Wikipedia / Killing tensor
Wikipedia / 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:
Definition
A Killing tensor is a tensor field (of some order m) on a (pseudo)-Riemannian manifold which is symmetric) and satisfies:
This equation is a generalization of Killing's equation for Killing vectors:
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 is a Killing tensor too.
Every Killing tensor corresponds to a constant of motion on geodesics 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, widely used in cosmology. It also has a Killing tensor
where a is the scale factor is the t-coordinate basis vector, and the −+++ signature convention is used.
Kerr metric
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. 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 vectors (of some order m) which is symmetric and satisfies
. This generalizes the equation for conformal Killing vectors, which states that
.
Every conformal Killing tensor corresponds to a constant of motion along null geodesics is constant along the geodesic.
.
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, 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 which are put below:
Conformal Killing–Yano tensors are a generalization of Killing–Yano tensors. It states that a Conformal Killing–Yano tensor of rank p with p-form if it fulfills:
is an asymmetrical tensor of rank p - 1., we get:
Closed Conformal Killing–Yano tensors and b is some p - 1 form. This follows the Hodge duality transformation result which is:
of a rank p of rank D - p and vice-versa.
An important property of Closed Conformal Killing–Yano tensor is that their wedge product 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:
Killing–Yano towers
Closed Conformal Killing–Yano tensor, the Killing–Yano tensor tower is defined as:
.
Bosonic and spinning string
In the invariances of tensionless Bosonic string is a killing vector.
The invariant of the tensionless spinning strings also involves super conformal Killing–Yano tensors.
Supersymmetries
For a bosonic particle falling in a geodesic background, the supersymmetric transformation with respect to Killing–Yano tensor can be derived. One such supersymmetry transform equation is as follows:
G-Structures
Killing–Yano tensors are also studied in G-Structures which are used in constructing supergravity solutions and also in holonomy manifolds.