Killing vector field

From Wikipedia, the free encyclopedia

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 $g_{\mu\nu}\,$ 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 Kroenecker 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.

[edit] 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 equation

$\nabla_i X_j + \nabla_j X_i = 0.$

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 $λ$ . 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.

[edit] 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

Misner, Thorne, Wheeler (1973). Gravitation. W H Freeman and Company. ISBN 0-7167-0344-0.

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Category: Riemannian geometry

Killing vector field

From Wikipedia, the free encyclopedia

[edit] Explanation

[edit] References

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