# Orbits of a Killing vector field

1. Mar 12, 2012

### praharmitra

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

2. Mar 12, 2012

### bcrowell

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

Last edited by a moderator: May 5, 2017
3. Mar 14, 2012

### Sam Gralla

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.

4. Mar 14, 2012

### praharmitra

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

5. Mar 14, 2012

### jfy4

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