Killing Vector Field - Explanation

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.

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