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Harrison spatial two-form differs from the seed by an integrable tail

Statement

Work in ZFC. Let an end chart on a smooth, oriented and time-oriented Lorentzian four-manifold of signature $(-, +, +, +)$ be $(t, x) : U_{e n d} \to I \times E$ , where $I$ is an open interval containing $0$ and $E = {x \in R^{3} : r = ∣ x ∣ > R_{0}}$ with $R_{0} > 0$ . Assume $T = \partial_{t}$ in this same chart and write $T^{♭} = g (T, \cdot)$ and $g_{0 i} = g (\partial_{t}, \partial_{x^{i}})$ . Supply a smooth complex function $σ$ , a smooth real one-form $W_{0}$ , and a smooth real two-form $F$ on this end, and put $F = F + i ⋆_{g} F$ , using the supplied orientation.

Fix $0 < q < 1$ and put $a_{q} = \sqrt{1 - q^{2}}$ . On the slice $t = 0$ assume the Cartesian component bounds $σ = 1 - 2 / r + O_{1} (r^{- 2})$ , $g_{0 i} = O_{1} (r^{- 2})$ , $(W_{0})_{i} = O (r^{- 3})$ , and $F_{i j} = O (r^{- 2})$ , where $1 \leq i, j \leq 3$ , $O = O_{0}$ , and every bound is uniform over spatial directions. Define the local expressions

$W_{q} = (1 - q)^{2} W_{0} - 4 q (1 - q) T^{♭}, B_{q} = (1 + q) - (1 - q) σ, D_{q} = ∣ B_{q} ∣^{2},$

and, on the open set where $D_{q} > 0$ , define the two-form by the explicit algebraic formula

$H_{q} = - a_{q} \frac{{\overline{B}}_{q}}{B_{q}} (F + \frac{W_{q} \land d σ}{2 D_{q}}) .$

For all sufficiently large $r$ on $t = 0$ , $D_{q} > 0$ and, writing $ι_{0} (x) = (0, x)$ , one has $ι_{0}^{*} H_{q} = - a_{q} ι_{0}^{*} F + E_{q}$ , where $E_{q}$ is a complex two-form on that spatial end with Cartesian coefficients $(E_{q})_{i j} = O_{q} (r^{- 3})$ . In particular the same identity holds after restriction to each sufficiently large spatial coordinate sphere. The constants and the starting radius may depend on this fixed positive $q$ ; no estimate uniform as $q ↓ 0$ is asserted.

Statement references

Differentiated asymptotic bounds and parameter uniformity

Complex self-dual forms and metric-dependent contraction

The inverse metric, musical isomorphisms and index raising

Real and imaginary parts, complex conjugation, and modulus

Coordinate formula for the differential

The wedge product of differential forms

The pullback of a differential form

The Axiom of Choice

Facts & Assumptions

[A1]

ZFC supplies the background choice assumption inherited by the geometric suppliers. The coefficient estimates below use only finite algebra in the supplied chart. The Axiom of Choice

[F1]

For the supplied smooth Lorentzian metric, the metric dual is $T^{♭} (v) = g (T, v)$ and $g_{0 i} = g (\partial_{t}, \partial_{x^{i}})$ . The inverse metric, musical isomorphisms and index raising

[F2]

On the supplied oriented Lorentzian four-manifold, a complex two-form has real and imaginary parts, and $F = F + i ⋆_{g} F$ uses the smooth metric Hodge star extended complex linearly. Complex self-dual forms and metric-dependent contraction

[F3]

The bounds $O_{m} (r^{- b})$ control all Cartesian derivatives of order at most $m$ by constants times $r^{- b - ∣ β ∣}$ , uniformly over directions; a fixed-parameter bound permits constants and starting radius to depend on $q$ . Differentiated asymptotic bounds and parameter uniformity

[F4]

For $z = u + i v$ , complex conjugation is $\overline{z} = u - i v$ and $∣ z ∣ = (u^{2} + v^{2})^{1 / 2}$ , so $∣ z ∣^{2} = z \overline{z}$ . Real and imaginary parts, complex conjugation, and modulus

[F5]

The differential of a smooth scalar has coordinate components given by its partial derivatives; for complex $σ$ this is applied separately to its real and imaginary parts. Coordinate formula for the differential

[F6]

The wedge product is the pointwise alternating product: for one-forms $α, β$ , $(α \land β)_{i j} = α_{i} β_{j} - α_{j} β_{i}$ , extended complex linearly. The wedge product of differential forms

[F7]

Restriction to a slice or sphere is pullback by its smooth inclusion, evaluated on its tangent vectors. In particular $d t$ pulls back to zero on $t = 0$ . The pullback of a differential form

Proof

Given: The supplied end chart, smooth fields, fixed parameter, local formulas and spatial bounds in the Statement; work in ZFC.

Goal: Establish eventual positivity and the spatial error estimate.

Proof technique: direct.

1.1givenF3F4algebra

Since $σ = 1 + O (r^{- 1})$ , the given formula gives $B_{q} = 2 q + O_{q} (r^{- 1})$ . Increase the radius until $∣ B_{q} - 2 q ∣ \leq q$ ; then $∣ B_{q} ∣ \geq q > 0$ and $D_{q} \geq q^{2}$ . Expanding $B_{q} {\overline{B}}_{q}$ gives $D_{q} = 4 q^{2} + O_{q} (r^{- 1})$ and $D_{q}^{- 1} = O_{q} (1)$ . Also ${\overline{B}}_{q} / B_{q} - 1 = ({\overline{B}}_{q} - B_{q}) / B_{q} = O_{q} (r^{- 1})$ , since $2 q$ is real. All these estimates are on $t = 0$ and are uniform in spatial directions.

1.2givenF1F3F5F6F7algebra

The adapted-chart hypothesis and the metric dual give $(T^{♭})_{i} = T^{♭} (\partial_{x^{i}}) = g (\partial_{t}, \partial_{x^{i}}) = g_{0 i}$ . Hence $W_{q, i} = (1 - q)^{2} (W_{0})_{i} - 4 q (1 - q) g_{0 i} = O_{q} (r^{- 2})$ . Differentiating the given spatial expansion yields $\partial_{i} σ = 2 x^{i} / r^{3} + O (r^{- 3}) = O (r^{- 2})$ . Therefore $(W_{q} \land d σ)_{i j} = W_{q, i} \partial_{j} σ - W_{q, j} \partial_{i} σ = O_{q} (r^{- 4})$ . The pullback to $t = 0$ contains exactly these spatial components; its time-component terms vanish on spatial tangent vectors.

2.1givenF2F3F7step 1.1step 1.2algebra∎

Use the explicitly defined local two-form on the nonzero-denominator set found in step 1.1 and set $E_{q} = ι_{0}^{*} (H_{q} + a_{q} F)$ . Its spatial coefficients are $(E_{q})_{i j} = - a_{q} (\frac{{\overline{B}}_{q}}{B_{q}} - 1) F_{i j} - \frac{a_{q}}{2 D_{q}} \frac{{\overline{B}}_{q}}{B_{q}} (W_{q} \land d σ)_{i j} .$ The first term is $O_{q} (r^{- 1}) O (r^{- 2}) = O_{q} (r^{- 3})$ and the second is $O_{q} (1) O_{q} (r^{- 4}) = O_{q} (r^{- 4})$ . This proves the asserted spatial estimate and its restriction to the coordinate spheres. The calculation uses only the stated bound on $g_{0 i}$ and imposes no leading angular-momentum expansion.

Depends on

Used by · 3

Dependency tree · 2 levels

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