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An asymptotically flat stationary end bounds null speed by Killing energy

Statement

Work in ZFC with stationary Lorentzian data $(M, g, N, T)$ . Specify the end chart as a diffeomorphism $(t, x) : U ⟶ I \times {x \in R^{3} : ∣ x ∣ > R_{0}}$ , where $I$ is a nonempty open interval, $R_{0} > 0$ , and $∣ \cdot ∣$ is the Euclidean spatial norm. Put $r = ∣ x ∣$ and $η = diag (- 1, 1, 1, 1)$ . Assume that the coordinate metric matrix tends uniformly to $η$ as $r \to \infty$ , that $T = c \partial_{t}$ for one fixed $c > 0$ , and that $T$ has the future orientation of the stationary end. These are supplied chart hypotheses, not consequences of stationarity alone.

For some $R_{*} > R_{0}$ , every nonzero future null vector $v = a \partial_{t} + \sum_{i = 1}^{3} w^{i} \partial_{x^{i}}$ at a point of $U_{*} = {p \in U : r (p) > R_{*}}$ has pointwise Killing energy $E (v) : = - g (T, v) > 0$ and satisfies $∣ w ∣ \leq C E (v)$ , with $C = 4 / c$ . Thus along any differentiable future null curve within this region, with parameter $λ$ and tangent $v = \dot{γ}$ , one has $∣ d x / d λ ∣ \leq C E (v)$ and $∣ d r / d λ ∣ \leq C E (v)$ . If the curve is an affine null geodesic, this energy is constant along its segment.

On $U_{*}$ , $T$ is future timelike and $t$ is temporal: its metric gradient $\nabla t = g^{- 1} (d t, \cdot)$ is past-directed timelike. For $t_{0} \in I$ and $R > R_{*}$ define the coordinate sphere $S_{t_{0}, R}$ by the embedding $ω \mapsto (t_{0}, R ω)$ and its stationary cylinder $C_{R}$ by $(s, ω) \mapsto (s, R ω)$ , where $ω \in S^{2} : = {ω \in R^{3} : ∣ ω ∣ = 1}$ and $s \in I$ , using the inverse chart to view these maps in $M$ . The sphere is spacelike, meaning that its induced metric is positive definite, and the cylinder is timelike, meaning that its induced metric has signature $(-, +, +)$ . Stationary here means tangent to $T$ and preserved wherever its local flow stays in the chart. No completeness of that flow or of geodesics is required for these pointwise conclusions.

Facts & Assumptions

[F1]

Stationary Lorentzian data give the smooth metric of signature $(-, +, +, +)$ , the future timelike orientation field $N$ , the Killing field $T$ , and the inverse-metric convention; a designated stationary end has future timelike $T$ . Stationary Lorentzian data and tensor conventions

[F2]

Killing energy is constant along an affine null geodesic segment. This interface is used only for that conservation assertion, not to establish positivity for arbitrary pointwise vectors. The stationary Killing energy is constant along every affine null segment

[F3]

At each tangent space take the future timelike component to be the one containing $N$ ; a future null vector is a nonzero null vector in its closure. Time orientation, future and past timelike/null/causal vectors, and the closed cones

[F4]

A smooth function is temporal when its metric gradient is past-directed timelike; equivalently for the orientation field $N$ , its gradient is timelike and its differential takes a positive value on $N$ . Chronology, causality, strong causality, global hyperbolicity and time functions

[F5]

A diffeomorphism is a bijective smooth map with smooth inverse; in particular it is a homeomorphism. Diffeomorphisms and local diffeomorphisms of manifolds

[F6]

A smooth embedding is an injective smooth immersion that is a homeomorphism onto its image with the subspace topology. Smooth embeddings

[A1]

The ambient set-theoretic convention is ZFC, including the axiom of choice inherited from the stationary-data and geodesic suppliers. The pointwise matrix argument below makes no additional choice assumption. The Axiom of Choice

Proof

Given: The stationary data and exterior product chart specified in the Statement.

Goal: Prove the pointwise energy and speed bounds and the geometric assertions on a sufficiently distant sub-end.

1.1F1givenchoosealgebra

Choose $R_{*} > R_{0}$ so that the Euclidean operator norm of $h = g - η$ is at most $ε = 1 / 10$ throughout $U_{*}$ . Write the coordinate matrix as $g = (\begin{array}{cc} g_{00} & b^{T} \\ b & H \end{array})$ . Then $H$ is positive definite, since $z^{T} H z \geq (1 - ε) ∣ z ∣^{2}$ , and $g_{00} \leq - 1 + ε < 0$ . In particular $\partial_{t}$ is timelike and is future because $T = c \partial_{t}$ has the designated future orientation. Solving the block equations for the inverse gives $g^{00} = (g_{00} - b^{T} H^{- 1} b)^{- 1} < 0$ ; hence the smooth vector $\nabla t$ is timelike, with $g (\nabla t, \nabla t) = g^{00}$ .

2.1F1F3F4step 1.1

No nonzero causal vector can have zero $t$ component: otherwise the positive definiteness of $H$ would give positive norm. Consequently $d t$ never vanishes on the timelike cone. On its connected future component it is positive, since $d t (T) = c > 0$ ; in particular $d t (N) > 0$ . On the closure of that component it is nonnegative by continuity, and on a nonzero future null vector it cannot be zero, so its time component $a$ is strictly positive. Thus $g (\nabla t, N) = d t (N) > 0$ , which makes the timelike gradient past-directed and proves that $t$ is temporal.

2.2F5F6givenstep 1.1algebra

The displayed maps for the sphere and cylinder are smooth embeddings in the supplied product chart. At $x = R ω$ , their spatial tangent plane is $P = {z \in R^{3} : z \cdot ω = 0}$ . Its induced metric is the positive definite restriction of $H$ , proving that the sphere is spacelike. The cylinder tangent space is $R \partial_{t} \oplus P$ . Positive definiteness on $P$ gives a unique $Y \in P$ with $g (Y, Z) = g (\partial_{t}, Z)$ for all $Z \in P$ . The vector $W = \partial_{t} - Y$ is orthogonal to $P$ , and $g (W, W) = g_{00} - g (Y, Y) < 0$ . Thus this tangent space has one negative and two positive directions, proving that the cylinder is timelike. The local flow of $T = c \partial_{t}$ preserves $r = R$ wherever defined in this chart.

3.1givenstep 1.1step 2.1algebra

For a nonzero future null vector write $v = (a, w)$ in these coordinates, where $a > 0$ by step 2.1. The null equation and $∣ h (v, v) ∣ \leq ε (a^{2} + ∣ w ∣^{2})$ give $(1 - ε) ∣ w ∣^{2} \leq (1 + ε) a^{2}$ , hence $∣ w ∣ \leq 2 a$ . Moreover $E (v) / c = - g (\partial_{t}, v) = a - h (\partial_{t}, v) \geq a - ε \sqrt{a^{2} + ∣ w ∣^{2}} \geq a (1 - \sqrt{5} / 10) \geq a / 2 > 0$ . Therefore $∣ w ∣ \leq 4 E (v) / c$ , proving positivity and the pointwise speed estimate without any geodesic premise.

4.1F2step 1.1step 2.1step 2.2step 3.1algebra∎

Along any differentiable future null curve in $U_{*}$ , $w = d x / d λ$ and $d r / d λ = (x / r) \cdot w$ , so $∣ d r / d λ ∣ \leq ∣ w ∣ \leq C E (v)$ with $C = 4 / c$ . For an affine null geodesic, [F2] additionally makes this energy constant on its segment. The radius and constants were fixed solely from the given end and $c$ ; all estimates concern existing vectors or curve segments, with no extension or completeness assertion. Together with steps 1.1, 2.1, 2.2 and 3.1 this proves every conclusion.

Depends on

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