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Uniform spatial components and the cylinder scale

Statement

Use the stationary Lorentzian and normalized Ernst conventions. Let the supplied stationary coordinate end be $(t, x) \in I \times {x \in R^{3} : r = ∣ x ∣ > R_{0}}$ , with $T = \partial_{t}$ and the mass-one bounds $g_{00} = - 1 + 2 / r + O_{6} (r^{- 2}), g_{0 i} = O_{6} (r^{- 2}), g_{i j} = δ_{i j} + O_{6} (r^{- 1}) .$ Suppose a smooth normalized Ernst potential $σ = f + i ψ$ , $f = - g (T, T) = - g_{00}$ on the end, is supplied on an open set $U$ containing this end. It satisfies $\nabla_{a} σ = 2 T^{b} (F + i ⋆_{g} F)_{b a}$ , with $F_{a b} = \nabla_{a} T_{b}$ , and $σ \to 1$ on the designated end (in particular $ψ \to 0$ ). Choose the same supplied pole $p \in U$ with $T (p) = 0$ for the Harrison data. For $q \in [0, 1]$ set $s_{q} = q + r^{- 1}$ , $D_{q} = ∣ 1 + q - (1 - q) σ ∣^{2}$ and $d_{q} = D_{q} (p)$ .

Suppose a smooth real stationary primitive $W_{0}$ is supplied on this end with $(W_{0})_{i} = O_{5} (r^{- 3})$ , and use exactly the family $W_{q} = (1 - q)^{2} W_{0} - 4 q (1 - q) T^{♭}$ . On one sufficiently large common end $V \subseteq U$ the denominator bounds give $D_{q} > 0$ for every $q$ . Assume on $V$ these are Harrison one-forms for the same $g, σ, p$ , satisfying the stationarity and differential conditions of the Harrison definition and $W_{q} (T) = d_{q} - D_{q}$ . Use the Harrison metric on $V$ defined from these same forms. These are supplied data, not an existence assertion.

On a sufficiently large common end the spatial coordinate components satisfy $(g_{q})_{i j} = \frac{D_{q}}{4} (δ_{i j} + O_{5} (r^{- 1})) + O_{5} (r^{- 4}) = s_{q}^{2} (δ_{i j} + E_{q i j}), E_{q i j} = O_{5} (r^{- 1}),$ uniformly for $q \in [0, 1]$ . Consequently the coordinate sections $t = c o n s t a n t$ are spacelike there. For $q = 0$ their metric is $r^{- 2} (δ_{i j} + O_{5} (r^{- 1}))$ ; for fixed $q > 0$ it is $q^{2} δ_{i j} + O_{5} (r^{- 1})$ . The last fixed-parameter remainder need not have constants independent of $q$ at zero.

Statement references

Stationary Lorentzian data and tensor conventions

Killing two-form and normalized Ernst notation

Differentiated asymptotic bounds and parameter uniformity

Mass-one Ernst decay and uniform Harrison denominator bounds

The differentiated spatial decay supplied by the Harrison lift

Stationary Harrison one-form and metric

Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons

Facts & Assumptions

[F1]

Differentiated asymptotic bounds and parameter uniformity supplies the differentiated convention and parameter uniformity. The explicit mass-one metric bounds and the supplied normalized Ernst equation and limit in the Statement meet the end hypotheses of Mass-one Ernst decay and uniform Harrison denominator bounds. It supplies $D_{q} = 4 s_{q}^{2} (1 + ε_{q})$ , $ε_{q} = O_{5} (r^{- 1})$ , positive bounds and $∣ \partial^{β} s_{q}^{- 1} ∣ \leq C_{β} s_{q}^{- 1} r^{- ∣ β ∣}$ for $∣ β ∣ \leq 5$ , uniformly on one end.

[F2]

The supplied stationary primitive bound $(W_{0})_{i} = O_{5} (r^{- 3})$ , the exact lift $W_{q} = (1 - q)^{2} W_{0} - 4 q (1 - q) T^{♭}$ , and $T_{i} = g_{0 i} = O_{6} (r^{- 2})$ meet the conditional hypotheses of The differentiated spatial decay supplied by the Harrison lift. Thus $(W_{q})_{i} = s_{q} Z_{q i}$ with $Z_{q i} = O_{5} (r^{- 2})$ uniformly. The original end also has $f = 1 + O_{6} (r^{- 1})$ and $g_{i j} = δ_{i j} + O_{6} (r^{- 1})$ . No primitive-existence conclusion is used.

[F3]

Stationary Lorentzian data and tensor conventions and Killing two-form and normalized Ernst notation fix the metric dual, connection, orientation, smooth time-orientation field $N$ with $g (N, N) < 0$ , and supplied potential equation used in the Statement. Stationary Harrison one-form and metric defines the Harrison covariant tensor from these same data, with the same pole $p \in U$ , on the domains $D_{q} > 0$ and for the supplied forms satisfying its differential, stationary and contraction conditions. Its local justification supplies the Lorentzian metric there and the identity $g_{q} = (D_{q} / 4) A_{q}^{*} g$ , where $A_{q} v = v + D_{q}^{- 1} W_{q} (v) T$ has smooth inverse $A_{q}^{- 1} v = v - d_{q}^{- 1} W_{q} (v) T$ and $d_{q} > 0$ . Existence of the forms is assumed here.

[F4]

In a time-oriented smooth Lorentzian four-manifold, a smooth embedded hypersurface is spacelike when the ambient metric restricts to a positive-definite quadratic form on each tangent space. Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons (Definition (a)).

[F5]

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 a quotient. Apply these rules repeatedly along coordinate lines, holding $q$ fixed. The supplied components are sufficiently differentiable through order five, and the denominators below are positive on the chosen common end.

Proof

Given: The stationary mass-one end, normalized potential, supplied primitive and exact Harrison family in the Statement.

Goal: Estimate the spatial block and its signature uniformly.

1.1F1F2F3F5givenalgebra

By repeated coordinatewise product differentiation in [F5], a derivative of a product is a finite sum of products whose derivative orders add to the total order; hence the decay exponents of differentiated bounds add. Increase the common radius in [F1] so $∣ ε_{q} ∣ \leq 1 / 2$ . The function $h_{q} = (1 + ε_{q})^{- 1}$ is bounded, and differentiating $(1 + ε_{q}) h_{q} = 1$ successively with [F5] gives $∣ \partial^{β} h_{q} ∣ \leq C_{β} r^{- ∣ β ∣}$ for $∣ β ∣ \leq 5$ : at each positive order solve for that derivative of $h_{q}$ , using $1 + ε_{q} \geq 1 / 2$ and the lower-order bounds. Thus $s_{q}^{2} / D_{q} = h_{q} / 4$ has these same uniform bounds. Also $s_{q} \leq 2$ for $r \geq 1$ ; differentiating $s_{q} s_{q}^{- 1} = 1$ and using the inverse derivative bounds in [F1] gives $∣ \partial^{β} s_{q} ∣ \leq C_{β} s_{q} r^{- ∣ β ∣}$ by the same successive product-rule argument. The spatial block of [F3] is $(g_{q})_{i j} = \frac{1}{4} (D_{q} g_{i j} + g_{0 i} (W_{q})_{j} + g_{0 j} (W_{q})_{i} - f D_{q}^{- 1} (W_{q})_{i} (W_{q})_{j})$ . By [F2] and the finite product rule, the mixed products are $s_{q} O_{5} (r^{- 4}) = O_{5} (r^{- 4})$ . The final product is $f (s_{q}^{2} / D_{q}) Z_{q i} Z_{q j} = O_{5} (r^{- 4})$ . This proves the first equality in the Statement.

2.1F1F3F4F5step 1.1algebra∎

By [F1] and the product rule in [F5], $∣ \partial^{β} s_{q}^{- 2} ∣ \leq C_{β} s_{q}^{- 2} r^{- ∣ β ∣}$ . Since $s_{q}^{- 2} \leq r^{2}$ , dividing the error $O_{5} (r^{- 4})$ from step 1.1 by $s_{q}^{2}$ gives $O_{5} (r^{- 2})$ : each derivative of total order $k \leq 5$ is a finite sum bounded by $C s_{q}^{- 2} r^{- 4 - k} \leq C r^{- 2 - k}$ . Meanwhile $D_{q} / (4 s_{q}^{2}) = 1 + O_{5} (r^{- 1})$ . The same product rule therefore gives $s_{q}^{- 2} (g_{q})_{i j} = δ_{i j} + E_{q i j}$ with $E_{q i j} = O_{5} (r^{- 1})$ uniformly. For a large common radius the operator norm of the symmetric error is below one half, so its quadratic form is positive definite. To meet the ambient time-orientation hypothesis in [F4], use the pullback and smooth inverse in [F3] to define $N_{q} = A_{q}^{- 1} N = N - d_{q}^{- 1} W_{q} (N) T$ on $V$ . This is a smooth vector field, and $g_{q} (N_{q}, N_{q}) = (D_{q} / 4) g (N, N) < 0$ since $D_{q} > 0$ . Thus $N_{q}$ time-orients $(V, g_{q})$ . The coordinate section is a smooth embedded hypersurface with tangent basis $\partial_{x^{1}}, \partial_{x^{2}}, \partial_{x^{3}}$ and induced metric matrix $((g_{q})_{i j})$ ; hence [F4] proves spacelikeness. At zero parameter put $s_{0} = r^{- 1}$ . At any fixed positive parameter use $s_{q}^{2} = q^{2} + 2 q / r + r^{- 2}$ , giving the asserted remaining expression.

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