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The pointwise static end bounds imply the original weighted hypotheses

Statement

Assume the Axiom of Countable Choice. Let $H_{i j} = h_{i j} - δ_{i j}$ and $v = V - 1 + m / r$ in the end of the regular static exterior data, and fix an end cutoff $R > R_{0}$ . For $4 < q < \infty$ and $0 < λ < 1$ , define $L_{α}^{q}$ on $r > R$ by finiteness of $\int_{r > R} ∣ f ∣^{q} r^{- α q - 3} d x$ . Then $\partial_{k} \partial_{ℓ} H_{i j} \in L_{- λ - 2}^{q}, \partial_{k} \partial_{ℓ} v \in L_{- λ - 3}^{q} .$ Thus the differentiated pointwise falloff supplies the weighted second-derivative hypotheses in Bunting–Masood-ul-Alam's Assumption 1 (p.148), for example with $q = 6$ and $λ = 1 / 2$ . The compact-core end topology and the zeroth- and first-derivative falloff in that asymptotic contract are also supplied by the regular static exterior data. This comparison concerns the end conditions; positivity of $m$ , the static vacuum equations and boundary conditions require their separate arguments.

Statement references

Complete regular static exterior data and its normalized end

Polar coordinates decompose Lebesgue measure into rⁿ⁻¹ dr dσ

The Axiom of Countable Choice (AC_ω)

Facts & Assumptions

[F1]

The regular static exterior data have a single Cartesian end outside a compact core, with $H = O_{2} (r^{- 1})$ and $v = O_{2} (r^{- 2})$ ; the $O_{2}$ convention gives uniform Cartesian derivative bounds through order two. Complete regular static exterior data and its normalized end

[F2]

In dimension three the polar integration formula applies to every nonnegative Borel function on $R^{3}$ , with volume element $r^{2} d r d σ$ and finite Borel measure $σ$ on $S^{2}$ . The cited theorem assumes Countable Choice. Polar coordinates decompose Lebesgue measure into rⁿ⁻¹ dr dσ

[A1]

Adopt the Axiom of Countable Choice to meet precisely the choice hypothesis of [F2]. The Axiom of Countable Choice (AC_ω)

[F3]

The imported asymptotic contract is Bunting–Masood-ul-Alam, p.148, Assumption 1 and its weighted-space definition, as recorded in the External input section and the source reference.

External input

Only the end-data formulation is imported here: outside a compact core the end is Euclidean in topology, $H = O (r^{- 1})$ , $\partial H = O (r^{- 2})$ , $V = 1 - m / r + v$ , $v = O (r^{- 2})$ , $\partial v = O (r^{- 3})$ , and for some $λ > 0$ and $4 < q < \infty$ one requires $\partial^{2} H \in L_{- λ - 2}^{q}$ and $\partial^{2} v \in L_{- λ - 3}^{q}$ . The weighted convention is $\int ∣ f ∣^{q} r^{- α q - 3} d x < \infty$ . These are the conditions read on p.148, Assumption 1, with $H$ here denoting the source's metric perturbation. The local proof below verifies these end conditions from [F1]; no static classification theorem or mass-positivity assertion is imported as a premise of this mathematical lemma.

Proof

Given: The Cartesian end and $O_{2}$ bounds of [F1], the polar integration theorem [F2] under [A1], and the external formulation [F3].

Goal: Verify the two weighted integrability assertions and the stated comparison of end conditions.

1.1givenF1F2A1

The differentiated bounds give $∣ \partial^{2} H ∣ \leq C r^{- 3}$ and $∣ \partial^{2} v ∣ \leq C r^{- 4}$ uniformly in angle for $r > R_{1}$ , where $R_{1} > R$ . Extend each nonnegative weighted component integrand by zero off $r > R_{1}$ ; smoothness on the end makes this extension Borel measurable. Apply [F2] with $n = 3$ , using [A1] at this invocation. The first weighted $q$ th power is at most $C^{q} σ (S^{2}) \int_{R_{1}}^{\infty} r^{- 3 q} r^{(λ + 2) q - 3} r^{2} d r = C^{q} σ (S^{2}) \int_{R_{1}}^{\infty} r^{q (λ - 1) - 1} d r$ . This last integral is $R_{1}^{q (λ - 1)} / (q (1 - λ)) < \infty$ , since $λ < 1$ . The finite sum of components does not affect convergence.

2.1givenF1F2A1step 1.1

For the lapse remainder the same polar formula bounds its weighted $q$ th power by $C^{q} σ (S^{2}) \int_{R_{1}}^{\infty} r^{- 4 q} r^{(λ + 3) q - 3} r^{2} d r = C^{q} σ (S^{2}) \int_{R_{1}}^{\infty} r^{q (λ - 1) - 1} d r < \infty$ . On the compact annulus $R \leq r \leq R_{1}$ all component second derivatives and weights are bounded, and its Cartesian volume is finite, also by [F2]. Adding the annular integrals proves both claims on $r > R$ .

3.1givenF1F3step 1.1step 2.1∎

The order-zero and order-one parts of [F1] give exactly $H = O (r^{- 1})$ , $\partial H = O (r^{- 2})$ , $v = O (r^{- 2})$ and $\partial v = O (r^{- 3})$ , while its compact-core chart gives the required end topology. The weight in [F3] agrees with the Statement, so steps 1.1 and 2.1 verify its second-derivative requirements. Choosing $q = 6$ and $λ = 1 / 2$ meets $4 < q < \infty$ and $λ > 0$ simultaneously. Enlarging the compact cutoff only discards a compact annulus, whose finite integral was checked in step 2.1. This establishes the asserted end comparison without claiming that a borderline $O (r^{- 2})$ second derivative would suffice.

Depends on

Used by · 1

Static vacuum Schwarzschild classification contract

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

G. L. Bunting and A. K. M. Masood-ul-Alam, Nonexistence of multiple black holes in asymptotically Euclidean static vacuum space-time, General Relativity and Gravitation 19 (1987), 147–154 · p.148, Assumption 1, equations (4)–(5), the following derivative conditions and definition of the weighted space

The asymptotic part of Assumption 1 requires a compact-core Euclidean end with metric perturbation H=O(r^-1), first derivatives O(r^-2), lapse V=1-m/r+v with v=O(r^-2) and first derivatives O(r^-3), and second derivatives of H in L^q_{-lambda-2} and of v in L^q_{-lambda-3}, for some lambda>0 and 4<q<infinity. Its convention is finiteness of the integral of |f|^q r^(-alpha q-3) against Cartesian volume.

Hypotheses: This is the specified end-data contract, not a classification theorem. It is read in the end chart outside a compact cutoff. The source separately states positivity of m; that assertion, the static vacuum equations and boundary assumptions are not derived by this integrability lemma.

Reading limits: In this adjudication, read the online transcription around Assumption 1 and visually inspected the retained original-paper image bunting-page-2.jpg (printed p.148), including both weighted subscripts, q range, first-derivative conditions and weight convention. The transcription corrupts the subscripts; the image supplies them. No classification proof or positive-mass proof was read or reconstructed for this assignment.

Sources

G. L. Bunting and A. K. M. Masood-ul-Alam, Nonexistence of multiple black holes in asymptotically Euclidean static vacuum space-time, General Relativity and Gravitation 19 (1987), 147–154 · p.148, Assumption 1, equations (4)–(5), the following derivative conditions and definition of the weighted space