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The stationary Killing energy is constant along every affine null segment

Statement

Work in ZFC. For an affine null geodesic in the stationary metric, $E = - g (T, \dot{γ})$ is constant. This conclusion needs no timelike assumption on T along the full segment. Where T is future timelike and the geodesic future null, E is strictly positive.

Facts & Assumptions

[F1]

The contraction of a Killing field with an affine geodesic tangent is constant (clause (b)). Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor

[F2]

The stationary field is a specified smooth Killing field, whether or not it is timelike in the interior. Stationary Lorentzian data and tensor conventions

[F3]

An affinely parametrized geodesic is a smooth curve on an interval with nonempty interior satisfying $D_{t} \dot{γ} = 0$ for the specified affine connection, with one-sided interpretation at an included endpoint. Geodesic of an affine connection

[F4]

A null tangent vector $w$ is nonzero and satisfies $g (w, w) = 0$ ; a nonzero causal vector is future-directed when $g (w, N) < 0$ . Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model

[F5]

An observer at $p$ is a tangent vector $n$ satisfying $g_{p} (n, n) = - 1$ and $g_{p} (n, N_{p}) < 0$ . Observers, rest spaces and the spatial projection at an event

[F6]

For an observer $n$ and a nonzero future causal vector $w$ at the same point, $- g_{p} (n, w) > 0$ (claim 1). Pairing of an observer with future causal vectors

[A1]

Work in ZFC, including the axiom of choice; this includes the countable choice used in the smooth Lorentzian interface. The Axiom of Choice

Proof

Given: The assumptions in the Statement.

Goal: Establish constancy on the affine segment and positivity at every point with the stated future timelike and future null hypotheses.

1.1F1F2F3givenalgebra

The stationary data in [F2] give an oriented time-oriented smooth Lorentzian four-manifold and its Levi–Civita connection, with $T$ Killing. The given affine null geodesic satisfies $D_{t} \dot{γ} = 0$ by [F3]. Thus clause (b) of [F1] applies and $g (T, \dot{γ})$ , hence $E = - g (T, \dot{γ})$ , is constant on its parameter interval. That clause requires no timelike condition on $T$ .

2.1F2F4F5F6givenstep 1.1algebra∎

At any point $p = γ (t)$ where $T_{p}$ is future timelike and $w = \dot{γ} (t)$ is future null, put $a = \sqrt{- g_{p} (T_{p}, T_{p})} > 0$ and $n = T_{p} / a$ . Bilinearity gives $g_{p} (n, n) = - 1$ and $g_{p} (n, N_{p}) = g_{p} (T_{p}, N_{p}) / a < 0$ , so $n$ is an observer by [F5]. The vector $w$ is nonzero future causal by [F4], so [F6] gives $- g_{p} (n, w) > 0$ . Consequently $E (t) = - g_{p} (T_{p}, w) = a [- g_{p} (n, w)] > 0$ . Together with step 1.1 this proves both assertions.

Depends on

Used by · 3

Dependency tree · 2 levels

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