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Zero stationary energy along null geodesics

Statement

Let $T$ be Killing for a smooth time-oriented Lorentzian metric, and let $γ : I \to E$ be a nonconstant affinely parametrized null geodesic in an open exterior $E$ . Its stationary energy $e = - g (T, \dot{γ})$ is constant. Its zero-energy condition $g (T, \dot{γ}) = 0$ is unchanged by any smooth regular reparametrization of its image. In particular zero energy is a property of the unparametrized geodesic. A timelike $T$ at a point excludes any nonzero null vector orthogonal to it; thus a zero-energy null geodesic lies where $T$ is spacelike, null or zero, not in its timelike region.

Use the four-dimensional signature $(-, +, +, +)$ conventions of stationary Lorentzian data. Here an affinely parametrized null geodesic means a smooth curve on an interval $I$ with nonempty interior satisfying $D_{s} \dot{γ} = 0$ , $\dot{γ} (s) \neq 0$ and $g_{γ (s)} (\dot{γ} (s), \dot{γ} (s)) = 0$ for every $s \in I$ , including any included endpoints with the one-sided derivative convention. A smooth regular reparametrization is $\tilde{γ} = γ \circ s$ , where $s : J \to I$ is a bijective smooth map of intervals with smooth inverse and $s^{'} (u) \neq 0$ everywhere; smoothness and derivatives at any included endpoints use smooth extensions and one-sided restrictions. Both increasing and decreasing changes are allowed. An unparametrized geodesic here is an equivalence class of regular parametrized curves (smooth curves with nowhere-zero velocity) under these changes, with at least one affinely parametrized geodesic representative; its other representatives need not be affinely parametrized. Equality of bare image sets is not the equivalence relation.

Facts & Assumptions

[F1]

Stationary Lorentzian data and tensor conventions supplies the smooth Lorentzian metric, signature, time-orientation field $N$ and metric-compatible connection. Geodesic of an affine connection defines affine geodesics by $D_{s} \dot{γ} = 0$ .

[F2]

Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor, clauses (a) and (b), gives the Killing equation $\nabla_{a} T_{b} + \nabla_{b} T_{a} = 0$ and conservation of the metric contraction along affine geodesics. Only these mathematical clauses are used.

[F3]

The Lorentzian interface imported by Stationary Lorentzian data and tensor conventions defines timelike, spacelike and null vectors; null vectors are nonzero and have zero squared norm. These pointwise conventions distinguish the zero vector from all three causal characters. The null-geodesic convention specified locally in the Statement combines this pointwise null condition with the affine-geodesic definition in [F1].

[F4]

The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a) applies to the smooth coordinate maps of the curve composed with a smooth parameter change.

[F5]

Classification of future and past causal vectors by a timelike field, clause 1, says that the orthogonal complement of the smooth timelike time-orientation field in a time-oriented Lorentzian four-manifold is positive definite. The field in that result may be any smooth timelike field on the domain where it is applied.

Proof

Given: The stated metric, Killing field and nonconstant affinely parametrized null geodesic, with the parameter-change conventions in the Statement.

Goal: Show energy conservation, parameter independence at zero energy and the location restriction.

1.1F1F2givenalgebra

Metric compatibility and the geodesic equation give $\frac{d}{d s} g (T, \dot{γ}) = g (\nabla_{\dot{γ}} T, \dot{γ}) + g (T, D_{s} \dot{γ}) = \frac{1}{2} {\dot{γ}}^{a} {\dot{γ}}^{b} (\nabla_{a} T_{b} + \nabla_{b} T_{a}) = 0$ . Thus the signed energy is constant on the interval, also directly by clause (b) of [F2].

1.2F3F4givenalgebra

For a permitted change $s : J \to I$ , apply [F4] in a chart around $γ (s (u))$ to obtain $\frac{d}{d u} (γ \circ s) (u) = s^{'} (u) \dot{γ} (s (u))$ . Bilinearity gives $g (T, {\tilde{γ}}^{'} (u)) = s^{'} (u) g (T, \dot{γ} (s (u)))$ . Since $s^{'} (u) \neq 0$ and $s$ is onto, the contraction vanishes along one representative exactly when it vanishes along the other. The new velocity remains nonzero and null, since its squared norm is $s^{'} (u)^{2} g (\dot{γ}, \dot{γ}) = 0$ . These identities extend to included endpoints by the one-sided conventions. Thus zero energy is a property of the specified equivalence class. Constancy itself was asserted only for affine parameters.

2.1F1F3F5givenstep 1.1step 1.2algebra∎

The timelike region $U = {p \in E : g_{p} (T_{p}, T_{p}) < 0}$ is open by smoothness. On $U$ , the restricted field $T ∣_{U}$ is itself a smooth time orientation, so apply [F5] to $(U, g ∣_{U}, T ∣_{U})$ . It gives positive definiteness of $g_{p}$ on $T_{p}^{⊥} = {v : g_{p} (v, T_{p}) = 0}$ for every $p \in U$ . Consequently every nonzero vector orthogonal to timelike $T_{p}$ has positive squared norm and cannot be null by [F3]. The velocity of the stated null geodesic is nonzero at every point by the null convention. Hence a zero-energy null geodesic never meets $U$ . Every remaining value of $T$ is spacelike, null or zero by [F3], which proves the location restriction together with the energy conclusions already obtained.

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