Killing vector field

However, for various special forms of , there are additional Killing vector fields.

In this coordinate system the Killing vector field has the components .

This can only be done in spacetimes with a timelike Killing vector field.

In the case that is identically zero, is called a Killing vector field.

Killing vector fields find extensive applications (including in classical mechanics) and are related to conservation laws.

The divergence of every Killing vector field vanishes.

A static spacetime is one in which a vorticity-free timelike Killing vector field can be found.

This group is generated by a six dimensional Lie algebra of Killing vector fields.

If is the square of the norm of the Killing vector field, , both and are independent of time (in fact ).

In rough terms, Killing vector fields preserve the distance between any two points of the manifold and often go by the name of isometries.

They are a special kind of spinor field related to Killing vector fields and Killing tensors.

Killing vector fields (Killings) preserve the metric, i.e. .

For a spherically symmetric spacetime , there are precisely 3 rotational Killing vector fields.

If the spacetime supports a Killing vector field tangent to a circle, then the angular momentum about the axis is conserved.

It turns out that the most general pp-wave spacetime has only one Killing vector field, the null geodesic congruence .

Alternatively, a stationary spacetime can be defined as a spacetime which possesses a timelike Killing vector field.

When the energy-momentum tensor represents an electromagnetic field, a Killing vector field does not necessarily preserve the electric and magnetic fields.

In this broader sense, a Killing vector field is the pushforward of a left 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.

The isometries referred to above are actually local flow diffeomorphisms of Killing vector fields and thus generate these vector fields.

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

The restriction of a Killing vector field to a geodesic is a Jacobi field in any Riemannian manifold.

In the usual coordinates, outside the Killing horizon, the Killing vector field is timelike, whilst inside it is spacelike.

The fact that our spacetime admits an irrotational timelike Killing vector field is in fact the defining characteristic of a static spacetime.

In a stationary spacetime with timelike killing vector field , the temperature satisfies instead the Tolman-Ehrenfest relation: .