de Sitter space and Holography

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1.2 Causal structure (Penrose diagram)Using a conformal factor Ω 2 = cos 2 T one may bring in thepoints at infinity to a finite position so that the whole spacetimeis shrink into a finite region called Penrose diagram.Since a conformal map does not change the causal structure,this may use to study the causal structure of the dS space,though the distances are highly distorted.We would like to study causal structure in various coordinatesin terms on the Penrose diagram.d ˜S 2 = cos 2 TdS 2 = −dT 2 + l 2 dΩ 2 d−1One may write this as follows−dT 2 + dθ 2 1 + sin 2 θ 1 dΩ 2 d−2,0 < θ 1 < πΙ + (T/l= π /2 = )( = 0) θ1North PoleSurface ofconstant T (or )Time line ofconstant ( θ 1 , θ2 , ... , θd-1 )South Pole ( θ1 = π )Ι − (T/l=- π /2 = − )20
INorth PoleSouth PoleI1) North and South poles are timelike line.2) Every point in the interior represents an S d−2 .3) A horizontal slice is an S d−1 .4) I − and I + are past and future null infinity. They are thesurfaces where all null geodesics originate and terminate.5) The dashed lines are the past and future horizons of anobserver at the south pole.Therefore no single observer can see entire spacetime.21
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 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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