Orbits of a Killing vector field

  • #1
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I was wondering what the orbits of a Killing vector field are. Do you have any good sources or reading material for this?
 

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  • #2
bcrowell
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http://www.lightandmatter.com/html_books/genrel/ch07/ch07.html#Section7.1 [Broken]
 
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  • #3
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Given a (not necessarily Killing) vector field [itex]V^\mu(x)[/itex], its "orbits" are solutions [itex]Z^\mu(\lambda)[/itex] to the differential equation [itex]\dot{Z}^\mu(\lambda) = V^\mu(Z(\lambda))[/itex], where the dot is a [itex]\lambda[/itex] derivative. Intuitively, an orbit just "follows the little arrows of the vector field".

That's all there is to it.
 
  • #4
311
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Given a (not necessarily Killing) vector field [itex]V^\mu(x)[/itex], its "orbits" are solutions [itex]Z^\mu(\lambda)[/itex] to the differential equation [itex]\dot{Z}^\mu(\lambda) = V^\mu(Z(\lambda))[/itex], where the dot is a [itex]\lambda[/itex] 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.
 
  • #5
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Hall's book on symmetries in GR is my favorite, I'd go so far as to say, I love it.
 

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