Proof of Killing Vectors Commutator Theorem

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In summary, the proof of the given theorem states that if A^{{\mu}} and B^{{\mu}} are Killing vectors, then their commutator C^{{\mu}}=[A,B]^{\mu} is also a Killing vector. This is shown by utilizing the metric g_{\mu \nu} and the covariant derivative, and finding a correlation between the two equations g_{\mu \alpha}A^{\alpha}_{;\nu}=-g_{\alpha\nu}A^{\alpha}_{;\mu} and g_{\mu \alpha}B^{\alpha}_{;\nu}=-g_{\alpha\nu}B^{\alpha}_{;\mu}. Further details on finding the
  • #1
Altabeh
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I was wondering if you could help me with the proof of the following theorem.

If [tex]A^{{\mu }}[/tex] and [tex]B^{{\mu }}[/tex] are Killing vectors, then so is their commutator [tex]C^{{\mu }}=[A,B]^{\mu}[/tex].

Thanks in advance
 
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  • #2
What have you tried?
 
  • #3
diazona said:
What have you tried?

Assuming [tex]g_{\mu \nu}[/tex] as our metric, I can write

[tex]g_{\mu \alpha}A^{\alpha}_{;\nu}=-g_{\alpha\nu}A^{\alpha}_{;\mu}[/tex] and
[tex]g_{\mu \alpha}B^{\alpha}_{;\nu}=-g_{\alpha\nu}B^{\alpha}_{;\mu}[/tex].

And I can correlate these two to each other, but I'm afraid about the commutator. I just need a clue as to how the commutator is written in terms of [tex]A^{\mu}[/tex] and [tex]B^{\mu}[/tex].
 

1. What is the Proof of Killing Vectors Commutator Theorem?

The Proof of Killing Vectors Commutator Theorem is a mathematical theorem that states that the commutator of two Killing vectors on a manifold is also a Killing vector. In other words, if two vector fields on a manifold preserve the metric and symmetries of the manifold, then their commutator will also preserve these properties.

2. Why is the Proof of Killing Vectors Commutator Theorem important?

The Proof of Killing Vectors Commutator Theorem is important because it provides a way to generate new Killing vectors from existing ones. This is useful in understanding and solving problems in physics, particularly in the field of general relativity.

3. What is the significance of Killing vectors in physics?

Killing vectors are important in physics because they represent symmetries in a system. In particular, they are used in general relativity to describe the symmetries of a spacetime, which can help in solving the Einstein field equations.

4. How is the Proof of Killing Vectors Commutator Theorem proven?

The Proof of Killing Vectors Commutator Theorem is proven using mathematical techniques such as Lie derivatives and tensor calculus. It involves showing that the commutator of two Killing vectors satisfies the Killing equation, which is the defining property of a Killing vector.

5. What are the applications of the Proof of Killing Vectors Commutator Theorem?

The Proof of Killing Vectors Commutator Theorem has various applications in physics, particularly in general relativity. It is used in deriving the geodesic equation, which describes the motion of particles in a curved spacetime, and in finding solutions to the Einstein field equations. It also has applications in other areas of mathematics, such as differential geometry and Lie theory.

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