Killing vector field

Wikipedia, the free encyclopedia - Cite This Source

In mathematics, a Killing vector field, named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric. Killing fields are the infinitesimal generators of isometries; that is, flows generated by Killing fields are continuous isometries of the manifold.

If the metric coefficients $g\_\{munu\},$ in some coordinate basis $dx^a,$ are independent of $x^K,$, then $x^mu\; =\; delta^mu\_K$ is automatically a Killing vector, where $delta^mu\_K$ is the Kronecker delta. (Misner, et al, 1973). For example if none of the metric coefficients are functions of time, the manifold must automatically have a time-like Killing vector.

Explanation

Specifically, a vector field X is a Killing field if the Lie derivative with respect to X of the metric g vanishes:

$mathcal\{L\}\_X\; g\; =\; 0.,$

In terms of the Levi-Civita connection, this is

$g(nabla\_Y\; X,\; Z)\; +\; g(Y,\; nabla\_Z\; X)\; =\; 0$

for all vectors Y and Z. In local coordinates, this amounts to the Killing equation

$nabla\_mu\; X\_nu\; +\; nabla\_nu\; X\_mu\; =\; 0.$

This condition is expressed in covariant form. Therefore it is sufficient to establish it in a preferred coordinate system in order to have it hold in all coordinate systems.

A Killing field is determined uniquely by a vector at some point and its gradient (i.e. all covariant derivatives of the field at the point).

The Lie bracket of two Killing fields is still a Killing field. The Killing fields on a manifold M thus form a Lie subalgebra of vector fields on M. This is the Lie algebra of the isometry group of the manifold.

For compact manifolds

• Negative Ricci curvature implies there are no nontrivial (nonzero) Killing fields.
• Nonpositive Ricci curvature implies that any Killing field is parallel. i.e. covariant derivative along any vector field is identically zero.
• If the sectional curvature is positive and the dimension of M is even, a Killing field must have a zero.

Killing vector fields can be generalized to conformal Killing vector fields defined by

$mathcal\{L\}\_X\; g\; =\; lambda\; g$

for some scalar $lambda$. The derivatives of one parameter families of conformal maps are conformal Killing fields. Another generalization is to conformal Killing tensor fields. These are symmetric tensor fields T such that the symmetrization of $nabla\; T$ vanishes.

References

• Jost, Jurgen (2002). Riemannian Geometry and Geometric Analysis. Berlin: Springer-Verlag. ISBN 3-540-42627-2..
• Adler, Ronald; Bazin, Maurice & Schiffer, Menahem (1975). Introduction to General Relativity (Second Edition). New York: McGraw-Hill. ISBN 0-07-000423-4.. See chapters 3,9
• Gravitation. W H Freeman and Company. ISBN 0-7167-0344-0.

Wikipedia, the free encyclopedia © 2001-2006 Wikipedia contributors (Disclaimer)
This article is licensed under the GNU Free Documentation License.
Last updated on Wednesday December 12, 2007 at 11:57:50 PST (GMT -0800)
View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation