Killing tensor

A Killing tensor, named after Wilhelm Killing, is a symmetric tensor, known in the theory of general relativity, ${\displaystyle K}$ that satisfies

${\displaystyle \nabla _{(\alpha }K_{\beta \gamma )}=0\,}$

where the parentheses on the indices refer to the symmetric part.

This is a generalization of a Killing vector. While Killing vectors are associated with continuous symmetries (more precisely, differentiable), and hence very common, the concept of Killing tensor arises much less frequently. The Kerr solution is the most famous example of a manifold possessing a Killing tensor.

Killing-Yano tensor

An antisymmetric tensor of order p, ${\displaystyle f_{a_{1}a_{2}...a_{p}}}$, is a Killing-Yano tensor fr:Tenseur de Killing-Yano if it satisfies the equation

${\displaystyle \nabla _{b}f_{ca_{2}...a_{p}}+\nabla _{c}f_{ba_{2}...a_{p}}=0\,}$.

While also a generalization of the Killing vector, it differs from the usual Killing tensor in that the covariant derivative is only contracted with one tensor index.

See also

• Killing form
• Killing vector field
• Wilhelm Killing

References