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Uniform transformed component estimates

PDF p. 10: (2.11)–(2.13); PDF p. 11: (2.11)–(2.13)

Statement

Adopt the stationary vacuum model and regular stationary spacetime data with mass parameter one and original stationary Cartesian end coordinates $(t, x^{i})$ , so $T = \partial_{t}$ and $r = ∣ x ∣$ . Assume also that the original smooth spacelike maximal exterior section $Σ_{1}$ is supplied, with end coordinates $x^{i}$ and an ADM mass limit that exists and equals one, as required by the mass-one normalization. Take the supplied normalized Ernst potential $σ = f + i ψ$ , where $f = - g (T, T)$ , and a supplied pole $p$ on the bifurcation sphere $S$ with $T_{p} = 0$ . For $q \in [0, 1]$ put $D_{q} = ∣ q (1 + σ) + (1 - σ) ∣^{2}$ and $d_{q} = D_{q} (p)$ . Use the supplied Harrison one-form family $W_{q} = (1 - q)^{2} W_{0} - 4 q (1 - q) T^{♭}$ , with spatial components $(W_{0})_{i} = O_{5} (r^{- 3})$ in the original chart, and the corresponding transformed metric $g_{q}$ on the end where $D_{q} > 0$ . This includes the radial choice in the denominator estimate; a supplied smooth stationary primitive with the same differentiated decay is also permitted, without equality to the radial integral. The supplied $W_{q}$ is assumed to satisfy the stationarity, exterior-derivative and contraction conditions of the Harrison one-form definition, and $g_{q}$ is the metric defined there.

Write $a = q + r^{- 1}$ and $B = q^{2} r^{- 1} + q r^{- 2} + r^{- 3}$ . Uniformly for $q \in [0, 1]$ and sufficiently large $r$ , $(g_{q})_{00} = - \frac{d_{q}^{2}}{16} a^{- 2} + a^{- 4} O_{5} (B), (g_{q})_{0 i} = d_{q} a^{- 1} O_{5} (r^{- 2}),$ $g_{0 i} (W_{q})_{j} = O_{5} (r^{- 4}), \frac{f (W_{q})_{i} (W_{q})_{j}}{D_{q}} = O_{5} (r^{- 4}),$ $(g_{q})_{i j} = \frac{D_{q}}{4} (δ_{i j} + O_{5} (r^{- 1})) + O_{5} (r^{- 4}) = a^{2} δ_{i j} + O_{5} (B) .$

Here $O_{5}$ denotes Cartesian spatial derivative bounds through order five with one radius and constants uniform in $q$ and angle. In particular, $O_{5} (B)$ means a remainder $R_{q}$ satisfying $∣ \partial^{β} R_{q} ∣ \leq C_{β} (q^{2} r^{- 1 - ∣ β ∣} + q r^{- 2 - ∣ β ∣} + r^{- 3 - ∣ β ∣})$ for $∣ β ∣ \leq 5$ . An expression such as $a^{- 4} O_{5} (B)$ denotes the product of the displayed factor and such a remainder; similarly for $a^{- 1} O_{5} (r^{- 2})$ .

Statement references

Stationary vacuum model and asymptotic normalization

Regular non-degenerate stationary vacuum spacetime

Stationary Lorentzian data and tensor conventions

Differentiated asymptotic bounds and parameter uniformity

Mass-one convention and Harrison denominator

Single-potential construction of the Harrison one-form family

Stationary Harrison one-form and metric

Coordinate components of the transformed metric

Uniform denominator and one-form estimates

Facts & Assumptions

[A1]

The model and regular data give the stationary end, $f = 1 + O_{6} (r^{- 1})$ , $g_{0 i} = O_{6} (r^{- 2})$ , and $g_{i j} = δ_{i j} + O_{6} (r^{- 1})$ in the stated chart. The differentiated estimates and their parameter uniformity have the meaning in the Statement. Stationary vacuum model and asymptotic normalization Regular non-degenerate stationary vacuum spacetime Stationary Lorentzian data and tensor conventions Differentiated asymptotic bounds and parameter uniformity

[A2]

The original smooth spacelike maximal exterior section $Σ_{1}$ is supplied, and the ADM mass limit of its end in $x^{i}$ exists and equals one. These are the section and normalization hypotheses of the component supplier, not existence or convergence conclusions of this item. Mass-one convention and Harrison denominator Coordinate components of the transformed metric

[A3]

The supplied family satisfies the Harrison one-form definition's stationarity, exterior-derivative and contraction conditions; in particular $W_{q} (T) = d_{q} - D_{q}$ , and $g_{q}$ is its defined metric on the positivity domain. Stationary Harrison one-form and metric

[F1]

For the specified potential and pole, and either permitted one-form choice with its stated spatial decay, $D_{q} = 4 a^{2} + O_{5} (B)$ , $c a^{2} \leq D_{q} \leq C a^{2}$ , $d_{q}$ is uniformly bounded, and $(W_{q})_{i}$ has derivatives bounded by $C_{β} (q r^{- 2 - ∣ β ∣} + r^{- 3 - ∣ β ∣})$ through order five, uniformly in $q$ . Mass-one convention and Harrison denominator Single-potential construction of the Harrison one-form family Uniform denominator and one-form estimates

[F2]

On a stationary chart where $D_{q} > 0$ , the transformed components are $(g_{q})_{00} = - d_{q}^{2} f / (4 D_{q})$ , $(g_{q})_{0 i} = \frac{d_{q}}{4} (g_{0 i} - f (W_{q})_{i} / D_{q})$ , and $(g_{q})_{i j} = \frac{1}{4} (D_{q} g_{i j} + g_{0 i} (W_{q})_{j} + g_{0 j} (W_{q})_{i} - f (W_{q})_{i} (W_{q})_{j} / D_{q})$ . Coordinate components of the transformed metric The same identities are derived directly in Proof 1.1 from the supplied one-form definition under the present local hypotheses; the proof does not require importing any additional global section hypothesis from that supplier.

[F3]

For real functions differentiable at a limit point of their common real domain, sums, scalar multiples and products are differentiable with the usual sum, scalar-multiple and product rules. Where the denominator is nonzero, their quotient is differentiable on the nonzero-denominator set and obeys the quotient rule. Sums, scalar multiples, products and quotients: (f+g)'(c) = f'(c) + g'(c), (α f)'(c) = α f'(c), (fg)'(c) = f'(c)g(c) + f(c)g'(c), and (f/g)'(c) = (f'(c)g(c) - f(c)g'(c))/g(c)² when g(c) ≠ 0

[F4]

For the supplied Harrison one-form, $A_{q} v = v + D_{q}^{- 1} W_{q} (v) T$ , $g_{q} = (D_{q} / 4) A_{q}^{*} g$ and $W_{q} (T) = d_{q} - D_{q}$ . Here $T^{♭} = g (T, \cdot)$ and $g (T, T) = - f$ . These are the defining algebraic identities on the positive-denominator domain; no exterior-access conclusion is part of this definition. Stationary Harrison one-form and metric

Proof

Given: The model, end, supplied original maximal exterior section and its ADM mass-one normalization, potential, pole, selected one-form family and transformed metric in the Statement.

Goal: Establish all displayed estimates with constants uniform for $q \in [0, 1]$ .

1.1givenF1F3F4algebra

First derive the exact component identities from the supplied data in F4. Evaluating the pullback on arbitrary tangent vectors gives $4 g_{q} = D_{q} g + T^{♭} \otimes W_{q} + W_{q} \otimes T^{♭} - f D_{q}^{- 1} W_{q} \otimes W_{q}$ . In the stationary chart, put $w_{0} = (W_{q})_{0} = d_{q} - D_{q}$ and use $g_{00} = - f$ . Its time-time component is $- f (D_{q} + 2 w_{0} + w_{0}^{2} / D_{q}) / 4 = - f d_{q}^{2} / (4 D_{q})$ . Its mixed component is $((D_{q} + w_{0}) g_{0 i} - f (1 + w_{0} / D_{q}) (W_{q})_{i}) / 4 = (d_{q} / 4) (g_{0 i} - f (W_{q})_{i} / D_{q})$ . Its spatial component is $(D_{q} g_{i j} + g_{0 i} (W_{q})_{j} + g_{0 j} (W_{q})_{i} - f (W_{q})_{i} (W_{q})_{j} / D_{q}) / 4$ . Thus the component formulas recorded in F2 hold directly under the stated one-form hypotheses, without a separate global maximality application. Apply [F3] on open Cartesian coordinate-line intervals in the end, with $q$ fixed, and iterate in coordinate directions through total order five. The supplied fields are smooth there; $D_{q} > 0$ by [F1] and $a > 0$ , so the quotient rule applies to their reciprocals. Repeated product differentiation gives finite sums of products of derivatives, with combinatorial coefficients independent of $q$ . In either permitted case the selected primitive has the denominator supplier's differentiated spatial decay in the same original chart; all its potential, pole and family data agree with the present data, so [F1] applies to this very family. For every spatial multi-index $β$ through order five, derivatives of $a$ obey $∣ \partial^{β} a ∣ \leq C_{β} a r^{- ∣ β ∣}$ : for positive order this follows from derivatives of $r^{- 1}$ and $r^{- 1} \leq a$ . Since $q \geq 0$ , $B \leq a^{2} r^{- 1} = q^{2} r^{- 1} + 2 q r^{- 2} + r^{- 3} \leq 2 B$ . The expansion in [F1] and this comparison, applied at each differentiated order, give $∣ \partial^{β} D_{q} ∣ \leq C_{β} a^{2} r^{- ∣ β ∣}$ . Differentiating $D_{q} D_{q}^{- 1} = 1$ inductively and using the lower bound in [F1] yields $∣ \partial^{β} D_{q}^{- 1} ∣ \leq C_{β} a^{- 2} r^{- ∣ β ∣}$ . Likewise derivatives of $a^{\pm k}$ have the corresponding bounds $C_{β} a^{\pm k} r^{- ∣ β ∣}$ for each fixed nonnegative integer $k$ . All constants are independent of $q$ .

2.1A1A2F1F2F3step 1.1algebra

By [A1] and [F1], $4 a^{2} f - D_{q} = O_{5} (B)$ : indeed $a^{2} (f - 1) = O_{5} (a^{2} r^{- 1}) = O_{5} (B)$ . Therefore $a^{4} (f / D_{q} - 1 / (4 a^{2})) = a^{2} (4 a^{2} f - D_{q}) / (4 D_{q}) = O_{5} (B)$ , by the product rule [F3] and step 1.1. Substitution in the first formula of [F2], with uniformly bounded $d_{q}$ , gives the time-time estimate.

2.2A1F1F2F3step 1.1algebra

The one-form bound in [F1] says $∣ (\partial^{β} W_{q})_{i} ∣ \leq C_{β} a r^{- 2 - ∣ β ∣}$ . The product rule [F3], [A1], and step 1.1 give $g_{0 i} (W_{q})_{j} = O_{5} (r^{- 4})$ and $f (W_{q})_{i} (W_{q})_{j} / D_{q} = O_{5} (r^{- 4})$ . They also give $a f (W_{q})_{i} / D_{q} = O_{5} (r^{- 2})$ and $a g_{0 i} = O_{5} (r^{- 2})$ , since $a \leq 2$ for $r \geq 1$ . The mixed component formula in [F2] consequently yields $(g_{q})_{0 i} = d_{q} a^{- 1} O_{5} (r^{- 2})$ .

3.1A1F1F2F3step 1.1step 2.2algebra∎

The spatial formula in [F2], [A1], and step 2.2 give $(g_{q})_{i j} = \frac{D_{q}}{4} (δ_{i j} + O_{5} (r^{- 1})) + O_{5} (r^{- 4})$ . By [F1], $D_{q} / 4 = a^{2} + O_{5} (B)$ ; moreover $D_{q} O_{5} (r^{- 1}) = O_{5} (a^{2} r^{- 1}) = O_{5} (B)$ and $r^{- 4} \leq B$ for $r \geq 1$ . The product rule [F3] establishes the same bounds after every permitted derivative, proving the final spatial estimate.

Source notes

The article states the uniform transformed time-time, mixed and spatial component estimates and the two intermediate spatial product estimates. Its citation records attribution only. The proof is a local application of the declared component and denominator results; it does not use the article's assertion as a premise. Its original radial case is retained. The extension to a supplied stationary primitive with the same differentiated spatial decay uses exactly the same estimates and is locally authored; it asserts neither equality of gauges nor the unresolved simultaneous smooth-global/exact-radial assertion.

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Sources

The uniqueness of smooth stationary black holes · PDF page 7, Section 2.1, opening conventions; pages 10–11, Section 2.3, equations (2.11)–(2.13)