de Sitter space and Holography

1.2 Causal structure (Penrose diagram)Using a conformal factor Ω 2 = cos 2 T one may bring in thepoints at infinity to a finite position so that the whole spacetimeis shrink into a finite region called Penrose diagram.Since a conformal map does not change the causal structure,this may use to study the causal structure of the dS space,though the distances are highly distorted.We would like to study causal structure in various coordinatesin terms on the Penrose diagram.d ˜S 2 = cos 2 TdS 2 = −dT 2 + l 2 dΩ 2 d−1One may write this as follows−dT 2 + dθ 2 1 + sin 2 θ 1 dΩ 2 d−2,0 < θ 1 < πΙ + (T/l= π /2 = )( = 0) θ1North PoleSurface ofconstant T (or )Time line ofconstant ( θ 1 , θ2 , ... , θd-1 )South Pole ( θ1 = π )Ι − (T/l=- π /2 = − )20