Killing vector field

A Killing vector field (named after the German mathematician Wilhelm Killing ) is a vector field on a Riemannian manifold that preserves the metric. Killing vector fields are the infinitesimal generators of isometries (see also Lie group ).

The same is true for pseudo- Riemannian manifolds, for example, in general relativity.

Definition and properties

A vector field X is a Killing vector field if the Lie derivative of the metric g with respect to X vanishes:

With the help of the Levi -Civita connection, this means pointwise

For all vectors Y and Z, or that a skew-symmetric endomorphism with respect to the tangent space.

In local coordinates this leads to the so-called Killing equation

A Killing field is on the whole space-time is uniquely determined by a vector at a point and the covariant derivatives of the vector at that point.

The Lie bracket of two Killing Fields is a Killing field again. The Killing fields of a manifold M thus form a Lie algebra on M. This is the Lie algebra of the so-called isometry group of the manifold (if M is complete).

A vector field is exactly then a Killing vector field if it is a Jacobi vector field along each geodesic.

Conserved quantities

Since Killing vector fields generate isometrics, there are in physics for every Killing vector field is a conserved quantity of the corresponding space-time. In the general theory of relativity Killing vector fields are therefore of great importance in the characterization of solutions of Einstein's field equations. The conserved quantity QX to a Killing vector field X is calculated here as

Where T is the energy -momentum tensor and | g | the amount of the 4x4 determinant of the metric tensor is. In the formula, Einstein summation convention has been used.

( The space-time itself is a four-dimensional pseudo- Riemannian manifold with a time coordinate x0 ( " upper indices " ) and three spatial coordinates x1, x2 and x3, with mixed signature, for example, according to the scheme (-, , , ) the Killing vector field also has four components;. g- matrix ( " 4x4" ) for example, has a negative and three positive eigenvalues ​​of the Lorentz transformations in the flat pseudo- Riemannian Minkowski space can be used as pseudo - spins. be construed and have as a determinant of the value one. results are also in non- flat spaces. )

Areas of integration and causality

The integration range in question, the formulas of the above type is, inter alia, therefore diffizil - not randomly missing above specifications - because you iA the limited nature of the cause in question space areas (see cause and effect or causal structure) and the lead time ( " retardation " ) take into account the causes and iA for all sizes must specify the respective arguments and the summation ranges explicitly. Also, the above intentionally not the case.

In fact, the integration range of the spatial coordinates of the full under the assumption that cause and effect are temporally infinitely far apart in the above formula. But you can choose instead of an arbitrary three-dimensional hypersurface, which is causally structured similarly. This also means that the formula does not apply to black holes.

Examples

If and only if the coefficients of the metrics are independent of the local coordinate on the basis of, a killing - vector field. In those same local coordinates it is then, where is the Kronecker delta is.

A set of independent Killing vector fields of the unit sphere with the induced metric in spherical coordinates are:

Corresponding to the rotation around the x -and y- and z - axis and in the quantum mechanics, except for a factor 1 / i, the components of the angular momentum operators.

All linear combinations of these vector fields do recover Killing vector fields represents the induced isometries are exactly the elements of the Drehgrupppe. The associated conservation law is the angular momentum.