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Stationary flow-box data for a nonzero Killing field

Statement

Let $(M, g, T)$ be stationary Lorentzian data, so $M$ is a smooth four-manifold and $T$ is a smooth Killing field with $L_{T} g = 0$ . On an open neighborhood $U$ of $x$ , let $ψ \in C^{\infty} (U, R)$ be a smooth scalar function and $W_{0} \in Ω^{1} (U)$ a smooth real covariant one-form. Assume both are stationary along $T$ , meaning $L_{T} ψ = 0$ and $L_{T} W_{0} = 0$ on $U$ .

If $T_{x} \neq 0$ , there are local coordinates $(t, x^{1}, x^{2}, x^{3})$ near $x$ , contained in $U$ , with $T = \partial_{t}$ . All coordinate components of $g, ψ, W_{0}$ are then independent of $t$ , whether $T_{x}$ is timelike, spacelike or null.

Their stationary two-jets at $x$ mean their values and coordinate derivatives through total order two in this chart, with every derivative involving $\partial_{t}$ equal to zero. In particular, for each component function $h$ of $g, ψ, W_{0}$ , these data are $h (x)$ , $\partial_{a} h (x)$ and $\partial_{a} \partial_{b} h (x)$ for $a, b \in {0, 1, 2, 3}$ , where $x^{0} = t$ ; the second derivatives are symmetric in $a, b$ . This is a chartwise description of the two-jets, not an assertion that component lists are invariant under changes of chart.

Facts & Assumptions

[F1]

The smooth four-manifold, metric, Killing field and local-flow definition of $L_{T} g$ are those of Stationary Lorentzian data and tensor conventions. No timelike or complete-flow hypothesis is part of stationarity here.

[F2]

A smooth $k$ -form is a smooth section of $⋀^{k} T^{*} U$ , with $Ω^{0} (U) = C^{\infty} (U)$ , by A smooth differential k-form. Thus $ψ$ is a scalar and $W_{0}$ is a covariant one-form, locally $W_{0} = (W_{0})_{a} d x^{a}$ . Their Lie derivatives use The Lie derivative of a function and The Lie derivative of a tensor field, respectively. The vanishing of both Lie derivatives is an explicit hypothesis, not a consequence of the Killing condition for $g$ .

[F3]

At a nonzero point of a smooth vector field, The flow-box theorem supplies coordinates straightening it to a coordinate vector field; the chart can be restricted to a product neighborhood contained in $U$ .

[F4]

For any smooth covariant $k$ -tensor $A$ and smooth vector field $X$ , The coordinate formula for the Lie derivative of a covariant tensor gives $(L_{X} A)_{i_{1} \dots i_{k}} = X^{j} \partial_{j} A_{i_{1} \dots i_{k}} + \sum_{a = 1}^{k} (\partial_{i_{a}} X^{j}) A_{i_{1} \dots j \dots i_{k}}$ . For a smooth scalar $h$ , Tensor Lie derivative agrees with X on functions and bracket on vector fields gives $L_{X} h = X h$ .

[F5]

Coordinate derivatives and their total order use Cᵏ maps and multi-index derivative notation in Euclidean space. Every component here is smooth, hence $C^{2}$ , so Clairaut--Schwarz theorem for continuous second partial derivatives gives $\partial_{a} \partial_{b} h = \partial_{b} \partial_{a} h$ for every component $h$ and all coordinate indices.

Proof

Given: The stationary Lorentzian data, smooth scalar and one-form, vanishing Lie derivatives and nonzero point in the Statement.

Goal: Construct the stated stationary chart and identify its two-jet data.

1.1givenF1F2F3F4

Apply [F3] at $x$ to arrange $T = \partial_{t}$ in a product chart contained in $U$ . The components of this vector field are constant, so all terms $\partial_{a} T^{b}$ in [F4] vanish. Applying its covariant-tensor formula to $g$ and $W_{0}$ , and its scalar formula to $ψ$ , gives $(L_{T} g)_{a b} = \partial_{t} g_{a b}$ , $(L_{T} W_{0})_{a} = \partial_{t} (W_{0})_{a}$ and $L_{T} ψ = \partial_{t} ψ$ . By [F1] and the stationary-field hypotheses in [F2], these expressions vanish throughout the chart. Thus the components are independent of the product coordinate $t$ .

2.1F5step 1.1∎

For each component function $h$ , differentiate the identity $\partial_{t} h = 0$ from step 1.1 in any coordinate direction to obtain $\partial_{a} \partial_{t} h = 0$ . By [F5], $\partial_{t} \partial_{a} h = \partial_{a} \partial_{t} h = 0$ as well, including $a = 0$ . The same mixed-partial identity makes all second derivatives symmetric. Consequently the component values, first derivatives and second derivatives through total order two are precisely the stationary two-jet data described in the Statement, with all time-containing derivatives zero. The construction required only $T_{x} \neq 0$ , so applies also at nonzero null or spacelike points.

Depends on

Used by · 3

Dependency tree · 2 levels

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