释义 |
Killing VectorsIf 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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 | (3) |
where is the Lie Derivative. An ordinary derivative can be replaced with a covariant derivative in aLie Derivative, so we can take as the definition
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 | (6) |
which gives Killing's Equation
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A Killing vector satisfies
 | (8) |
 | (9) |
 | (10) |
where is the Ricci Tensor and is the Riemann Tensor.
A 2-sphere with Metric
 | (11) |
has three Killing vectors, given by the angular momentum operators 
The Killing vectors in Euclidean 3-space are
In Minkowski Space, there are 10 Killing vectors
The first group is Translation, the second Rotation, and the final corresponds to a ``boost .''
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