If any set of points is displaced by where all distance relationships are unchanged (i.e., there is an
Isometry), then the Vector field is called a Killing vector.
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so let
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where is the Lie Derivative. An ordinary derivative can be replaced with a covariant derivative in a
Lie Derivative, so we can take as the definition
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which gives Killing's Equation
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A Killing vector satisfies
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where is the Ricci Tensor and is the Riemann Tensor.
A 2-sphere with Metric
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has three Killing vectors, given by the angular momentum operators
The Killing vectors in Euclidean 3-space are
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In Minkowski Space, there are 10 Killing vectors
The first group is Translation, the second Rotation, and the final corresponds to a
``boost .''
© 1996-9 Eric W. Weisstein
1999-05-26