Killing Vectors

If any set of points is displaced by where all distance relationships are unchanged (i.e., there is an Isometry), then the Vector field is called a Killing vector.

 (1)

so let
 (2)

 (3)

 (4)

where is the Lie Derivative. An ordinary derivative can be replaced with a covariant derivative in a Lie Derivative, so we can take as the definition
 (5)

 (6)

which gives Killing's Equation
 (7)

A Killing vector satisfies
 (8)

 (9)

 (10)

where is the Ricci Tensor and is the Riemann Tensor.

A 2-sphere with Metric

 (11)

has three Killing vectors, given by the angular momentum operators
 (12) (13) (14)

The Killing vectors in Euclidean 3-space are
 (15) (16) (17) (18) (19) (20)

In Minkowski Space, there are 10 Killing vectors
 (21) (22) (23) (24)

The first group is Translation, the second Rotation, and the final corresponds to a boost .''