De Sitter Space and Some Related Matters
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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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De sitter space and some related matters

De Sitter Space and Some Related Matters

Rong-Gen Cai (蔡荣根)

Institute of Theoretical Physics

Chinese Academy of Sciences


De sitter space and some related matters

Contents:

  • What is de Sitter Space?

  • Why de Sitter Space?

  • C. Some Related Matters (Puzzles)


De sitter space and some related matters

  • What is de Sitter Space?

  • (W. de Sitter, 1917)

1) Λ<0

anti-de Sitter

2) Λ=0

Minkowski

3) Λ>0

de Sitter

(Willem de Sitter,1872-1934)


De sitter space and some related matters

Definitions:

1):

Conformal Flat


De sitter space and some related matters

Topology

2):

A four dimensional de Sitter space is a hyperboloid

embedded in a five dimensional Minkowski space!


De sitter space and some related matters

The coordinates often used

i) The globe coordinates:


De sitter space and some related matters

ii) The planar coordinates:

O-gauge!


De sitter space and some related matters

The Penrose Diagram of de Sitter Space in the Planar Coordinates


De sitter space and some related matters

iii) The static coordinates:


De sitter space and some related matters

B. Why the de Sitter Space?

  • Maximally symmetric curved space

  • 2) Inflation model for the early universe

  • Inflation Model⊕Dark Matter⊕ Dark Energy

  • 22% ⊕ 73%

  • 3) Current accelerated expanding universe


De sitter space and some related matters

astro-phys/9812133


De sitter space and some related matters

astro-phys/9812133


De sitter space and some related matters

astro-phys/9812133


De sitter space and some related matters

astro-phys/9812133


De sitter space and some related matters

C. Some Related Matters (Puzzles)

  • The CC problem

  • ----------the Cosmological Constant problem

QFT


De sitter space and some related matters

(2) What is the statistical degrees of freedom

of the de Sitter space?

Area of cosmological horizon

(G. W. Gibbons and S. Hawking, 1977)


De sitter space and some related matters

(3) The cosmological constant has any relation to SUSY?

In general:

Fitting data:

why not ?

(T. Banks, 2000)

is a critical limit of M theory!


De sitter space and some related matters

(4) The vacuum for QFT in de Sitter Space?

αvacuum

What is the vacuum in the inflation model?

(Bunch-Davies Vacuum, Trans-Planck Physics)


De sitter space and some related matters

(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.


De sitter space and some related matters

(6) The cosmological constant has any relation

to inflation model?

(T. Banks and W. Fischler,2003)

Cosmological Entropy Bound:

(Cai,JCAP 0402(2004)007)


De sitter space and some related matters

(7) How to define conserved quantities for

asymptotically de Sitter space?

  • AD mass

  • (L. Abbott and S. Deser, 1982)

  • Surface counterm method

  • (V. Balasubramanian et al, 2001)

(8) Are there corresponding descriptions for

thermodynamics of black hole horizon and

cosmological horizon in terms of CFTs?


De sitter space and some related matters

(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).


De sitter space and some related matters

D. Defining conserved charges

in asymptotically dS spaces

As an example, consider an (n+2)-dimensional SdS spacetime

Narirai Black Holes


De sitter space and some related matters

Path integral method to quantum gravity

For (asymptotically) dS space:


De sitter space and some related matters

The action:

A finite action could be obtained as:

Counterterms


De sitter space and some related matters

The counterterms:


De sitter space and some related matters

Beyond the cosmological horizon:


De sitter space and some related matters

The Brown-York “Tensor”:

For a Killing vector, there is a conserved charge!


De sitter space and some related matters

The conserved mass for the Killing vector


De sitter space and some related matters

For the SdS spacetime:

e.g.


De sitter space and some related matters

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.


De sitter space and some related matters

Topological dS spaces:

(Cai,Myung and Zhang, PRD65, 2002)


De sitter space and some related matters

(Cai,Myung and Zhang, PRD65, 2002)


De sitter space and some related matters

E. Thermodynamics of black hole horizon

and cosmological horizon in dS space

  • Black Hole Horizon: r_+

  • (2) Cosmological Horizon: r_c


De sitter space and some related matters

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)


De sitter space and some related matters

(1) Cosmological horizon in SdS spacetime:

(Cai, PLB525,2002)


De sitter space and some related matters

(2) Black Hole horizon in SdS spacetime

(Cai, NPB628, 2002)


De sitter space and some related matters

F. Dyanamics of a Brane in SdS Spacetime

For a closed FRW universe with a positive CC:

If , then

(E. Verlinde, 2000)


De sitter space and some related matters

If , we introduce

Then

(Cai & Mung, PRD67,2003)


De sitter space and some related matters

The dynamics of the brane is governed by

The equation of motion:


De sitter space and some related matters

Consider a radial timelike geodesic satisfying

then the reduced metric on the brane:


De sitter space and some related matters

Define

Case 1:

The Penrose diagram for the SdS spacetime


De sitter space and some related matters

Case 2:


De sitter space and some related matters

Case 3:


De sitter space and some related matters

Holography on the brane:

Suppose

Then

and


De sitter space and some related matters

On the brane, one has

Entropy density

Energy density

Temperature


De sitter space and some related matters

In particular, one has

It coincides with the Friedmann equation when

the brane crosses the black hole horizon!


De sitter space and some related matters

Thanks !


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