Killing vector fields

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 will not distort distances on the object.

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.

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.

A typical use of the Killing Field is to express a symmetry in General relativity (in which the geometry of spacetime as distorted by gravitational fields is viewed as a 4-dimensional Riemanian manifold). In a static configuration, in which nothing changes with time, the time vector will be a Killing vector, and thus the Killing field will point in the direction of forward motion in time.

...
Wikipedia