Orbits of a Killing vector field

• praharmitra
In summary, the orbits of a Killing vector field are solutions to a differential equation that follows the arrows of the vector field. A good source for further reading on this topic is Hall's book on symmetries in GR.
praharmitra
I was wondering what the orbits of a Killing vector field are. Do you have any good sources or reading material for this?

http://www.lightandmatter.com/html_books/genrel/ch07/ch07.html#Section7.1

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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.

Sam Gralla said:
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.

1. What is a Killing vector field?

A Killing vector field is a vector field on a manifold that preserves the metric of that manifold. In other words, it is a vector field that generates a one-parameter group of isometries, meaning it preserves the distances and angles between points on the manifold.

2. How are Killing vector fields related to orbits?

Killing vector fields are closely related to orbits because they generate the orbits of isometries on a manifold. In other words, the flow of a Killing vector field maps points to other points on the manifold, preserving the metric along the way.

3. What are the properties of orbits of a Killing vector field?

The orbits of a Killing vector field have several properties, including being closed, having constant length, and being geodesic. Additionally, the orbits form a foliation of the manifold, meaning they cover the entire manifold without intersecting.

4. How are the orbits of a Killing vector field useful in physics?

In physics, the orbits of a Killing vector field are useful in studying symmetries and conservation laws. For example, in general relativity, the existence of a Killing vector field can imply the existence of a conserved quantity, such as energy or angular momentum.

5. Are there any applications of the orbits of a Killing vector field outside of physics?

Yes, the orbits of a Killing vector field have applications in other areas such as differential geometry and dynamical systems. They are also used in computer graphics and animation to create smooth and realistic movements in virtual objects.

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