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De Sitter space is the submanifold described by the hyperboloid of one sheet
where is some nonzero constant with its dimension being that of length. The metric on de Sitter space is the metric induced from the ambient Minkowski metric. The induced metric is nondegenerate and has Lorentzian signature. (Note that if one replaces with in the above definition, one obtains a hyperboloid of two sheets. The induced metric in this case is positive-definite, and each sheet is a copy of hyperbolic n-space. For a detailed proof, see Minkowski space § Geometry.)
(using the sign convention for the Riemann curvature tensor). De Sitter space is an Einstein manifold since the Ricci tensor is proportional to the metric:
This means de Sitter space is a vacuum solution of Einstein's equation with cosmological constant given by
where forming a with the standard metric . Then the metric of the de Sitter space reads
where
is the standard hyperbolic metric.
Closed slicing
Let
where s describe a . Then the metric reads:
Changing the time variable to the conformal time via we obtain a metric conformally equivalent to Einstein static universe:
These coordinates, also known as "global coordinates" cover the maximal extension of de Sitter space, and can therefore be used to find its Penrose diagram.[5]
dS slicing
Let
where s describe a . Then the metric reads:
where
is the metric of an dimensional de Sitter space with radius of curvature in open slicing coordinates. The hyperbolic metric is given by:
This is the analytic continuation of the open slicing coordinates under and also switching and because they change their timelike/spacelike nature.
Nomizu, Katsumi (1982), "The Lorentz–Poincaré metric on the upper half-space and its extension", Hokkaido Mathematical Journal, 11 (3): 253–261, doi:10.14492/hokmj/1381757803
Susskind, L.; Lindesay, J. (2005), An Introduction to Black Holes, Information and the String Theory Revolution:The Holographic Universe, p. 119(11.5.25)
External links
Simplified Guide to de Sitter and anti-de Sitter Spaces A pedagogic introduction to de Sitter and anti-de Sitter spaces. The main article is simplified, with almost no math. The appendix is technical and intended for readers with physics or math backgrounds.