de Sitter space and Holography
de Sitter space and Holography de Sitter space and Holography
In d dimensions the Einstein equations with positivecosmological constant can be derived from the actionI= I bulk + I GH∫1− d d x √ −g(R + 2Λ) + 116πG8πGM∫∂Md d−1 x √ hKhere• I GH is Gibbons-Hawking surface term which is needed toget a well-defined Euler-Lagrange variation.• M is d-dimensional Manifold with Newton’s constant Gwith spatial Euclidean boundary ∂M.• g µν is the bulk metric.• h µν and K are induced metric and the trace of the extrinsiccurvature of the boundary. The extrinsic curvature is definedby K µν = −∇ (µ n ν) where n ν is outward pointing unit vector.• A useful length scale in the model is given byl =√(d − 1)(d − 2)2ΛFor example in the vacuum dS solution, l is the radius of dSspace.32
In general this action is divergent when evaluated on asolution of the equations of motion due to infinite volume ofthe spacetime.For example in the case of dS space and in the inflationarycoordinates one has (d = 3)I ∼ 18πG∫( ) −1d 2 xe 2t/l lwhich diverges as t → ∞.The divergence can be canceled by adding local boundarycounterterms that do not affect the equations of motion.In our case we haveI total = I + 18πG∫∂Md 2 x √ h 1 lFor dS space with two boundaries at t → ±∞, ∂M ± , we haveI total = I + 18πG∫∂M +d 2 x √ h 1 l + 18πG∫∂M −d 2 x √ h 1 lwhich has the same solution as the previous action but is finite.33
- 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 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
In general this action is divergent when evaluated on asolution of the equations of motion due to infinite volume ofthe <strong>space</strong>time.For example in the case of dS <strong>space</strong> <strong>and</strong> in the inflationarycoordinates one has (d = 3)I ∼ 18πG∫( ) −1d 2 xe 2t/l lwhich diverges as t → ∞.The divergence can be canceled by adding local boundarycounterterms that do not affect the equations of motion.In our case we haveI total = I + 18πG∫∂Md 2 x √ h 1 lFor dS <strong>space</strong> with two boundaries at t → ±∞, ∂M ± , we haveI total = I + 18πG∫∂M +d 2 x √ h 1 l + 18πG∫∂M −d 2 x √ h 1 lwhich has the same solution as the previous action but is finite.33
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