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 pseudo-Riemannian manifold that preserves the metric tensor. Killing vector fields are the infinitesimal generators of isom ...
, for
symmetric
Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
tensor field
In mathematics and physics, a tensor field is a function assigning a tensor to each point of a region of a mathematical space (typically a Euclidean space or manifold) or of the physical space. Tensor fields are used in differential geometry, ...
s instead of just
vector field
In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...
s. It is a concept in
Riemannian and
pseudo-Riemannian geometry
In mathematical physics, 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 the r ...
, and is mainly used in the theory of
general relativity
General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
. 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 locally 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 conn ...
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 M ...
.
Definition and properties
In the following definition, parentheses around tensor indices are notation for symmetrization. For example:
:
Definition
A Killing tensor is a
tensor field
In mathematics and physics, a tensor field is a function assigning a tensor to each point of a region of a mathematical space (typically a Euclidean space or manifold) or of the physical space. Tensor fields are used in differential geometry, ...
(of some order ''m'') on a
(pseudo)-Riemannian manifold which is
symmetric
Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
(that is,
) and satisfies:
:
This equation is a generalization of Killing's equation for
Killing vector
In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a pseudo-Riemannian manifold that preserves the metric tensor. Killing vector fields are the infinitesimal generators of isom ...
s:
:
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
In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...
of Killing tensors is a Killing tensor. A symmetric product of Killing tensors is also a Killing tensor; that is, if
and
are Killing tensors, then
is a Killing tensor too.
Every Killing tensor corresponds to a
constant of motion In mechanics, a constant of motion is a physical quantity 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 tha ...
on
geodesic
In geometry, a geodesic () is a curve representing in some sense the locally 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 conn ...
s. More specifically, for every geodesic with tangent vector
, the quantity
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 metric (FLRW; ) is a metric that describes a homogeneous, isotropic, expanding (or otherwise, contracting) universe that is path-connected, but not necessarily simply connected. The general form o ...
, widely used in
cosmology
Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe, the cosmos. The term ''cosmology'' was first used in English in 1656 in Thomas Blount's ''Glossographia'', with the meaning of "a speaking of the wo ...
, has spacelike Killing vectors corresponding to its spatial symmetries, in particular rotations around arbitrary axes and in the flat case for
translations along
,
, and
. It also has a Killing tensor
:
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 ( isotropically). The result of uniform sc ...
,
is the ''t''-coordinate basis vector, and the −+++
signature
A signature (; from , "to sign") is a 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. Signatures are often, but not always, Handwriting, handwritt ...
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 gen ...
, 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 or line (geometry). An object is said to be ''axially symmetric'' if its appearance is unchanged if transformed around an axis. The main types of axial symmetry are ''reflection symmetry'' and ''rotatio ...
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.
Killing–Yano tensor
An antisymmetric tensor of order ''p'',
, is a Killing–Yano tensor
:fr:Tenseur de Killing-Yano if it satisfies the equation
:
.
While also a generalization of the
Killing vector
In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a pseudo-Riemannian manifold that preserves the metric tensor. Killing vector fields are the infinitesimal generators of isom ...
, it differs from the usual Killing tensor in that the
covariant derivative
In mathematics and physics, covariance is a measure of how much two variables change together, and may refer to:
Statistics
* Covariance matrix, a matrix of covariances between a number of variables
* Covariance or cross-covariance between ...
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
(of some order ''m'') which is symmetric and satisfies
:
for some symmetric tensor field
.
This generalizes the equation for conformal Killing vectors, which states that
:
for some scalar field
.
Every conformal Killing tensor corresponds to a constant of motion along
null geodesic
In general relativity, a geodesic generalizes the notion of a "straight line" to curved spacetime. Importantly, the world line of a particle free from all external, non-gravitational forces is a particular type of geodesic. In other words, a fre ...
s. More specifically, for every null geodesic with tangent vector
, the quantity
is constant along the geodesic.
The property of being a conformal Killing tensor is preserved under conformal transformations in the following sense. If
is a conformal Killing tensor with respect to a metric
, then
is a conformal Killing tensor with respect to the conformally equivalent metric
, for all positive-valued
.
See also
*
Killing form
In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. Cartan's criteria (criterion of solvability and criterion of semisimplicity) sho ...
*
Killing vector field
In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a pseudo-Riemannian manifold that preserves the metric tensor. Killing vector fields are the infinitesimal generators of isom ...
*
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 M ...
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