Killing vector field

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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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