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:
“Young children constantly invent new explanations to account for complex processes. And since their inventions change from week to week, furnishing the “correct” explanation is not quite so important as conveying a willingness to discuss the subject. Become an “askable parent.””
—Ruth Formanek (20th century)
“Are cans constitutionally iffy? Whenever, that is, we say that we can do something, or could do something, or could have done something, is there an if in the offing—suppressed, it may be, but due nevertheless to appear when we set out our sentence in full or when we give an explanation of its meaning?”
—J.L. (John Langshaw)
“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 mind’s 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)