de Sitter space and Holography

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• One can always decompose the metric h µν on the equaltime surfaces as followsh νν dx µ dx ν = N ρ dρ 2 + σ ab (dφ a + U a dρ)(dφ b + U b dρ)where φ a are angular variables parameterizing closed surfaces.• Suppose the boundary has an isometry generated by aKilling vector ξ µ . One can show that T µν ξ ν is divergencelessand therefore we can define conserved charge associated to ξ µ .• Consider n µ be the unit normal on a surface of fixed ρ.Then the conserved charge is defined∮Q = d d−2 φ √ σn µ ξ ν T µνΣPhysically this means that a collection of observers on thehypersurface whose metric is h µν would all observe the samevalue of Q provided this surface had an isometry generated byξ µ .Since there is no globally timelike Killing vector, it is difficultto see how the mass can be defined in dS space. There ishowever an proposal for this using the above construction.Consider the case where ρ is the coordinate associated withthe asymptotic Killing vector that is timelike inside the staticpatch but spacelike at I − .36
The mass of an asymptotically dS space is∮M = d d−2 φ √ σN ρ n µ n ν T µνΣOne may also define momenta as∮J a =Σd d−2 φ √ σσ ab n ν T bν 37
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 38 and 39: Consider the Brown-York stress tens
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

de Sitter space and Holography

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