# Helical killing vector

A killing vector X is defined as a vector field that satisfies the relation

$$\mathcal{L}_X g_{\mu\nu}=0.$$

which basically means that if one were to transport the metric along this vector, there would be no change in the metric

I have often read that killing vectors of binary systems are approximated as helical killing vectors. However as the math is too abstract for me, I don't see the reasoning behind this nor do I understand what it means so I was hoping someone could tell me, why should the killing vectors be helical and what does it mean?

• Think about the motion of the binary system itself: It's approximately helical. – Danu Sep 7 '15 at 11:49
• So can I say that the killing vector is the path of the binary system? Or is that a misunderstanding? – Horus Sep 7 '15 at 11:58
• No, but it is a vector field that generates a flow, so that the metric is preserved, so the flow lines must be a helix in space-time (as the metric of the space will be approximately stationary in the co-rotating frame). (Note: A binary system in GR does not really have such a Killing vector, as the gravitational radiation destroys the symmetry – it is only approximately a Killing vector). – Sebastian Riese Sep 7 '15 at 15:42
• Hmm okay I think I get it. – Horus Sep 9 '15 at 10:24