de Sitter space and Holography

1.3 Thermal propertyConsider a circle with radius l : X 2 1 + X 2 2 = l 2 .X 1 = l cosθ,X 2 = l sinθThe geodesic distance between two points on the circle is givenbyd(θ,θ ′ ) = l(θ − θ ′ )Let’s consider a quantity P(X,X ′ ) = 1 l 2 δ ij X i X ′ jwhich is1l 2δij X i X ′ j = cosθ cosθ ′ + sinθ sinθ ′ = cos(θ − θ ′ )Sod(θ,θ ′ ) = l cos −1 P(X,X ′ )One may generalize it for higher sphere or dS space. In thecase of the dS space given by η µν X µ X ν = l 2 one hasP(X,X ′ ) = 1 l 2ηµν X µ X ′ ν, d(1,2) = l cos −1 P(X,X ′ )25