Killing tensor

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. 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:

${\displaystyle T_{(\alpha \beta \gamma )}={\frac {1}{6}}(T_{\alpha \beta \gamma }+T_{\alpha \gamma \beta }+T_{\beta \alpha \gamma }+T_{\beta \gamma \alpha }+T_{\gamma \alpha \beta }+T_{\gamma \beta \alpha })}$

Definition

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:

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

Properties

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]}

Examples

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.

FLRW metric

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 ${\displaystyle k=1}$ translations along ${\displaystyle x}$, ${\displaystyle y}$, and ${\displaystyle z}$. 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]

Kerr metric

Main article: Carter constant

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.

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.

Conformal Killing tensor

Conformal Killing tensors are a generalization of Killing tensors and conformal Killing vectors. A conformal Killing tensor is a tensor field ${\displaystyle K}$ (of some order m) which is symmetric and satisfies ^[4]

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

for some symmetric tensor field ${\displaystyle k}$. This generalizes the equation for conformal Killing vectors, which states that

${\displaystyle \nabla _{\alpha }K_{\beta }+\nabla _{\beta }K_{\alpha }=\lambda g_{\alpha \beta }}$

for some scalar field ${\displaystyle \lambda }$.

Every conformal Killing tensor corresponds to a constant of motion along null geodesics. More specifically, for every null geodesic with tangent vector ${\displaystyle v^{\alpha }}$, the quantity ${\displaystyle K_{\beta _{1}\cdots \beta _{m}}v^{\beta _{1}}\cdots v^{\beta _{m}}}$ is constant along the geodesic. ^[4]

The property of being a conformal Killing tensor is preserved under conformal transformations in the following sense. If ${\displaystyle K_{\beta _{1}\cdots \beta _{m}}}$ is a conformal Killing tensor with respect to a metric ${\displaystyle g_{\alpha \beta }}$, then ${\displaystyle {\tilde {K}}_{\beta _{1}\cdots \beta _{m}}=u^{m}K_{\beta _{1}\cdots \beta _{m}}}$ is a conformal Killing tensor with respect to the conformally equivalent metric ${\displaystyle {\tilde {g}}_{\alpha \beta }=ug_{\alpha \beta }}$, for all positive-valued ${\displaystyle u}$. ^[7]

See also

• Killing form
• Killing vector field
• Wilhelm Killing

References

1. Carroll 2003 , pp. 136–137
2. Wald 1984 , p. 444
3. Carroll 2003 , p. 344
4. Walker, Martin; Penrose, Roger (1970), "On Quadratic First Integrals of the Geodesic Equations for Type {22} Spacetimes" (PDF), Communications in Mathematical Physics, 18 (4): 265–274, doi:10.1007/BF01649445, S2CID   123355453
5. Carroll 2003 , pp. 262–263
6. Wald 1984 , p. 321
7. Dairbekov, N. S.; Sharafutdinov, V. A. (2011), "On conformal Killing symmetric tensor fields on Riemannian manifolds", Siberian Advances in Mathematics, 21: 1–41, arXiv: 1103.3637 , doi:10.3103/S1055134411010019

• Carroll, Sean (2003), Spacetime and Geometry: An Introduction to General Relativity, ISBN   0-8053-8732-3
• Wald, Robert M. (1984), General Relativity, Chicago: University of Chicago Press, ISBN   0-226-87033-2