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
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