de Sitter space and Holography
de Sitter space and Holography de Sitter space and Holography
3. dS/CFT correspondenceFrom what we have learned in AdS/CFT correspondence onemay hope that some kind of holography can also be appliedhere and could help us to understand the quantum gravity ondS.There is a naive observation:Consider a AdS space with radius l, under l → il one getsΛ −→ −ΛAdS −→ dSSO(2,d) −→ SO(1, d + 1)Gravity on dS is dual to a Euclidean CFT.One can make this statement more precise which is in factwhat is known as dS/CFT correspondence.30
How to define conserved charges?The deviation of the metric and other fields near spatialinfinity from the vacuum provides a way to define conservedcharges like mass, angular momentum....Equivalently the conserved charges can be computed fromthe asymptotic symmetries of a space time.For example the eigenvalue of an asymptotic timelike Killingvector will give the mass.There are two basic problems to apply this definition for dSspace (spacetime which is asymptotically dS):1. There is no spatial infinity.2. There is no globally defined asymptotic timelike Killingvector.Fortunately there is a way to proceed generalizing the Brown-York construction to define stress tensor on the Euclideanboundary and by using this quantity to define mass or othercharges for spaces which are asymptotically dS.31
- 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 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
3. dS/CFT correspon<strong>de</strong>nceFrom what we have learned in AdS/CFT correspon<strong>de</strong>nce onemay hope that some kind of holography can also be appliedhere <strong>and</strong> could help us to un<strong>de</strong>rst<strong>and</strong> the quantum gravity ondS.There is a naive observation:Consi<strong>de</strong>r a AdS <strong>space</strong> with radius l, un<strong>de</strong>r l → il one getsΛ −→ −ΛAdS −→ dSSO(2,d) −→ SO(1, d + 1)Gravity on dS is dual to a Eucli<strong>de</strong>an CFT.One can make this statement more precise which is in factwhat is known as dS/CFT correspon<strong>de</strong>nce.30
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