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Killing vectors for Anti desitter space in 3+1

  1. Jul 15, 2009 #1
    I'm trying to derive the lie algebra for Anti desitter space in 3+1 I know it's O(4,1) and I think I understand that but I'm taking different approach. I tried to derive the conformal killing vectors by solving Killing's equation, but when I got my solution the conformal factor vanished :confused: .

    The solution I got for the Killing vectors is:

    \xi _t = - r\sqrt {1 + \left( {r/R} \right)^2 } \sin \theta \left( {A\sin \phi \cos \left( {t/R} \right) - B\cos \phi \cos (t/R) + C\sin \phi \sin (t/R)} \right)

    \xi _r = \frac{R}
    {{\sqrt {1 + \left( {r/R} \right)^2 } }}\sin \theta \left( {A\sin \phi \sin (t/R) - B\cos \phi \sin (t/R) - C\sin \phi \cos (t/R) + D\cos \phi \cos \left( {t/R} \right)} \right)

    \xi _\theta = rR\sqrt {1 + \left( {r/R} \right)^2 } \cos \theta \left( {A\sin \phi \sin \left( {t/R} \right) - B\cos \phi \sin \left( {t/R} \right) - C\sin \phi \cos \left( {t/R} \right) + D\cos \phi \cos \left( {t/R} \right)} \right)

    \xi _\phi = rR\sqrt {1 + \left( {r/R} \right)^2 } \sin \theta \left( {A\cos \phi \sin \left( {t/R} \right) + B\sin \phi \sin \left( {t/R} \right) - C\cos \phi \cos \left( {t/R} \right) - D\sin \phi \cos \left( {t/R} \right)} \right)

    These do satisfy the killing equation, but I feel like i"m missing something.

    they only has 4 unknowns A, B, C, D not 10 like SO(1,4), also I don't understand how Teitelbohm and Henneaux got their lie algebra in their paper asymptotically anti de sitter spaces.

    Any help will be much appreciated.
  2. jcsd
  3. Jul 19, 2009 #2
    To find the isometry group of AdS(n,m) consider its embedding in M(n,m+1) (Minkowski spacewith one extra time dimension). We can consider AdS as the set of points in M such that x2 = -R2.
    Now consider the isometries of M that leave AdS invarient; it is not hard to show that this is O(n,m+1); hence O(n,m+1)< Iso(AdS(n,m));
    As dim(O(n,m+1)) = (N2-N)/2 = (n+m)(n+m+1)/2; N = n+m+1
    wehave that O(n,m+1) = Iso(adS(n,m)), Ads space is maximally symmetric.

    To compute a basis for the lie algebra (in some coord's) use co-ord patch of the embeding of AdS and then compute which vector will be pushed forward to give all lin indep killing vectors in the embedding
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