Concept

Killing tensor

Summary
In mathematics, a Killing tensor or Killing tensor field is a generalization of a Killing vector, for symmetric tensor fields instead of just vector fields. It is a concept in pseudo-Riemannian geometry, and is mainly used in the theory of general relativity. Killing tensors satisfy an equation similar to Killing's equation for Killing vectors. Like Killing vectors, every Killing tensor corresponds to a quantity which is conserved along geodesics. However, unlike Killing vectors, which are associated with symmetries (isometries) of a manifold, Killing tensors generally lack such a direct geometric interpretation. Killing tensors are named after Wilhelm Killing. Definition and properties In the following definition, parentheses around tensor indices are notation for symmetrization. For example: :T_{(\alpha\beta\gamma)} = \frac{1}{6}(T_{\alpha\beta\gamma} + T_{\alpha\gamma\beta} + T_{\beta\alpha\gamma} + T_{\beta\gamma\alpha} + T_{\gamma\alpha\beta} + T_{\gamma\beta\alpha
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