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The differentiated spatial decay supplied by the Harrison lift

Statement

Use the fixed mass-one end and the positive weight $s_{q} = q + r^{- 1}$ of the preceding denominator lemma. Suppose a supplied smooth stationary primitive has $(W_{0})_{i} = O_{5} (r^{- 3})$ on this end, and use the chosen family $W_{q} = (1 - q)^{2} W_{0} - 4 q (1 - q) T^{♭}, T_{i} = g_{0 i} = O_{6} (r^{- 2}) .$ Then $(W_{q})_{i} = s_{q} Z_{q i}$ with $Z_{q i} = O_{5} (r^{- 2})$ uniformly for $q \in [0, 1]$ . This is the precise meaning of $(W_{q})_{i} = (q + r^{- 1}) O_{5} (r^{- 2})$ here. At $q = 0$ it gives $O_{5} (r^{- 3})$ . It supplies no extra power of decay at $q = 0$ : the stronger expression $(q + r^{- 1}) O (r^{- 3})$ would require $(W_{0})_{i} = O (r^{- 4})$ .

Existence of a primitive with this end choice is a separate input: the declared primitive and construction suppliers supply it under their own precise GR/SBS and original maximality hypotheses. This conditional calculation does not relax those hypotheses or construct a new primitive.

Facts & Assumptions

[F1]

Differentiated asymptotic bounds and parameter uniformity fixes Cartesian componentwise differentiated bounds. Mass-one Ernst decay and uniform Harrison denominator bounds supplies the weight and uniform inverse derivative bounds, as well as the end metric estimates.

[F2]

Stationary Harrison one-form and metric specifies the conditions on a supplied Harrison one-form, $T^{♭}$ and the Harrison tensor convention; it does not select a unique $W_{q}$ . The polynomial parameter lift displayed in the Statement is the chosen family assumed here and used by the construction interface in [F3].

[F3]

Smooth invariant Harrison primitive near the horizon and at infinity supplies one primitive with spatial $O_{5} (r^{- 3})$ radial end gauge under its full recorded hypotheses. Regular Harrison Einstein–Maxwell family uses the displayed polynomial lift. These existence interfaces are not inferred from decay notation.

[F4]

Sum, product, quotient, and chain rules from difference quotients supplies the scalar sum, product, quotient and chain rules at points of differentiability, with nonzero denominator for quotients. Apply these rules successively on coordinate lines of the exterior chart; the spatial derivatives hold $q$ fixed. Repeated finite product differentiation is valid through order five for the supplied component bounds.

Proof

**Given:**The supplied primitive and end bounds in the Statement.

Goal: Establish the claimed uniform weighted decay and its exact implication at zero parameter.

1.1F1F4algebra

On the exterior chart, the difference quotient of $r = ∣ x ∣$ in coordinate direction $j$ is $(2 x^{j} + h) / (r (x + h e_{j}) + r (x))$ , so $\partial_{j} r = x^{j} / r$ . Applying [F4] repeatedly gives $∣ \partial^{β} r^{\pm 1} ∣ \leq C_{β} r^{\pm 1 - ∣ β ∣}$ for $∣ β ∣ \leq 5$ : the product and quotient rules give $\partial_{j} r^{a} = a x^{j} r^{a - 2}$ for integer $a$ , and each further derivative of the resulting finite monomials lowers their radial order by one. By [F1], $∣ \partial^{β} s_{q}^{- 1} ∣ \leq C_{β} s_{q}^{- 1} r^{- ∣ β ∣}$ . Since $q$ is spatially constant and $q / s_{q} \leq 1$ , it follows that $∣ \partial^{β} (q / s_{q}) ∣ \leq C_{β} r^{- ∣ β ∣}$ . The product rule applied to $r^{- 1} s_{q}^{- 1}$ gives the same bound for $\partial^{β} (r^{- 1} / s_{q})$ , using $r^{- 1} / s_{q} \leq 1$ . These constants are independent of $q \in [0, 1]$ .

2.1givenF1F2F3F4step 1.1algebra

Divide the displayed spatial formula by $s_{q}$ . The first term is $(1 - q)^{2} (r^{- 1} / s_{q}) [r (W_{0})_{i}]$ , and the second is $- 4 (1 - q) (q / s_{q}) g_{0 i}$ . By the radial bounds in step 1.1 and the iterated product rule in [F4], every derivative of order $k \leq 5$ of $r (W_{0})_{i}$ is a finite sum of terms of size $O (r^{1 - a} r^{- 3 - (k - a)}) = O (r^{- 2 - k})$ , with $0 \leq a \leq k$ . Thus this bracketed array is $O_{5} (r^{- 2})$ . The quotient bounds from step 1.1, $g_{0 i} = O_{6} (r^{- 2})$ , and $∣ 1 - q ∣ \leq 1$ give, by the same finite product rule, $Z_{q i} = O_{5} (r^{- 2})$ with one radius and constants for all $q \in [0, 1]$ .

3.1F1step 2.1algebra∎

At zero parameter the displayed family equals the supplied $W_{0}$ and $s_{0} = r^{- 1}$ . The proved bound is exactly $O_{5} (r^{- 3})$ . In contrast $s_{0} O (r^{- 3}) = O (r^{- 4})$ . An $O (r^{- 3})$ hypothesis alone does not imply $O (r^{- 4})$ : the scalar function $r^{- 3}$ has the former order and $r^{4} r^{- 3} = r$ is unbounded. This observation is only about logical strength of estimates; it is not a counterexample satisfying all geometric hypotheses of any article claim.

Depends on

Used by · 3

Dependency tree · 2 levels

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