de Sitter space and Holography

For pure dS space where the energy-momentum tensor ofthe matter field vanishes, one may consider the spacetime as asolution of the Einstein equation for an empty spacetime witha positive constant vacuum energyT vacµν = Λ8πG g µν → R µν − 1 2 g µνR = −g µν ΛTherefore the only non-trivial component of the Einsteinequation isR = 2dd − 2 Λ > 0An interesting observation about the dS d space is itsembedding into the flat d + 1 dimensional space time. Ind + 1 dimensional Minkowski spacetime the Einstein equationis trivially satisfied0 = d+1 R ≡ g AB R AB forA, B = 0,...,d= R + R ddIf we set R dd = − 2dd−2Λ( which means positive constantcurvature of embedded space) one getsR = 2dd − 2 Λ (dS Space) 6