de Sitter space and Holography
de Sitter space and Holography de Sitter space and Holography
Comparing with the Einstein equation we had which was writtenin terms of the cosmological constant one findsΛ =(d − 1)(d − 2)2l 2Therefore the dS d in the flat d + 1 dimensional Minkowskispace is a hyperboloidX0X0l 2 +( X0 )2Sd-1lX , X , ..., X1 2 d8
Let us know study different coordinates systems with willbe used later. The different coordinates systems are good fordifferent purpose.Global coordinates: τ, θ i , i = 1, · · · ,d − 1This is the simplest one and can be found by just looking atthe constraint. In fact different coordinates systems correspondto the different way one can solve the constraint.Let’s decompose the constraint as follows−X 2 0 + (X 2 1 + · · · + X 2 d) = l 2so the solution isX 0 = l sinh τ l , X i = lω i cosh τ l, for i = 1, ...,dfor −∞ < τ < ∞. Here ∑ i ω2 i = 1 parameterize a S d−1sphereω 1 = cos θ 1 0 ≤ θ 1 < πω 2 = sinθ 1 cosθ 2 0 ≤ θ 2 < π· · · = · · ·ω d = sinθ 1 · · ·sinθ d−1 0 ≤ θ d−1 < 2π9
- Page 1 and 2: de Sitter space and HolographyThird
- Page 3: 1. Historical review and motivation
- Page 7: 5. Technologies:There are a number
- 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 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
Let us know study different coordinates systems with willbe used later. The different coordinates systems are good fordifferent purpose.Global coordinates: τ, θ i , i = 1, · · · ,d − 1This is the simplest one <strong>and</strong> can be found by just looking atthe constraint. In fact different coordinates systems correspondto the different way one can solve the constraint.Let’s <strong>de</strong>compose the constraint as follows−X 2 0 + (X 2 1 + · · · + X 2 d) = l 2so the solution isX 0 = l sinh τ l , X i = lω i cosh τ l, for i = 1, ...,dfor −∞ < τ < ∞. Here ∑ i ω2 i = 1 parameterize a S d−1sphereω 1 = cos θ 1 0 ≤ θ 1 < πω 2 = sinθ 1 cosθ 2 0 ≤ θ 2 < π· · · = · · ·ω d = sinθ 1 · · ·sinθ d−1 0 ≤ θ d−1 < 2π9
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