TITLE: Why are Killing fields relevant in physics?
QUESTION [33 upvotes]: I'm taking a course on General Relativity and the notes that I'm following define a Killing vector field $X$  as those  verifying:
$$\mathcal{L}_Xg~=~ 0.$$
They seem to be very important in physics but I don't understand why yet because that definition is the only thing that I have so far. I'm not very familiar with the Lie derivative so I don't know how to interpret that equation so far. 
What is the meaning of the Lie derivative of the metric being 0? Why are Killing fields relevant physically?

REPLY [1 votes]: This is an answer that just addresses some of the mathematical aspects of the question, which you may or may not find useful. 
A vector field $X$ on a manifold $M$ is an infinitesimal generator of an (active) diffeomorphism $\phi: M \rightarrow M$; in differential geometry this would be a section of its tangent bundle $X \in \Gamma(TM)$.
Now, I suspect that the physical relevance is this: general covariance is the suggestion that physically relevant properties are invariant under diffeomorphisms; but presumably not all diffeomorphisms count; we would need it to preserve the metric structure, i.e. it is isometric.
Then, that $X$,  is a vector field generating such an isometric diffeomorphism, is understood when it is a Killing vector field ie $L_X g=0$.
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