HELP with Killing Vectors in AdS

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In summary, the conversation discussed finding the Killing vectors of the AdS metric. The process involves using the Christoffel symbols and the Killing equation to solve for components of the vector field on the manifold. These components correspond to functions on the manifold, and the resulting vector field is given by X = X^a \partial_a.
  • #1
llorgos
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Hi and I am sorry if you find my question naive.

I have to find the Killing vectors of the AdS metric

[itex]ds_{d+1}^{2} = \frac{dz^2 - dt^2 + dx^idx^i}{z^2} [/itex]

I have found the Christoffel symbols. If I use the Killing's equation [itex]\nabla_{a}X^{b} + \nabla_{b}X^{a} = 0[/itex] I find a set of differential equations. Ok, then supposing I can solve them I get components of vectors, e.g. [itex]X_{z} = ze^{c}[/itex]. So this is a component of the Killing vector?

I am quite confused and I would appreciate if someone could explain in simple steps how to proceed.

Thank you very much for your help and patience.
 
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  • #2
The metric you have, the ##ds_{d+1}^2##, gives you the components of the metric, ##g_{ab}##, which you can just read off. Feeding this into the Christoffel symbols and the Killing equation gives a system of differential equations which you solve for ##X^a##. I think you've got this far.

I think you may be confused because the ##X^a##'s are functions? Correct me if I'm wrong.

These ##X^a## should be functions on the manifold, since they correspond to the components of a vector field on it. Thus, the Killing vector field is just (locally, that is, in the coordinate system specified) ##X=X^a\partial_a##, where ##\partial_a## is the coordinate frame (I'm not sure how physicists do their notation).
 
  • #3
Yes. I get the [itex]X_a[/itex]'s or [itex]X^a[/itex]'s. I know they are funcitons on the Manifold. The thing is, do I just say, ok, the the vector field is just [itex]X = X^a \partial_a[/itex]?
Is it that simple?
 
  • #4
Yep. It's that simple.
 
  • #5
Ok. Thank you very much. Let's see if I can make any progress.
 

1. What are Killing vectors in AdS?

Killing vectors in AdS refer to the set of vector fields that preserve the AdS metric under Lie derivative. In other words, they represent the symmetries of the AdS space.

2. Why are Killing vectors important in AdS?

Killing vectors play a crucial role in understanding the symmetries and dynamics of AdS space. They are used to construct conserved quantities, such as energy and angular momentum, and to solve the equations of motion for particles and fields in AdS.

3. How do you find Killing vectors in AdS?

The standard method for finding Killing vectors in AdS is to use the Killing equation, which is a set of differential equations that must be satisfied by a Killing vector. These equations can be solved for various boundary conditions to obtain different Killing vectors.

4. Can Killing vectors in AdS be used to solve the AdS/CFT correspondence?

Yes, Killing vectors in AdS have been used in the AdS/CFT correspondence to study the symmetries of the AdS space and their dual conformal field theories. They have also been used to investigate the holographic dualities between AdS and other spaces.

5. Are there any applications of Killing vectors in AdS outside of theoretical physics?

Yes, Killing vectors in AdS have been applied in various areas of mathematics, such as differential geometry and Lie algebra. They have also been used in computer graphics to generate visualizations of AdS space and to study the behavior of particles and fields in curved spacetimes.

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