Tolman–Ehrenfest relation

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= Killing vector
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= Killing field
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A <Killing vector field> describes a direction in which geometry has a continuous <symmetry>: moving points together along its <flow of a vector field> preserves the <metric tensor>, including its pairing of <tangent vectors>. Such a <symmetry> gives a <geodesic conserved quantity from a Killing vector>, often a component of <momentum> or <angular momentum>. The direction may vary with position: <rotation> provides the basic example.

On a <Riemannian manifold> or <Lorentzian manifold> with <metric tensor> $g$, a smooth <vector field> $K$ is a <Killing vector field> when its <Lie derivative> of $g$ vanishes. If $\Phi_s$ is its local <flow of a vector field>, the equivalent statements are
$$
\mathcal L_Kg=0\quad\Longleftrightarrow\quad\Phi_s^*g=g
\quad\Longleftrightarrow\quad\nabla_{(\mu}K_{\nu)}=0.
$$
The middle equality uses the <pullback of a covariant tensor>: the local flow maps are <isometries> on their domains. The last equality is the <Killing equation>, using the <Levi-Civita connection>. The infinitesimal condition holds throughout an open domain and integrates to exact local <isometries>; it is stronger than making the first <derivatives> of $g$ vanish at a single point. See https://davidtong.org/pdfs/teaching/general-relativity/gr.pdf#page=167[Tong, section 4.3.1, equation (4.29)].

For example, on the interval $0<x<1$ with <Euclidean metric> $dx^2$, $K=\partial_x$ translates points by $x\mapsto x+s$. This is a local <isometry>, but a point reaches the end of the interval in finite flow parameter. A local <flow of a vector field> can be incomplete; see <Killing fields on incomplete manifolds need not generate global isometries>. The full-plane <translation> and <rotation> examples under <Euclidean Killing vector field> instead have flows for every real $s$.


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