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 Riemannian and pseudo-Riemannian geometry, and is mainly used in the theory of general relativity.
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.
In the following definition, parentheses around tensor indices are notation for symmetrization.
(of some order m) on a (pseudo)-Riemannian manifold which is symmetric (that is,
) and satisfies:[1][2] This equation is a generalization of Killing's equation for Killing vectors: Killing vectors are a special case of Killing tensors.
A linear combination of Killing tensors is a Killing tensor.
A symmetric product of Killing tensors is also a Killing tensor; that is, if
[1] Every Killing tensor corresponds to a constant of motion on geodesics.
More specifically, for every geodesic with tangent vector
[1][2] Since Killing tensors are a generalization of Killing vectors, the examples at Killing vector field § Examples 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, in particular rotations around arbitrary axes and in the flat case for
It also has a Killing tensor where a is the scale factor,
is the t-coordinate basis vector, and the −+++ signature convention is used.
[3] The Kerr metric, describing a rotating black hole, has two independent Killing vectors.
One Killing vector corresponds to the time translation symmetry of the metric, and another corresponds to the axial symmetry about the axis of rotation.
In addition, as shown by Walker and Penrose (1970), there is a nontrivial Killing tensor of order 2.
[4][5][6] The constant of motion corresponding to this Killing tensor is called the Carter constant.
An antisymmetric tensor of order p,
, is a Killing–Yano tensor fr:Tenseur de Killing-Yano if it satisfies the equation 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.
Conformal Killing tensors are a generalization of Killing tensors and conformal Killing vectors.
(of some order m) which is symmetric and satisfies[4] for some symmetric tensor field
This generalizes the equation for conformal Killing vectors, which states that for some scalar field
Every conformal Killing tensor corresponds to a constant of motion along null geodesics.
More specifically, for every null geodesic with tangent vector
[4] The property of being a conformal Killing tensor is preserved under conformal transformations in the following sense.
is a conformal Killing tensor with respect to a metric
is a conformal Killing tensor with respect to the conformally equivalent metric