In mathematics, a Killing tensor or Killing tensor field is a generalization of a

Killing vector In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric tensor, metric. Killing fields are the Lie gro ...

, for

symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...

tensor field In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis ...

s instead of just vector fields. It is a concept in

pseudo-Riemannian geometry In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which t ...

, and is mainly used in the theory of

general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...

. 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 In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...

s. However, unlike Killing vectors, which are associated with symmetries (

isometries In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...

) of a

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...

, Killing tensors generally lack such a direct geometric interpretation. Killing tensors are named after

Wilhelm Killing Wilhelm Karl Joseph Killing (10 May 1847 – 11 February 1923) was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups, and non-Euclidean geometry. Life Killing studied at the University of Mü ...

Definition and properties

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

T_ = \frac(T_ + T_ + T_ + T_ + T_ + T_)

Definition

A Killing tensor is a

K

(of some order ''m'') on a (pseudo)-Riemannian manifold which is

(that is,

K_ = K_

) and satisfies: :

\nabla_K_ = 0

This equation is a generalization of Killing's equation for

s: :

\nabla_K_ = \frac (\nabla_K_ + \nabla_K_) = 0

Properties

Killing vectors are a special case of Killing tensors. Another simple example of a Killing tensor is the

metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...

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

S_

and

T_

are Killing tensors, then

S_T_

is a Killing tensor too. Every Killing tensor corresponds to a

constant of motion In mechanics, a constant of motion is a quantity that is conserved throughout the motion, imposing in effect a constraint on the motion. However, it is a ''mathematical'' constraint, the natural consequence of the equations of motion, rather than ...

s. More specifically, for every geodesic with tangent vector

u^\alpha

, the quantity

K_ u^ \cdots u^

is constant along the geodesic.

Examples

Since Killing tensors are a generalization of Killing vectors, the examples at 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 The Friedmann–Lemaître–Robertson–Walker (FLRW; ) metric is a metric based on the exact solution of Einstein's field equations of general relativity; it describes a homogeneous, isotropic, expanding (or otherwise, contracting) universe tha ...

, widely used in

cosmology Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe. The term ''cosmology'' was first used in English in 1656 in Thomas Blount (lexicographer), Thomas Blount's ''Glossographia'', and in 1731 taken up in ...

, has spacelike Killing vectors corresponding to its spatial symmetries. It also has a Killing tensor :

K_ = a^2 (g_ + U_U_)

where ''a'' is the

scale factor In affine geometry, uniform scaling (or isotropic scaling) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a '' scale factor'' that is the same in all directions. The result of uniform scaling is similar ...

U^ = (1,0,0,0)

is the ''t''-coordinate basis vector, and the −+++

signature A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...

convention is used.

Kerr metric

The

Kerr metric The Kerr metric or Kerr geometry describes the geometry of empty spacetime around a rotating uncharged axially symmetric black hole with a quasispherical event horizon. The Kerr metric is an exact solution of the Einstein field equations of ge ...

, describing a rotating black hole, has two independent Killing vectors. One Killing vector corresponds to the

time translation symmetry Time translation symmetry or temporal translation symmetry (TTS) is a mathematical transformation in physics that moves the times of events through a common interval. Time translation symmetry is the law that the laws of physics are unchanged ( ...

of the metric, and another corresponds to the

axial symmetry Axial symmetry is symmetry around an axis; an object is axially symmetric if its appearance is unchanged if rotated around an axis.

about the axis of rotation. In addition, as shown by Walker and Penrose (1970), there is a nontrivial Killing tensor of order 2. The constant of motion corresponding to this Killing tensor is called the

Carter constant The Carter constant is a conserved quantity for motion around black holes in the general relativistic formulation of gravity. Its SI base units are kg2⋅m4⋅s−2. Carter's constant was derived for a spinning, charged black hole by Australian ...

Killing-Yano tensor

An antisymmetric tensor of order ''p'',

f_

, is a Killing-Yano tensor :fr:Tenseur de Killing-Yano if it satisfies the equation :

\nabla_b f_ + \nabla_c f_ = 0\,

. While also a generalization of the

, it differs from the usual Killing tensor in that the

covariant derivative In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a different ...

is only contracted with one tensor index.

References

* *{{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) Riemannian geometry