de Sitter space and Holography

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Consider the Brown-York stress tensor T µν for a spacetimewhich is asymptotically dS. One may ask the followingquestions:1. What would be the boundary conditions for the metric ifwe want the stress tensor to be finite?2. What is the most general diffeomorphism which preservesthis boundary conditions?The first question can be answered by perturbing dS spaceand computing T µν and then one may answer to the secondquestion which will beThe conformal group of the (d − 1)-dimensional Euclideanspace.This is one of the main hints of the dS/CFT correspondencewhich saysQuantum gravity on dS d is dual to a (d − 1)-dimensionalEuclidean conformal field theory residing on the past boundaryI − of dS d .This CFT may be non-unitary!To see how this works, let’s consider an explicit example;dS 3 .38
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
Page 1 and 2: de Sitter space and HolographyThird
Page 3: 1. Historical review and motivation
Page 7 and 8: 5. Technologies:There are a number
Page 9 and 10: Let us know study different coordin
Page 11 and 12: Conformal coordinates: T, θ i , i
Page 13 and 14: Planer (Inflationary) coordinates:
Page 15 and 16: The corresponding metric isdS 2 =
Page 17 and 18: It is easy to see thatX 0 + X d =
Page 20 and 21: 1.2 Causal structure (Penrose diagr
Page 22 and 23: IINorth PoleOSouth PoleNorth PoleOS
Page 24 and 25: The entire region of the dS space i
Page 26 and 27: For example• In static coordinate
Page 28 and 29: An observer moving along a timelike
Page 30 and 31: 3. dS/CFT correspondenceFrom what w
Page 32 and 33: In d dimensions the Einstein equati
Page 34 and 35: One can generalize it for a general
Page 36 and 37: • One can always decompose the me
Page 40 and 41: Using the definition of Brown-York
Page 42 and 43: Using this diffeomorphism one may s
Page 44 and 45: lim dtdtr→∞∫I ′ dφdφ ′
Page 46 and 47: dS static patch (spatial)dg =0 g =1
Page 48 and 49: • Lower dimensional Plank mass is
Page 50 and 51: One can use AdS/CFT correspondence
Page 52 and 53: Under this conformal map one has:
Page 54 and 55: Conformal AnomalyAdS/CFT instructs
Page 56 and 57: • Naively one would think that gr
Page 58: This talk is based on the following

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