de Sitter space and Holography

• One can always decompose the metric h µν on the equaltime surfaces as followsh νν dx µ dx ν = N ρ dρ 2 + σ ab (dφ a + U a dρ)(dφ b + U b dρ)where φ a are angular variables parameterizing closed surfaces.• Suppose the boundary has an isometry generated by aKilling vector ξ µ . One can show that T µν ξ ν is divergencelessand therefore we can define conserved charge associated to ξ µ .• Consider n µ be the unit normal on a surface of fixed ρ.Then the conserved charge is defined∮Q = d d−2 φ √ σn µ ξ ν T µνΣPhysically this means that a collection of observers on thehypersurface whose metric is h µν would all observe the samevalue of Q provided this surface had an isometry generated byξ µ .Since there is no globally timelike Killing vector, it is difficultto see how the mass can be defined in dS space. There ishowever an proposal for this using the above construction.Consider the case where ρ is the coordinate associated withthe asymptotic Killing vector that is timelike inside the staticpatch but spacelike at I − .36