{{Short description|Tensor in general relativity}} In mathematics, a '''Killing tensor''' or '''Killing tensor field''' is a generalization of a [[Killing vector]], for [[Symmetric tensor|symmetric]] [[tensor field]]s instead of just [[vector field]]s. It is a concept in [[Riemannian geometry|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 [[geodesic]]s. However, unlike Killing vectors, which are associated with symmetries ([[isometries]]) of a [[pseudo-Riemannian manifold|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: :<math>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})</math>
===Definition=== A Killing tensor is a [[tensor field]] <math>K</math> (of some order ''m'') on a [[Pseudo-Riemannian manifold|(pseudo)-Riemannian manifold]] which is [[symmetric tensor|symmetric]] (that is, <math>K_{\beta_1 \cdots \beta_m} = K_{(\beta_1 \cdots \beta_m)}</math>) and satisfies:<ref name="carroll-definition">{{harvnb|Carroll|2003|pp=136–137}}</ref><ref name="wald-definition">{{harvnb|Wald|1984|p=444}}</ref> :<math>\nabla_{(\alpha}K_{\beta_1 \cdots \beta_m)} = 0</math> This equation is a generalization of Killing's equation for [[Killing vector]]s: :<math>\nabla_{(\alpha}K_{\beta)} = \frac{1}{2} (\nabla_{\alpha}K_{\beta} + \nabla_{\beta}K_{\alpha}) = 0</math>
===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 <math>S_{\alpha_1 \cdots \alpha_l}</math> and <math>T_{\beta_1 \cdots \beta_m}</math> are Killing tensors, then <math>S_{(\alpha_1 \cdots \alpha_l}T_{\beta_1 \cdots \beta_m)}</math> is a Killing tensor too.<ref name="carroll-definition" />
Every Killing tensor corresponds to a [[constant of motion]] on [[geodesic]]s. More specifically, for every geodesic with tangent vector <math>u^\alpha</math>, the quantity <math>K_{\beta_1 \cdots \beta_m} u^{\beta_1} \cdots u^{\beta_m}</math> is constant along the geodesic.<ref name="carroll-definition" /><ref name="wald-definition" />
==Examples== Since Killing tensors are a generalization of Killing vectors, the examples at {{Section link|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 [[Physical cosmology|cosmology]], has spacelike Killing vectors corresponding to its spatial symmetries, in particular rotations around arbitrary axes and in the flat case for <math>k=1</math> translations along <math>x</math>, <math>y</math>, and <math>z</math>. It also has a Killing tensor :<math>K_{\mu\nu} = a^2 (g_{\mu\nu} + U_{\mu}U_{\nu})</math> where ''a'' is the [[Scale factor (cosmology)|scale factor]], <math>U^{\mu} = (1,0,0,0)</math> is the ''t''-coordinate [[Basis (linear algebra)|basis vector]], and the −+++ [[metric signature|signature]] convention is used.<ref>{{harvnb|Carroll|2003|p=344}}</ref>
===Kerr metric=== {{Main|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 [[Roger Penrose|Penrose]] (1970), there is a nontrivial Killing tensor of order 2.<ref name="walker-penrose-1970">{{citation |title=On Quadratic First Integrals of the Geodesic Equations for Type {22} Spacetimes |first1=Martin |last1=Walker |first2=Roger |last2=Penrose |author-link2=Roger Penrose |url=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 |journal=Communications in Mathematical Physics |volume=18 |number=4 |date=1970 |pages=265–274|doi=10.1007/BF01649445 |s2cid=123355453 }}</ref><ref>{{harvnb|Carroll|2003|pp=262–263}}</ref><ref>{{harvnb|Wald|1984|p=321}}</ref> The constant of motion corresponding to this Killing tensor is called the [[Carter constant]].
==Conformal Killing tensor== '''Conformal Killing tensors''' are a generalization of Killing tensors and [[conformal Killing vector]]s. A conformal Killing tensor is a tensor field <math>K</math> (of some order ''m'') which is symmetric and satisfies<ref name="walker-penrose-1970" /> :<math>\nabla_{(\alpha}K_{\beta_1 \cdots \beta_m)} = k_{(\beta_1 \cdots \beta_{m-1}} g_{\beta_m \alpha)}</math> for some symmetric tensor field <math>k</math>. This generalizes the equation for conformal Killing vectors, which states that :<math>\nabla_\alpha K_\beta + \nabla_\beta K_\alpha = \lambda g_{\alpha \beta}</math> for some scalar field <math>\lambda</math>.
Every conformal Killing tensor corresponds to a constant of motion along [[null geodesic]]s. More specifically, for every null geodesic with tangent vector <math>v^\alpha</math>, the quantity <math>K_{\beta_1 \cdots \beta_m} v^{\beta_1} \cdots v^{\beta_m}</math> is constant along the geodesic.<ref name="walker-penrose-1970" />
The property of being a conformal Killing tensor is preserved under conformal transformations in the following sense. If <math>K_{\beta_1 \cdots \beta_m}</math> is a conformal Killing tensor with respect to a metric <math>g_{\alpha \beta}</math>, then <math>\tilde{K}_{\beta_1 \cdots \beta_m} = u^m K_{\beta_1 \cdots \beta_m}</math> is a conformal Killing tensor with respect to the conformally equivalent metric <math>\tilde{g}_{\alpha \beta} = u g_{\alpha \beta}</math>, for all positive-valued <math>u</math>.<ref>{{citation |title=On conformal Killing symmetric tensor fields on Riemannian manifolds |first1=N. S. |last1=Dairbekov |first2=V. A. |last2=Sharafutdinov |journal=Siberian Advances in Mathematics |volume=21 |date=2011 |pages=1–41|doi=10.3103/S1055134411010019 |arxiv=1103.3637 }}</ref>
==Killing–Yano tensor== An [[antisymmetric tensor]] of order ''p'', <math>f_{a_1 a_2 ... a_p}</math>, is a Killing–Yano tensor [[:fr:Tenseur de Killing-Yano]] if it satisfies the equation :<math>\nabla_b f_{c a_2 ... a_p} + \nabla_c f_{b a_2 ... a_p} = 0\,</math>. 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.
Killing–Yano tensors are the square root of Killing tensors because of satisfying certain theorems<ref name=":0">https://www.nbi.dk/~obers/MSc_PhD_files/KillingYanoProject_Dennis_final.pdf</ref> which are put below:
* For a Killing–Yano tensor, <math>f_{\mu_1\ldots\mu_p}</math>, the Killing tensor of rank 2 is <math>k_{\mu\nu} = f_{\mu\mu_2\ldots\mu_p} f_\nu^{{\mu_2}\ldots\mu_p}</math> * <math>f_{\mu\mu_2\ldots\mu_p} p^{\mu_p}</math> is parallel transported along the geodesic with tangent vector, <math>p^{\mu_p}</math>
'''Conformal Killing–Yano tensors''' are a generalization of Killing–Yano tensors.<ref name=":0" /> It states that a Conformal Killing–Yano tensor of rank ''p'' is totally antisymmetric tensor <math>k_{\mu_2...\mu_p}</math> with p-form if it fulfills:
<math>\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]}</math>
where <math>\bar k_{\mu_2\ldots\mu_p}</math> is an asymmetrical tensor of rank ''p - 1.'' By doing a contraction of <math>\mu</math> and <math>\mu_1</math>, we get:
<math>\bar k_{\mu_2\ldots\mu_p} = \frac 1 {D - p + 1} \nabla_\mu k_{\mu_2\ldots\mu_p}^\mu</math>
'''Closed Conformal Killing–Yano tensors''' are a special case of Conformal Killing–Yano tensor when <math>\nabla_\mu k_{\mid{\mu_2\ldots\mu_p}\mid} = 0</math> where <math>k = db</math> and ''b'' is some p - 1 form.<ref name=":0" /> This follows the [[Hodge star operator|Hodge duality]] transformation result which is:
The Hodge dual <math>k\star</math> of a rank ''p'' Closed Conformal Killing–Yano tensor <math>k</math> is a Killing–Yano tensor <math>f \equiv \star k</math> of rank ''D - p'' and vice-versa.
An important property of Closed Conformal Killing–Yano tensor is that their [[Exterior algebra|wedge product]] is a Closed Conformal Killing–Yano tensor of higher rank. In other words, <math>k \equiv a \land b</math> is a Closed Conformal Killing–Yano tensor of rank p + q, where <math>a</math> is a Closed Conformal Killing–Yano tensor of rank p and <math>b</math> 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:<ref>{{Cite web |last=Jezierski |first=Jacek |last2=Łukasik |first2=Maciej |date=2005-10-12 |title=Conformal Yano-Killing tensor for the Kerr metric and conserved quantities |url=https://arxiv.org/abs/gr-qc/0510058v2 |access-date=2026-05-08 |website=arXiv.org |language=en}}</ref>
<math>Y = r^3 sin\,\theta\,d\theta \,\land d\phi + O(1) = \star(\tau_0 \land D)</math>
where <math>D</math> is a dilation vector field with value <math>x^\mu {\partial\over\partial x^\mu}</math> and <math>\tau</math> is a Killing field with value <math>{\partial\over\partial x^\mu}</math>
==== 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 <math>n^{th}</math> Closed Conformal Killing–Yano tensor, the Killing–Yano tensor tower is defined as:
<math>h^{(j)} \equiv h^{\land j} = \land_{n=1}^j h</math>
where <math>h^{(j)}</math> is a Closed Conformal Killing–Yano tensor of rank <math>2j</math>.<ref name=":0" />
==== Bosonic and spinning string ==== In the invariances of tensionless [[Bosonic string theory|Bosonic string]], the expression for the field equation for tension in the string vanishes when <math>K</math> is a killing vector.<ref name=":1">{{Cite journal |last=Lindström |first=Ulf |last2=Sarıoğlu |first2=Özgür |date=2022-06-10 |title=Tensionless strings and Killing(-Yano) tensors |url=https://www.sciencedirect.com/science/article/pii/S0370269322002222 |journal=Physics Letters B |volume=829 |article-number=137088 |doi=10.1016/j.physletb.2022.137088 |issn=0370-2693|arxiv=2202.06542 }}</ref>
<math>\nabla_{(\mu} K_{\nu)} = \lambda G_{\mu\nu}</math>
The [[Invariant (mathematics)|invariant]] of the tensionless spinning strings also involves super conformal Killing–Yano tensors.<ref name=":1" />
==== Supersymmetries ==== For a bosonic particle falling in a geodesic background, the [[Supersymmetry|supersymmetric]] transformation with respect to Killing–Yano tensor can be derived.<ref name=":2">{{Cite journal |last=Santillan |first=Osvaldo P. |date=2012-04-01 |title=Hidden symmetries and supergravity solutions |url=https://pubs.aip.org/jmp/article/53/4/043509/232323/Hidden-symmetries-and-supergravity-solutions |journal=Journal of Mathematical Physics |language=en |volume=53 |issue=4 |doi=10.1063/1.3698087 |issn=0022-2488|hdl=20.500.12110/paper_00222488_v53_n4_p_Santillan |hdl-access=free }}</ref> One such supersymmetry transform equation is as follows:
<math>\delta x^\mu = -i \in \xi^\mu</math>
==== G-Structures ==== Killing–Yano tensors are also studied in [[G-structure on a manifold|G-Structures]] which are used in constructing supergravity solutions and also in [[Holonomy|holonomy manifolds]].<ref name=":2" />
==See also== *[[Killing form]] *[[Killing vector field]] *[[Wilhelm Killing]] *[[Kentaro Yano (mathematician)]]
==References== {{Reflist}} *{{citation |last=Carroll |first=Sean |author-link=Sean M. Carroll |title=Spacetime and Geometry: An Introduction to General Relativity |date=2003 |isbn=0-8053-8732-3}} *{{citation |last=Wald |first=Robert M. |author-link=Robert Wald |title=General Relativity |date=1984 |publisher=University of Chicago Press |location=Chicago |isbn=0-226-87033-2 |title-link=General Relativity (book)}}
[[Category:Riemannian geometry]]