# 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:

In terms of the Levi-Civita connection, this is

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

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

- The vector field on a circle that points clockwise and has the same length at each point is a Killing vector field, since moving each point on the circle along this vector field simply rotates the circle.
- If the metric coefficients in some coordinate basis are independent of , then is automatically a Killing vector, where is the Kronecker delta.
^{[1]}

To prove this, let us assume

Then and

Now let us look at the Killing condition

and from

The Killing condition becomes

that is , which is true.- The physical meaning is, for example, that, if none of the metric coefficients is a function of time, the manifold must automatically have a time-like Killing vector.
- In layman's terms, if an object doesn't transform or "evolve" in time (when time passes), time passing won't change the measures of the object. Formulated like this, the result sounds like a tautology, but one has to understand that the example is very much contrived: Killing fields apply also to much more complex and interesting cases.

## 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

- 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 j field is identically zero.
- If the sectional curvature is positive and the dimension of
*M*is even, a Killing field must have a zero.

The divergence of every Killing vector field vanishes.

If is a Killing vector field and is a harmonic vector field, then is a harmonic function.

If is a Killing vector field and is a harmonic p-form, then

### 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 the equation is satisfied. This aids in analytically studying motions in a spacetime with symmetries.^{[2]}

## Generalizations

- Killing vector fields can be generalized to
*conformal*Killing vector fields defined by for some scalar The derivatives of one parameter families of conformal maps are conformal Killing fields. - Killing
*tensor*fields are symmetric tensor fields*T*such that the trace-free part of the symmetrization of vanishes. Examples of manifolds with Killing tensors include the rotating black hole and the FRW cosmology.^{[3]} - Killing vector fields can also be defined on any (possibly nonmetric) manifold M if we take any Lie group G acting on it instead of the group of isometries.
^{[4]}In this broader sense, a Killing vector field is the pushforward of a right invariant vector field on G by the group action. If the group action is effective, then the space of the Killing vector fields is isomorphic to the Lie algebra of G.

## See also

- Affine vector field
- Curvature collineation
- Homothetic vector field
- Killing form
- Killing horizon
- Killing spinor
- Killing tensor
- Matter collineation
- Spacetime symmetries

## Notes

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

## 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 ed.). New York: McGraw-Hill. ISBN 0-07-000423-4..*See chapters 3,9*