De Sitter Space and Some Related Matters. Rong-Gen Cai ( 蔡荣根 ) Institute of Theoretical Physics Chinese Academy of Sciences. Contents:. What is de Sitter Space? Why de Sitter Space? C. Some Related Matters (Puzzles). What is de Sitter Space? (W. de Sitter, 1917).
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Rong-Gen Cai (蔡荣根)
Institute of Theoretical Physics
Chinese Academy of Sciences
(Willem de Sitter,1872-1934)
A four dimensional de Sitter space is a hyperboloid
embedded in a five dimensional Minkowski space!
i) The globe coordinates:
iii) The static coordinates: Coordinates
B. CoordinatesWhy the de Sitter Space?
C. CoordinatesSome Related Matters (Puzzles)
(2) What is the statistical degrees of freedom Coordinates
of the de Sitter space?
Area of cosmological horizon
(G. W. Gibbons and S. Hawking, 1977)
why not ?
(T. Banks, 2000)
is a critical limit of M theory!
(4) The vacuum for QFT in de Sitter Space? Coordinates
What is the vacuum in the inflation model?
(Bunch-Davies Vacuum, Trans-Planck Physics)
(5) Is there the dS/CFT correspondence? Coordinates
(A. Strominger, 2001)
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.
(6) The cosmological constant has any relation Coordinates
to inflation model?
(T. Banks and W. Fischler,2003)
Cosmological Entropy Bound:
(7) How to define conserved quantities for Coordinates
asymptotically de Sitter space?
(8) Are there corresponding descriptions for
thermodynamics of black hole horizon and
cosmological horizon in terms of CFTs?
(KKLT Model, hep-th/0301240)
“de Sitter Vacua in String Theory”
(10) Entropy of black hole-de-Sitter spacetime?
(This can be derived only for the lukewarm black hole)
Cai,Ji and Soh, CQG15,2783 (1998),
Cai and Guo, PRD69, 104025 (2004).
D. Defining conserved charges Coordinates
in asymptotically dS spaces
As an example, consider an (n+2)-dimensional SdS spacetime
Narirai Black Holes
Path integral method to quantum gravity Coordinates
For (asymptotically) dS space:
The action: Coordinates
A finite action could be obtained as:
The counterterms: Coordinates
Beyond the cosmological horizon: Coordinates
The Brown-York “Tensor”: Coordinates
For a Killing vector, there is a conserved charge!
The conserved mass for the Killing vector Coordinates
For the SdS spacetime: Coordinates
A Conjecture for Mass Bound in dS Spaces? Coordinates
(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.
Topological dS spaces: Coordinates
(Cai,Myung and Zhang, PRD65, 2002)
(Cai,Myung and Zhang, PRD65, 2002) Coordinates
E. Thermodynamics of black hole horizon Coordinates
and cosmological horizon in dS space
Cardy-Verlinde Formula Coordinates
------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)
(1) Cosmological horizon in SdS spacetime: Coordinates
(2) Black Hole horizon in SdS spacetime Coordinates
(Cai, NPB628, 2002)
F. Dyanamics of a Brane in SdS Spacetime Coordinates
For a closed FRW universe with a positive CC:
If , then
(E. Verlinde, 2000)
If , we introduce Coordinates
(Cai & Mung, PRD67,2003)
The dynamics of the brane is governed by Coordinates
The equation of motion:
Consider a radial timelike geodesic satisfying Coordinates
then the reduced metric on the brane:
The Penrose diagram for the SdS spacetime
Case 2: Coordinates
Case 3: Coordinates
Holography on the brane: Coordinates
On the brane, one has Coordinates
In particular, one has Coordinates
It coincides with the Friedmann equation when
the brane crosses the black hole horizon!
Thanks ! Coordinates