de Sitter space and Holography
de Sitter space and Holography de Sitter space and Holography
One can generalize it for a general space which isasymptotically dS. For d =, 3,4, 5 the counterterms are givenbyI ct = 18πG∫∂M +d 2 x √ hL ct + 18πG∫∂M −d 2 x √ hL ctwhereL ct = d − 2l−l 22(d − 3) R• The second term is present for d > 3.• R is the intrinsic curvature of the boundary.• The calculations are preformed by cutting off the dS spaceat finite time and then sending the surface to infinity.The total action is thenI total = I bulk + I GH + I ctWe can now compute the Euclidean boundary stress tensorwhich measures the response of the spacetime to changes ofthe boundary metric (Brown-York prescription).Consider a spacetime with the metricds 2 = g ij dx i dx j 34
One may rewrite it as followsds 2 = g ij dx i dx j = −N 2 t dt 2 + h µν (dx µ + V µ dt)(dx ν + V ν dt)So h µν is the metric induced on surface with fixed time.Suppose u µ is the future pointing unit normal to this surface,then the extrinsic curvature is given byK µν = −h i µ∇ i u νThe stress tensor associated to the boundary is given byT µν = − 2 √hδI total=18πG[δh µνwhere G µν is the Einstein tensor of h.K µν − Kh µν − d − 2 ]h µν − lGµνl d − 3In global coordinates ( on the boundary t → −∞)T θθ = −l16πG , T φφ = − l16πG sin2 θ35
- 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 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
One may rewrite it as followsds 2 = g ij dx i dx j = −N 2 t dt 2 + h µν (dx µ + V µ dt)(dx ν + V ν dt)So h µν is the metric induced on surface with fixed time.Suppose u µ is the future pointing unit normal to this surface,then the extrinsic curvature is given byK µν = −h i µ∇ i u νThe stress tensor associated to the boundary is given byT µν = − 2 √hδI total=18πG[δh µνwhere G µν is the Einstein tensor of h.K µν − Kh µν − d − 2 ]h µν − lGµνl d − 3In global coordinates ( on the boundary t → −∞)T θθ = −l16πG , T φφ = − l16πG sin2 θ35
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