de Sitter space and Holography

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Any null geodesic with respect to the conformal metric is alsonull in the conformally rescaled metricd ˜S 2 = cos 2 T l dS2 = −dT 2 + l 2 dΩ 2 d−1Therefore this coordinates system is useful for studying thecausal structure of the dS space.12
Planer (Inflationary) coordinates: t, x i , i = 1, · · · ,d − 1Consider the constraint and decompose into two parts(−X0 2 + Xd) 2 + (X1 2 + · · · + Xd−1) 2 = l 2One may also consider a situation−X0 2 + Xd 2 = l 2 − x 2 e −2t/lX1 2 + · · · + Xd−1 2 = x 2 e −2t/l (x 2 = x i x i )which may solve as followsX 0X 0X i= l sinh t l − x22 e−t/l= l cosh t l − x22 e−t/l= x i e −t/lfor −∞ < x i < ∞ and −∞ < t < ∞.13
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: Conformal coordinates: T, θ i , i
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 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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