Concept
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. In the following definition, parentheses around tensor indices are notation for symmetrization. For example: A Killing tensor is a tensor field (of some order m) on a (pseudo)-Riemannian manifold which is symmetric (that is, ) and satisfies: This equation is a generalization of Killing's equation for Killing vectors: Killing vectors are a special case of Killing tensors. Another simple example of a Killing tensor is the metric tensor itself. A linear combination of Killing tensors is a Killing tensor. A symmetric product of Killing tensors is also a Killing tensor; that is, if and are Killing tensors, then is a Killing tensor too. Every Killing tensor corresponds to a constant of motion on geodesics. More specifically, for every geodesic with tangent vector , the quantity is constant along the geodesic. Since Killing tensors are a generalization of Killing vectors, the examples at are also examples of Killing tensors. The following examples focus on Killing tensors not simply obtained from Killing vectors. The Friedmann–Lemaître–Robertson–Walker metric, widely used in cosmology, has spacelike Killing vectors corresponding to its spatial symmetries. It also has a Killing tensor where a is the scale factor, is the t-coordinate basis vector, and the −+++ signature convention is used. Carter constant The Kerr metric, describing a rotating black hole, has two independent Killing vectors.