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).
De Sitter Space and Some Related Matters
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!
The coordinates often used
i) The globe coordinates:
ii) The planar coordinates:
The Penrose Diagram of de Sitter Space in the Planar Coordinates
iii) The static coordinates:
B. Why the de Sitter Space?
C. Some Related Matters (Puzzles)
(2) What is the statistical degrees of freedom
of the de Sitter space?
Area of cosmological horizon
(G. W. Gibbons and S. Hawking, 1977)
(3) The cosmological constant has any relation to SUSY?
why not ?
(T. Banks, 2000)
is a critical limit of M theory!
(4) The vacuum for QFT in de Sitter Space?
What is the vacuum in the inflation model?
(Bunch-Davies Vacuum, Trans-Planck Physics)
(5) Is there the dS/CFT correspondence?
(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
to inflation model?
(T. Banks and W. Fischler,2003)
Cosmological Entropy Bound:
(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?
(9) The de Sitter space can be realized in string theory?
(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
in asymptotically dS spaces
As an example, consider an (n+2)-dimensional SdS spacetime
Narirai Black Holes
Path integral method to quantum gravity
For (asymptotically) dS space:
A finite action could be obtained as:
Beyond the cosmological horizon:
The Brown-York “Tensor”:
For a Killing vector, there is a conserved charge!
The conserved mass for the Killing vector
For the SdS spacetime:
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.
Topological dS spaces:
(Cai,Myung and Zhang, PRD65, 2002)
(Cai,Myung and Zhang, PRD65, 2002)
E. Thermodynamics of black hole horizon
and cosmological horizon in dS space
------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:
(2) Black Hole horizon in SdS spacetime
(Cai, NPB628, 2002)
F. Dyanamics of a Brane in SdS Spacetime
For a closed FRW universe with a positive CC:
If , then
(E. Verlinde, 2000)
If , we introduce
(Cai & Mung, PRD67,2003)
The dynamics of the brane is governed by
The equation of motion:
Consider a radial timelike geodesic satisfying
then the reduced metric on the brane:
The Penrose diagram for the SdS spacetime
Holography on the brane:
On the brane, one has
In particular, one has
It coincides with the Friedmann equation when
the brane crosses the black hole horizon!