De Sitter space
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n-dimensional de Sitter space is the geometry with Lorentzian signature with the automorphism group <math>O(n,1)<math>, the orthochronous Lorentz group (in one dimension higher) according to the Erlangen program.
It is also a cosmological model for the universe.
De Sitter spacetime is the maximally symmetric solution of the vacuum Einstein equations with a cosmological constant, <math>G_{ab} + \Lambda g_{ab} = 0<math>.
We now concentrate on four dimensional spacetime.
In static coordinates, the de Sitter metric takes the form
<math> ds^2 = -(1-\Lambda r^2/3)dt^2+(1-\Lambda r^2/3)^{-1}dr^2+r^2 d\Omega_{n-2}^2. <math>
There is a cosmological horizon at <math>r = \sqrt{3/\Lambda}<math>.