section 3.2, equations (3.12)–(3.18).

The source defines componentwise Cartesian differentiated decay notation and constructs the parameter one-form from a normalized stationary primitive by a polynomial formula, obtaining its curl and contraction identities.

Hypotheses: The source works with its stationary vacuum data and a normalized primitive at a zero of the stationary field. This local calculation assumes the displayed primitive identities directly; it does not assume the vacuum equations to derive them.

Reading limits: Consulted the named HTML passages, including the formulas and their adjacent explanations. Only notation and attribution are taken from them. The conditional algebra and uniform derivative estimate are proved locally; no primitive existence theorem or field-equation conclusion is imported.

Sources

Alexandru F. Radu, Rigidity of stationary black holes with small ADM angular momentum · Section 1.1, differentiated notation after (1.1); Section 3.2, equations (3.12)–(3.18).