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# Desitter - Anti-de Sitter space

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In mathematics and physics, n-dimensional anti-de Sitter space is a maximally symmetric Lorentzian manifold with constant negative scalar curvature. Anti-de Sitter space and de Sitter space are named after Willem de Sitter, professor of astronomy at Leiden University and director of the Leiden Observatory.

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### Desitter - Anti-de Sitter space

1. 1. De Sitter Space and Some Related Matters Rong-Gen Cai (蔡荣根) Institute of Theoretical Physics Chinese Academy of Sciences
2. 2. Contents: A. What is de Sitter Space? B. Why de Sitter Space? C. Some Related Matters (Puzzles)
3. 3. A. What is de Sitter Space? (W. de Sitter, 1917) 1 2 0R g R gµn µn µn- - L = (Willem de Sitter,1872-1934) 1) Λ<0 anti-de Sitter 2) Λ=0 Minkowski 3) Λ>0 de Sitter
4. 4. Definitions: 1): 1 12 ( )R R g g g gµnab µa nb µb na= - 0Cµnab = Conformal Flat
5. 5. 2): 3 R S´ Topology A four dimensional de Sitter space is a hyperboloid embedded in a five dimensional Minkowski space! 2 2 2 2 2 2 0 1 2 3 4z z z z z l- + + + + = 2 2 2 2 2 2 0 1 2 3 4ds dz dz dz dz dz= - + + + +
6. 6. The coordinates often used i) The globe coordinates: 0 1 2 3 4 sinh( / ), cosh( / )cos , cosh( / )sin cos , cosh( / )sin sin cos , cosh( / )sin sin sin , z l t l z l t l z l t l z l t l z l t l c c q c q f c q f = = = = = 2 2 2 2 2 2 2 2 2 cosh ( / )[ sin ( sin )]ds dt l t l d d dc c q q f= - + + +
7. 7. ii) The planar coordinates: / 21 0 2 / 21 4 2 / , sinh( / ) | | / , cosh( / ) | | / , t l t l t l i i z l t l e x l z l t l e x l z e x = + = - = 3 2 2 2 / 2 1 ( ) i t l i i ds dt e dx = = = - + å , it x-¥ < < ¥ 1,2,3i = 0 4 0z z+ > O-gauge!
8. 8. The Penrose Diagram of de Sitter Space in the Planar Coordinates
9. 9. iii) The static coordinates: 2 2 1/2 0 2 2 1/2 1 2 3 4 ( ) sinh( / ), ( ) cosh( / ), sin cos , sin sin , cos z l r t l z l r t l z r z r z r q f q f q = - = - = = = 0 1 0z z+ > 2 2 2 2 2 2 1 2 2 2 2 2 (1 / ) (1 / ) ( sin )ds r l dt r l dr r d dq q f- = - - + - + +
10. 10. B. Why the de Sitter Space? 1) Maximally symmetric curved space 2) Inflation model for the early universe Inflation Model⊕Dark Matter ⊕ Dark Energy 22% ⊕ 73% 3) Current accelerated expanding universe 0?L >
11. 11. astro-phys/9812133
12. 12. astro-phys/9812133
13. 13. astro-phys/9812133
14. 14. astro-phys/9812133
15. 15. C. Some Related Matters (Puzzles) (1) The CC problem ----------the Cosmological Constant problem 3 4 29 3 ~ (10 ) ~10 /ev g cm- - L 4 19 4 ~ ( ) ~ (10 )QGE GevL 4 3 4 ~ ( ) ~ (10 )SUSYE Gev QFT
16. 16. (2) What is the statistical degrees of freedom of the de Sitter space? 4 A GS = Area of cosmological horizon (G. W. Gibbons and S. Hawking, 1977) 2 2 2 2 2 2 1 2 2 2 2 2 (1 / ) (1 / ) ( sin )ds r l dt r l dr r d dq q f- = - - + - + +
17. 17. (3) The cosmological constant has any relation to SUSY? In general: Fitting data: (T. Banks, 2000) 1/8a = is a critical limit of M theory!4 / 0pML ® 4 ~ ( / )SUSY p pM M M a L 1/ 4a ¹why not ?
18. 18. (4) The vacuum for QFT in de Sitter Space? α vacuum What is the vacuum in the inflation model? (Bunch-Davies Vacuum, Trans-Planck Physics)
19. 19. (5) Is there the dS/CFT correspondence? (A. Strominger, 2001) However, hep-th/0202163 by L. Dyson, J. Lindesay & L.Susskind dS complementarity precludes the existence of the appropriate limits. We find that the limits exist only in approximations in which the entropy of the de Sitter Space is infinite. The reason that the correlators exist in quantum field theory in the de Sitter Space background is traced to the fact that horizon entropy is infinite in QFT.
20. 20. (6) The cosmological constant has any relation to inflation model? (T. Banks and W. Fischler,2003) Cosmological Entropy Bound: 1/S G< L 0(65)N < (Cai, JCAP 0402(2004)007)
21. 21. (7) How to define conserved quantities for asymptotically de Sitter space? (8) Are there corresponding descriptions for thermodynamics of black hole horizon and cosmological horizon in terms of CFTs? a) AD mass (L. Abbott and S. Deser, 1982) b) Surface counterm method (V. Balasubramanian et al, 2001)
22. 22. (9) The de Sitter space can be realized in string theory? (10) Entropy of black hole-de-Sitter spacetime? BH COHS S S= + (KKLT Model, hep-th/0301240) “de Sitter Vacua in String Theory” (This can be derived only for the lukewarm black hole) Cai,Ji and Soh, CQG15,2783 (1998), Cai and Guo, PRD69, 104025 (2004).
23. 23. D. Defining conserved charges in asymptotically dS spaces 2 2 1 2 1 2 2 2 1 2 2 2 (1 ) (1 )n n n n m mr r nr l r l ds dt dr r dw w - - - = - - - + - - + W 16 n n G n p w = W As an example, consider an (n+2)-dimensional SdS spacetime Narirai Black Holes
24. 24. Path integral method to quantum gravity For (asymptotically) dS space:
25. 25. The action: A finite action could be obtained as: Counterterms
26. 26. The counterterms:
27. 27. Beyond the cosmological horizon: 2 2 1 2 1 2 2 2 1 2 2 2 ( 1 ) ( 1 )n n n n m mr r nr l r l ds dt dr r dw w - - - = - + + - - + + + W 0>
28. 28. The Brown-York “Tensor”: For a Killing vector, there is a conserved charge!
29. 29. The conserved mass for the Killing vector t ¶ ¶
30. 30. For the SdS spacetime: 1 2 , 1 2 1 ( ) 2 2 ( 1) ( ) n p n n p l M m p d w p - + - G = - - G + 4 ,mM = - 5 23 32 l GM mp= - e.g.
31. 31. A Conjecture for Mass Bound in dS Spaces? (V. Balasubramanian et al, 2001) For an asymptotically dS spaqce if its mass is beyond the mass of a pure dS space, there must be a singularity.
32. 32. Topological dS spaces: 2 2 1 2 1 2 2 2 1 2 2 22 2 ( ) ( )n n Gm Gmr r nr l r l ds dt dr r dx- - - = - - + - + 8 nmV M p= (Cai,Myung and Zhang, PRD65, 2002)
33. 33. 2 2 1 2 1 2 2 2 1 2 22 2 ( 1 ) ( 1 )n n i jGm Gmr r ijr l r l ds dt dr r g dx dx- - - = - - + - + - + - + 4 4 mV M p = (Cai,Myung and Zhang, PRD65, 2002) 2 5 2 3 1 8 8 ( ) Vl Gm M G lp = +
34. 34. E. Thermodynamics of black hole horizon and cosmological horizon in dS space 2 2 1 2 1 2 2 2 1 2 2 2 (1 ) (1 )n n n n m mr r nr l r l ds dt dr r dw w - - - = - - - + - - + W (1)Black Hole Horizon: r_+ (2) Cosmological Horizon: r_c
35. 35. Cardy-Verlinde Formula ------An Entropy Formula for a CFT (J. Cardy, 1986, E. Verlinde, 2000) in (n+1) dimensions (Cai, PRD 63, 2001; Cai, Myung & Ohta, CQG18, 2001, Cai & Zhang, PRD64, 2001)
36. 36. (1) Cosmological horizon in SdS spacetime: ( 1)cE n E nTS= + - (Cai, PLB525,2002)
37. 37. (2) Black Hole horizon in SdS spacetime (Cai, NPB628, 2002)
38. 38. F. Dyanamics of a Brane in SdS Spacetime For a closed FRW universe with a positive CC: If , then 2 11/ 0nl + = 1 2 ( 1) 4 BH n E R V n n G R p + = - (E. Verlinde, 2000)
39. 39. 2 11/ 0nl + ¹If , we introduce Then (Cai & Mung, PRD67,2003)
40. 40. The dynamics of the brane is governed by The equation of motion:
41. 41. Consider a radial timelike geodesic satisfying then the reduced metric on the brane:
42. 42. Define Case 1: The Penrose diagram for the SdS spacetime
43. 43. Case 2:
44. 44. Case 3:
45. 45. Holography on the brane: Suppose Then and
46. 46. On the brane, one has Entropy density Energy density Temperature
47. 47. In particular, one has It coincides with the Friedmann equation when the brane crosses the black hole horizon!
48. 48. Thanks !