Killing vector field

In mathematics, a Killing vector field (often just Killing 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. More simply, the flow generates a symmetry, in the sense that moving each point on an object the same distance in the direction of the Killing vector field will not distort distances on the object.

Definition

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.

Examples

Properties

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 if M is complete.

For compact manifolds

The divergence of every Killing vector field vanishes.

If X is a Killing vector field and Y is a harmonic vector field, then g(X,Y) is a harmonic function.

If X is a Killing vector field and \omega is a harmonic p-form, then \mathcal{L}_{X} \omega = 0 \,.

Geodesics

Each Killing vector corresponds to a quantity which is conserved along geodesics. This conserved quantity is the metric product between the Killing vector and the geodesic tangent vector. That is, along a geodesic with some affine parameter \lambda,
the equation \frac d {d\lambda} (K_\mu \frac{dx^\mu}{d\lambda}) = 0 is satisfied. This aids in analytically studying motions in a spacetime with symmetries.[2]

Generalizations

See also

Notes

  1. ↑ Misner, Thorne, Wheeler (1973). Gravitation. W H Freeman and Company. ISBN 0-7167-0344-0.
  2. ↑ Carrol, Sean (2004). An Introduction to General Relativity Spacetime and Geometry. Addison Wesley. pp. 133–139.
  3. ↑ Carrol, Sean (2004). An Introduction to General Relativity Spacetime and Geometry. Addison Wesley. pp. 263, 344.
  4. ↑ Choquet-Bruhat, Yvonne; DeWitt-Morette, Cécile (1977), Analysis, Manifolds and Physics, Amsterdam: Elsevier, ISBN 978-0-7204-0494-4

References

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