Killing tensor

A Killing tensor is a tensor field ${\displaystyle K}$ (of some order m) on a (pseudo)-Riemannian manifold which is symmetric (that is, ${\displaystyle K_{\beta _{1}\cdots \beta _{m}}=K_{(\beta _{1}\cdots \beta _{m})}}$) and satisfies:[1][2]

${\displaystyle \nabla _{(\alpha }K_{\beta _{1}\cdots \beta _{m})}=0}$

This equation is a generalization of Killing's equation for Killing vectors:

${\displaystyle \nabla _{(\alpha }K_{\beta )}={\frac {1}{2}}(\nabla _{\alpha }K_{\beta }+\nabla _{\beta }K_{\alpha })=0}$

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 ${\displaystyle S_{\alpha _{1}\cdots \alpha _{l}}}$ and ${\displaystyle T_{\beta _{1}\cdots \beta _{m}}}$ are Killing tensors, then ${\displaystyle S_{(\alpha _{1}\cdots \alpha _{l}}T_{\beta _{1}\cdots \beta _{m})}}$ is a Killing tensor too.[1]

Every Killing tensor corresponds to a constant of motion on geodesics. More specifically, for every geodesic with tangent vector ${\displaystyle u^{\alpha }}$, the quantity ${\displaystyle K_{\beta _{1}\cdots \beta _{m}}u^{\beta _{1}}\cdots u^{\beta _{m}}}$ is constant along the geodesic.[1][2]

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

${\displaystyle K_{\mu \nu }=a^{2}(g_{\mu \nu }+U_{\mu }U_{\nu })}$

where a is the scale factor, ${\displaystyle U^{\mu }=(1,0,0,0)}$ 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.