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  "body": "\n## Summary\n\nA Killing vector field on a [[pseudo-Riemannian manifold|GH6CXWFNDJ]] is a vector field along which the metric does not change.\nThat is, the [[Lie derivative|PJWK0PKN6Q]] of the metric along it is zero.\nIts local flow consists of [[isometries|R9YWQS15AA]], so Killing fields are the infinitesimal symmetries of the metric.\nThey form a Lie algebra, which is the Lie algebra of the isometry group when the manifold is complete.\nIn physics they give conserved quantities along geodesics.\n\n## Explanations\n\n### L1 human\n\nImagine sliding every point of a space a little along a vector field.\nIf the slide never stretches or shrinks any distance, the vector field is a Killing field.\nOn the round sphere, the field that turns everything around the north–south axis is a Killing field, since a rotation keeps distances.\nIn the plane, the constant field $(1, 0)$ is a Killing field, because translating the plane keeps distances, while the field $(x, y)$ is not, because its flow is a scaling.\n",
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