de Sitter space and Holography

Consider dS 3 space in the planar coordinates (l = 1)ds 2 = −dt 2 + e −2t dzd¯zConsider a perturbation around this metric g ij + γ ij so thatds 2 = g ij dx i dx j = −dt 2 + e −2t dzd¯z + γ ij dx i dx jwhich can be recast into the following formds 2 = −Ndt 2 + h µν (dx µ + V µ dt)(dx ν + V ν dt)Therefor the induced metric readsh zz = γ zz , h z¯z = 1 2 e2t + γ z¯z , h¯z¯z = γ¯z¯zThe outward pointing unit normal vector is also given bySo we findn µ = (1 − γ tt, 0, 0)2K zz= −∂ z γ tz + 1 2 ∂ tγ zzK z¯z= − 1 2 e−2t (1 + γ tt2 ) − 1 2 (∂¯zγ tz + ∂ z γ t¯z − ∂ t γ z¯z )K = −2 − γ tt + 4e 2t γ z¯z − 2e 2t (∂¯z γ tz + ∂ z γ t¯z − ∂ t γ z¯z )39