# Killing tensor

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Tensor in general relativity

In mathematics, a **Killing tensor** or **Killing tensor field** is a generalization of a [Killing vector](/source/Killing_vector), for [symmetric](/source/Symmetric_tensor) [tensor fields](/source/Tensor_field) instead of just [vector fields](/source/Vector_field). It is a concept in [Riemannian](/source/Riemannian_geometry) and [pseudo-Riemannian geometry](/source/Pseudo-Riemannian_geometry), and is mainly used in the theory of [general relativity](/source/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](/source/Geodesic). However, unlike Killing vectors, which are associated with symmetries ([isometries](/source/Isometries)) of a [manifold](/source/Pseudo-Riemannian_manifold), Killing tensors generally lack such a direct geometric interpretation. Killing tensors are named after [Wilhelm Killing](/source/Wilhelm_Killing).

## Definition and properties

In the following definition, parentheses around tensor indices are notation for symmetrization. For example:

- T ( α β γ ) = 1 6 ( T α β γ + T α γ β + T β α γ + T β γ α + T γ α β + T γ β α ) {\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](/source/Tensor_field) K {\displaystyle K} (of some order *m*) on a [(pseudo)-Riemannian manifold](/source/Pseudo-Riemannian_manifold) which is [symmetric](/source/Symmetric_tensor) (that is, K β 1 ⋯ β m = K ( β 1 ⋯ β m ) {\displaystyle K_{\beta _{1}\cdots \beta _{m}}=K_{(\beta _{1}\cdots \beta _{m})}} ) and satisfies:[1][2]

- ∇ ( α K β 1 ⋯ β m ) = 0 {\displaystyle \nabla _{(\alpha }K_{\beta _{1}\cdots \beta _{m})}=0}

This equation is a generalization of Killing's equation for [Killing vectors](/source/Killing_vector):

- ∇ ( α K β ) = 1 2 ( ∇ α K β + ∇ β K α ) = 0 {\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](/source/Metric_tensor) itself. A [linear combination](/source/Linear_combination) of Killing tensors is a Killing tensor. A symmetric product of Killing tensors is also a Killing tensor; that is, if S α 1 ⋯ α l {\displaystyle S_{\alpha _{1}\cdots \alpha _{l}}} and T β 1 ⋯ β m {\displaystyle T_{\beta _{1}\cdots \beta _{m}}} are Killing tensors, then S ( α 1 ⋯ α l T β 1 ⋯ β m ) {\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](/source/Constant_of_motion) on [geodesics](/source/Geodesic). More specifically, for every geodesic with tangent vector u α {\displaystyle u^{\alpha }} , the quantity K β 1 ⋯ β m u β 1 ⋯ u β m {\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](/source/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](/source/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric), widely used in [cosmology](/source/Physical_cosmology), has spacelike Killing vectors corresponding to its spatial symmetries, in particular rotations around arbitrary axes and in the flat case for k = 1 {\displaystyle k=1} translations along x {\displaystyle x} , y {\displaystyle y} , and z {\displaystyle z} . It also has a Killing tensor

- K μ ν = a 2 ( g μ ν + U μ U ν ) {\displaystyle K_{\mu \nu }=a^{2}(g_{\mu \nu }+U_{\mu }U_{\nu })}

where *a* is the [scale factor](/source/Scale_factor_(cosmology)), U μ = ( 1 , 0 , 0 , 0 ) {\displaystyle U^{\mu }=(1,0,0,0)} is the *t*-coordinate [basis vector](/source/Basis_(linear_algebra)), and the −+++ [signature](/source/Metric_signature) convention is used.[3]

### Kerr metric

Main article: [Carter constant](/source/Carter_constant)

The [Kerr metric](/source/Kerr_metric), describing a rotating black hole, has two independent Killing vectors. One Killing vector corresponds to the [time translation symmetry](/source/Time_translation_symmetry) of the metric, and another corresponds to the [axial symmetry](/source/Axial_symmetry) about the axis of rotation. In addition, as shown by Walker and [Penrose](/source/Roger_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](/source/Carter_constant).

## Conformal Killing tensor

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

- ∇ ( α K β 1 ⋯ β m ) = k ( β 1 ⋯ β m − 1 g β m α ) {\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 k {\displaystyle k} . This generalizes the equation for conformal Killing vectors, which states that

- ∇ α K β + ∇ β K α = λ g α β {\displaystyle \nabla _{\alpha }K_{\beta }+\nabla _{\beta }K_{\alpha }=\lambda g_{\alpha \beta }}

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

Every conformal Killing tensor corresponds to a constant of motion along [null geodesics](/source/Null_geodesic). More specifically, for every null geodesic with tangent vector v α {\displaystyle v^{\alpha }} , the quantity K β 1 ⋯ β m v β 1 ⋯ v β m {\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 K β 1 ⋯ β m {\displaystyle K_{\beta _{1}\cdots \beta _{m}}} is a conformal Killing tensor with respect to a metric g α β {\displaystyle g_{\alpha \beta }} , then K ~ β 1 ⋯ β m = u m K β 1 ⋯ β m {\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 g ~ α β = u g α β {\displaystyle {\tilde {g}}_{\alpha \beta }=ug_{\alpha \beta }} , for all positive-valued u {\displaystyle u} .[7]

## Killing–Yano tensor

An [antisymmetric tensor](/source/Antisymmetric_tensor) of order *p*, f a 1 a 2 . . . a p {\displaystyle f_{a_{1}a_{2}...a_{p}}} , is a Killing–Yano tensor [fr:Tenseur de Killing-Yano](https://fr.wikipedia.org/wiki/Tenseur_de_Killing-Yano) if it satisfies the equation

- ∇ b f c a 2 . . . a p + ∇ c f b a 2 . . . a p = 0 {\displaystyle \nabla _{b}f_{ca_{2}...a_{p}}+\nabla _{c}f_{ba_{2}...a_{p}}=0\,} .

While also a generalization of the [Killing vector](/source/Killing_vector), it differs from the usual Killing tensor in that the [covariant derivative](/source/Covariant_derivative) is only contracted with one tensor index.

Killing–Yano tensors are the square root of Killing tensors because of satisfying certain theorems[8] which are put below:

- For a Killing–Yano tensor, f μ 1 … μ p {\displaystyle f_{\mu _{1}\ldots \mu _{p}}} , the Killing tensor of rank 2 is k μ ν = f μ μ 2 … μ p f ν μ 2 … μ p {\displaystyle k_{\mu \nu }=f_{\mu \mu _{2}\ldots \mu _{p}}f_{\nu }^{{\mu _{2}}\ldots \mu _{p}}}

- f μ μ 2 … μ p p μ p {\displaystyle f_{\mu \mu _{2}\ldots \mu _{p}}p^{\mu _{p}}} is parallel transported along the geodesic with tangent vector, p μ p {\displaystyle p^{\mu _{p}}}

**Conformal Killing–Yano tensors** are a generalization of Killing–Yano tensors.[8] It states that a Conformal Killing–Yano tensor of rank *p* is totally antisymmetric tensor k μ 2 . . . μ p {\displaystyle k_{\mu _{2}...\mu _{p}}} with p-form if it fulfills:

∇ μ k μ 1 … μ p = ∇ [ μ k μ 1 … μ p ] + p g μ [ μ 1 k μ 2 … μ p ] ¯ {\displaystyle \nabla _{\mu }k_{\mu _{1}\ldots \mu _{p}}=\nabla _{[\mu k_{\mu _{1}\ldots \mu _{p}}]}+\,}{\bar {p\,g_{\mu [\mu _{1}}{k}_{\mu _{2}\ldots \mu _{p}]}}}

where k ¯ μ 2 … μ p {\displaystyle {\bar {k}}_{\mu _{2}\ldots \mu _{p}}} is an asymmetrical tensor of rank *p - 1.* By doing a contraction of μ {\displaystyle \mu } and μ 1 {\displaystyle \mu _{1}} , we get:

k ¯ μ 2 … μ p = 1 D − p + 1 ∇ μ k μ 2 … μ p μ {\displaystyle {\bar {k}}_{\mu _{2}\ldots \mu _{p}}={\frac {1}{D-p+1}}\nabla _{\mu }k_{\mu _{2}\ldots \mu _{p}}^{\mu }}

**Closed Conformal Killing–Yano tensors** are a special case of Conformal Killing–Yano tensor when ∇ μ k ∣ μ 2 … μ p ∣ = 0 {\displaystyle \nabla _{\mu }k_{\mid {\mu _{2}\ldots \mu _{p}}\mid }=0} where k = d b {\displaystyle k=db} and *b* is some p - 1 form.[8] This follows the [Hodge duality](/source/Hodge_star_operator) transformation result which is:

The Hodge dual k ⋆ {\displaystyle k\star } of a rank *p* Closed Conformal Killing–Yano tensor k {\displaystyle k} is a Killing–Yano tensor f ≡ ⋆ k {\displaystyle f\equiv \star k} of rank *D - p* and vice-versa.

An important property of Closed Conformal Killing–Yano tensor is that their [wedge product](/source/Exterior_algebra) is a Closed Conformal Killing–Yano tensor of higher rank. In other words, k ≡ a ∧ b {\displaystyle k\equiv a\land b} is a Closed Conformal Killing–Yano tensor of rank p + q, where a {\displaystyle a} is a Closed Conformal Killing–Yano tensor of rank p and b {\displaystyle b} is a Closed Conformal Killing–Yano tensor of rank q.

### Examples

#### Kerr spacetime

A particle moving in Kerr spacetime is closely related to Conformal Killing–Yano tensors of rank two. There is one solution for a rotating blackhole described by Kerr Metric in which the asymptomatic tensor looks like as follows:[9]

Y = r 3 s i n θ d θ ∧ d ϕ + O ( 1 ) = ⋆ ( τ 0 ∧ D ) {\displaystyle Y=r^{3}sin\,\theta \,d\theta \,\land d\phi +O(1)=\star (\tau _{0}\land D)}

where D {\displaystyle D} is a dilation vector field with value x μ ∂ ∂ x μ {\displaystyle x^{\mu }{\partial \over \partial x^{\mu }}} and τ {\displaystyle \tau } is a Killing field with value ∂ ∂ x μ {\displaystyle {\partial \over \partial x^{\mu }}}

#### Killing–Yano towers

From the wedge product property of Closed Conformal Killing–Yano tensors, many Conformal Killing–Yano tensors can be constructed which is known as Killing–Yano tensor tower. For a n t h {\displaystyle n^{th}} Closed Conformal Killing–Yano tensor, the Killing–Yano tensor tower is defined as:

h ( j ) ≡ h ∧ j = ∧ n = 1 j h {\displaystyle h^{(j)}\equiv h^{\land j}=\land _{n=1}^{j}h}

where h ( j ) {\displaystyle h^{(j)}} is a Closed Conformal Killing–Yano tensor of rank 2 j {\displaystyle 2j} .[8]

#### Bosonic and spinning string

In the invariances of tensionless [Bosonic string](/source/Bosonic_string_theory), the expression for the field equation for tension in the string vanishes when K {\displaystyle K} is a killing vector.[10]

∇ ( μ K ν ) = λ G μ ν {\displaystyle \nabla _{(\mu }K_{\nu )}=\lambda G_{\mu \nu }}

The [invariant](/source/Invariant_(mathematics)) of the tensionless spinning strings also involves super conformal Killing–Yano tensors.[10]

#### Supersymmetries

For a bosonic particle falling in a geodesic background, the [supersymmetric](/source/Supersymmetry) transformation with respect to Killing–Yano tensor can be derived.[11] One such supersymmetry transform equation is as follows:

δ x μ = − i ∈ ξ μ {\displaystyle \delta x^{\mu }=-i\in \xi ^{\mu }}

#### G-Structures

Killing–Yano tensors are also studied in [G-Structures](/source/G-structure_on_a_manifold) which are used in constructing supergravity solutions and also in [holonomy manifolds](/source/Holonomy).[11]

## See also

- [Killing form](/source/Killing_form)

- [Killing vector field](/source/Killing_vector_field)

- [Wilhelm Killing](/source/Wilhelm_Killing)

- [Kentaro Yano (mathematician)](/source/Kentaro_Yano_(mathematician))

## References

1. ^ [***a***](#cite_ref-carroll-definition_1-0) [***b***](#cite_ref-carroll-definition_1-1) [***c***](#cite_ref-carroll-definition_1-2) [Carroll 2003](#CITEREFCarroll2003), pp. 136–137

1. ^ [***a***](#cite_ref-wald-definition_2-0) [***b***](#cite_ref-wald-definition_2-1) [Wald 1984](#CITEREFWald1984), p. 444

1. **[^](#cite_ref-3)** [Carroll 2003](#CITEREFCarroll2003), p. 344

1. ^ [***a***](#cite_ref-walker-penrose-1970_4-0) [***b***](#cite_ref-walker-penrose-1970_4-1) [***c***](#cite_ref-walker-penrose-1970_4-2) Walker, Martin; [Penrose, Roger](/source/Roger_Penrose) (1970), ["On Quadratic First Integrals of the Geodesic Equations for Type {22} Spacetimes"](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-18/issue-4/On-quadratic-first-integrals-of-the-geodesic-equations-for-type/cmp/1103842577.pdf) (PDF), *Communications in Mathematical Physics*, **18** (4): 265–274, [doi](/source/Doi_(identifier)):[10.1007/BF01649445](https://doi.org/10.1007%2FBF01649445), [S2CID](/source/S2CID_(identifier)) [123355453](https://api.semanticscholar.org/CorpusID:123355453)

1. **[^](#cite_ref-5)** [Carroll 2003](#CITEREFCarroll2003), pp. 262–263

1. **[^](#cite_ref-6)** [Wald 1984](#CITEREFWald1984), p. 321

1. **[^](#cite_ref-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](/source/ArXiv_(identifier)):[1103.3637](https://arxiv.org/abs/1103.3637), [doi](/source/Doi_(identifier)):[10.3103/S1055134411010019](https://doi.org/10.3103%2FS1055134411010019)

1. ^ [***a***](#cite_ref-:0_8-0) [***b***](#cite_ref-:0_8-1) [***c***](#cite_ref-:0_8-2) [***d***](#cite_ref-:0_8-3) [https://www.nbi.dk/~obers/MSc_PhD_files/KillingYanoProject_Dennis_final.pdf](https://www.nbi.dk/~obers/MSc_PhD_files/KillingYanoProject_Dennis_final.pdf)

1. **[^](#cite_ref-9)** Jezierski, Jacek; Łukasik, Maciej (2005-10-12). ["Conformal Yano-Killing tensor for the Kerr metric and conserved quantities"](https://arxiv.org/abs/gr-qc/0510058v2). *arXiv.org*. Retrieved 2026-05-08.

1. ^ [***a***](#cite_ref-:1_10-0) [***b***](#cite_ref-:1_10-1) Lindström, Ulf; Sarıoğlu, Özgür (2022-06-10). ["Tensionless strings and Killing(-Yano) tensors"](https://www.sciencedirect.com/science/article/pii/S0370269322002222). *Physics Letters B*. **829** 137088. [arXiv](/source/ArXiv_(identifier)):[2202.06542](https://arxiv.org/abs/2202.06542). [doi](/source/Doi_(identifier)):[10.1016/j.physletb.2022.137088](https://doi.org/10.1016%2Fj.physletb.2022.137088). [ISSN](/source/ISSN_(identifier)) [0370-2693](https://search.worldcat.org/issn/0370-2693).

1. ^ [***a***](#cite_ref-:2_11-0) [***b***](#cite_ref-:2_11-1) Santillan, Osvaldo P. (2012-04-01). ["Hidden symmetries and supergravity solutions"](https://pubs.aip.org/jmp/article/53/4/043509/232323/Hidden-symmetries-and-supergravity-solutions). *Journal of Mathematical Physics*. **53** (4). [doi](/source/Doi_(identifier)):[10.1063/1.3698087](https://doi.org/10.1063%2F1.3698087). [hdl](/source/Hdl_(identifier)):[20.500.12110/paper_00222488_v53_n4_p_Santillan](https://hdl.handle.net/20.500.12110%2Fpaper_00222488_v53_n4_p_Santillan). [ISSN](/source/ISSN_(identifier)) [0022-2488](https://search.worldcat.org/issn/0022-2488).

- [Carroll, Sean](/source/Sean_M._Carroll) (2003), *Spacetime and Geometry: An Introduction to General Relativity*, [ISBN](/source/ISBN_(identifier)) [0-8053-8732-3](https://en.wikipedia.org/wiki/Special:BookSources/0-8053-8732-3)

- [Wald, Robert M.](/source/Robert_Wald) (1984), [*General Relativity*](/source/General_Relativity_(book)), Chicago: University of Chicago Press, [ISBN](/source/ISBN_(identifier)) [0-226-87033-2](https://en.wikipedia.org/wiki/Special:BookSources/0-226-87033-2)

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