# Orbits of a Killing vector field

I was wondering what the orbits of a Killing vector field are. Do you have any good sources or reading material for this?

bcrowell
Staff Emeritus
Gold Member
http://www.lightandmatter.com/html_books/genrel/ch07/ch07.html#Section7.1 [Broken]

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Given a (not necessarily Killing) vector field $V^\mu(x)$, its "orbits" are solutions $Z^\mu(\lambda)$ to the differential equation $\dot{Z}^\mu(\lambda) = V^\mu(Z(\lambda))$, where the dot is a $\lambda$ derivative. Intuitively, an orbit just "follows the little arrows of the vector field".

That's all there is to it.

Given a (not necessarily Killing) vector field $V^\mu(x)$, its "orbits" are solutions $Z^\mu(\lambda)$ to the differential equation $\dot{Z}^\mu(\lambda) = V^\mu(Z(\lambda))$, where the dot is a $\lambda$ derivative. Intuitively, an orbit just "follows the little arrows of the vector field".

That's all there is to it.

Thanks Sam. I'll do some calculations with this definition and come back if I have further clarifications.

Hall's book on symmetries in GR is my favorite, I'd go so far as to say, I love it.