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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