Explanation
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.
Read more about this topic: Killing Vector Field
Famous quotes containing the word explanation:
“What causes adolescents to rebel is not the assertion of authority but the arbitrary use of power, with little explanation of the rules and no involvement in decision-making. . . . Involving the adolescent in decisions doesnt mean that you are giving up your authority. It means acknowledging that the teenager is growing up and has the right to participate in decisions that affect his or her life.”
—Laurence Steinberg (20th century)
“How strange a scene is this in which we are such shifting figures, pictures, shadows. The mystery of our existenceI have no faith in any attempted explanation of it. It is all a dark, unfathomed profound.”
—Rutherford Birchard Hayes (18221893)
“Natural selection, the blind, unconscious, automatic process which Darwin discovered, and which we now know is the explanation for the existence and apparently purposeful form of all life, has no purpose in mind. It has no mind and no minds eye. It does not plan for the future. It has no vision, no foresight, no sight at all. If it can be said to play the role of the watchmaker in nature, it is the blind watchmaker.”
—Richard Dawkins (b. 1941)