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| In [[mathematics]] and [[physics]], a '''de Sitter space''' is the analog in [[Minkowski space]], or spacetime, of a sphere in ordinary, [[Euclidean space]]. The ''n''-dimensional '''de Sitter space''', denoted <math>dS_n</math>, is the [[Lorentzian manifold]] analog of an [[n-sphere|''n''-sphere]] (with its canonical [[Riemannian metric]]); it is maximally [[symmetric]], has constant positive [[scalar curvature|curvature]], and is [[simply-connected]] for ''n'' at least 3.
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| In the language of [[general relativity]], de Sitter space is the maximally symmetric, [[vacuum solution]] of [[Einstein's field equations]] with a positive (repulsive) [[cosmological constant]] <math>\Lambda</math> (corresponding to a positive vacuum energy density and negative pressure). When ''n'' = 4 (3 space dimensions plus time), it is a cosmological model for the physical universe; see [[de Sitter universe]].
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| De Sitter space was discovered by [[Willem de Sitter]], and, at the same time, independently by [[Tullio Levi-Civita]].
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| More recently it has been considered as the setting for [[Special Relativity]] rather than using [[Minkowski space]], since a [[group contraction]] reduces the [[isometry]] group of de Sitter space to the [[Poincaré group]], allowing a unification of the [[spacetime]] [[Translation_(geometry)|translation subgroup]] and [[Lorentz transformation|Lorentz transformation subgroup]] of the Poincaré group into a [[simple group]] rather than a [[semi-simple group]]. This alternate formulation of Special Relativity is called [[de Sitter relativity]].
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| ==Definition==
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| De Sitter space can be defined as a [[submanifold]] of a [[Minkowski space]] of one higher [[dimension]]. Take Minkowski space '''R'''<sup>1,''n''</sup> with the standard [[metric tensor|metric]]:
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| :<math>ds^2 = -dx_0^2 + \sum_{i=1}^n dx_i^2.</math>
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| De Sitter space is the submanifold described by the [[hyperboloid]] of one sheet
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| :<math>-x_0^2 + \sum_{i=1}^n x_i^2 = \alpha^2</math>
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| where <math>\alpha</math> is some positive constant with dimensions of length. The [[metric tensor|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 <math>\alpha^2</math> with <math>-\alpha^2</math> 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 space|hyperbolic ''n''-space]].)
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| De Sitter space can also be defined as the [[quotient space (topology)|quotient]] ''O''(1,''n'')/''O''(1,''n''−1) of two [[indefinite orthogonal group]]s, which shows that it is a non-Riemannian [[symmetric space]].
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| [[Topology|Topologically]], de Sitter space is '''R''' × ''S''<sup>''n''−1</sup> (so that if ''n'' ≥ 3 then de Sitter space is [[simply-connected]]).
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| ==Properties==
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| The [[isometry group]] of de Sitter space is the [[Lorentz group]] O(1,''n''). The metric therefore then has ''n''(''n''+1)/2 independent [[Killing vector]]s and is maximally symmetric. Every maximally symmetric space has constant curvature. The [[Riemann curvature tensor]] of de Sitter is given by
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| :<math>R_{\rho\sigma\mu\nu} = {1\over \alpha^2}(g_{\rho\mu}g_{\sigma\nu} - g_{\rho\nu}g_{\sigma\mu})</math>
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| De Sitter space is an [[Einstein manifold]] since the [[Ricci tensor]] is proportional to the metric:
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| :<math>R_{\mu\nu} = \frac{n-1}{\alpha^2}g_{\mu\nu}</math>
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| This means de Sitter space is a vacuum solution of Einstein's equation with cosmological constant given by
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| :<math>\Lambda = \frac{(n-1)(n-2)}{2\alpha^2}.</math>
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| The [[scalar curvature]] of de Sitter space is given by | |
| :<math>R = \frac{n(n-1)}{\alpha^2} = \frac{2n}{n-2}\Lambda.</math>
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| For the case ''n'' = 4, we have Λ = 3/α<sup>2</sup> and ''R'' = 4Λ = 12/α<sup>2</sup>.
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| ==Static coordinates==
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| We can introduce [[static spacetime|static coordinates]] <math>(t, r, \ldots)</math> for de Sitter as follows:
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| :<math>x_0 = \sqrt{\alpha^2-r^2}\sinh(t/\alpha)</math>
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| :<math>x_1 = \sqrt{\alpha^2-r^2}\cosh(t/\alpha)</math>
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| :<math>x_i = r z_i \qquad\qquad\qquad\qquad\qquad 2\le i\le n.</math>
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| where <math>z_i</math> gives the standard embedding the (''n''−2)-sphere in '''R'''<sup>''n''−1</sup>. In these coordinates the de Sitter metric takes the form:
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| :<math>ds^2 = -\left(1-\frac{r^2}{\alpha^2}\right)dt^2 + \left(1-\frac{r^2}{\alpha^2}\right)^{-1}dr^2 + r^2 d\Omega_{n-2}^2.</math>
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| Note that there is a [[Observable universe#Cosmological horizon|cosmological horizon]] at <math>r = \alpha</math>.
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| ==Flat slicing==
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| Let
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| :<math>x_0 = \alpha \sinh(t/\alpha) + r^2 e^{t/\alpha}/2\alpha,</math>
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| :<math>x_1 = \alpha \cosh(t/\alpha) - r^2 e^{t/\alpha}/2\alpha,</math>
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| :<math>x_i = e^{t/\alpha}y_i, \qquad 2 \leq i \leq n</math>
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| where <math>r^2=\sum_i y_i^2</math>. Then in the <math>(t,y_i)</math> coordinates metric reads:
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| :<math>ds^{2} = -dt^{2} + e^{2t/\alpha} dy^{2}</math>
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| where <math>dy^2=\sum_i dy_i^2</math> is the flat metric on <math>y_i</math>'s.
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| ==Open slicing==
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| Let
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| :<math>x_0 = \alpha \sinh(t/\alpha) \cosh\xi,</math>
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| :<math>x_1 = \alpha \cosh(t/\alpha),</math>
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| :<math>x_i = \alpha z_i \sinh(t/\alpha) \sinh\xi, \qquad 2 \leq i \leq n</math>
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| where <math>\sum_i z_i^2 = 1</math> forming a <math>S^{n-2}</math> with the standard metric <math>\sum_i dz_i^2 = d\Omega_{n-2}^2</math>. Then the metric of the de Sitter space reads
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| :<math>ds^2 = -dt^2 + \alpha^2 \sinh^2(t/\alpha) dH_{n-1}^2,</math>
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| where
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| :<math>dH_{n-1}^2 = d\xi^2 + \sinh^2\xi d\Omega_{n-2}^2</math>
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| is the metric of a Euclidean hyperbolic space. | |
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| ==Closed slicing==
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| Let
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| :<math>x_0 = \alpha \sinh(t/\alpha),</math>
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| :<math>x_i = \alpha \cosh(t/\alpha) z_i, \qquad 1 \leq i \leq n</math>
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| where <math>z_i</math>s describe a <math>S^{n-1}</math>. Then the metric reads:
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| :<math>ds^2 = -dt^2 + \alpha^2 \cosh^2(t/\alpha) d\Omega_{n-1}^2.</math>
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| Changing the time variable to the conformal time via <math>\tan(\eta/2)=\tanh(t/2\alpha)</math> we obtain a metric conformally equivalent to Einstein static universe:
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| :<math>ds^2 = \frac{\alpha^2}{\cos^2\eta}(-d\eta^2 + d\Omega_{n-1}^2).</math>
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| This serves to find the [[Penrose diagram]] of de Sitter space.{{clarify|date=November 2012}}
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| ==dS slicing==
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| Let
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| :<math>x_0 = \alpha \sin(\chi/\alpha) \sinh(t/\alpha) \cosh\xi,</math>
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| :<math>x_1 = \alpha \cos(\chi/\alpha),</math>
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| :<math>x_2 = \alpha \sin(\chi/\alpha) \cosh(t/\alpha),</math>
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| :<math>x_i = \alpha z_i \sin(\chi/\alpha) \sinh(t/\alpha) \sinh\xi, \qquad 3 \leq i \leq n</math> | |
| where <math>z_i</math>s describe a <math>S^{n-3}</math>. Then the metric reads:
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| :<math>ds^2 = d\chi^2 + \sin^2(\chi/\alpha) ds_{dS,\alpha,n-1}^2,</math>
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| where
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| :<math>ds_{dS,\alpha,n-1}^2 = -dt^2 + \alpha^2 \sinh^2(t/\alpha) dH_{n-2}^2</math>
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| is the metric of an <math>n-1</math> dimensional de Sitter space with radius of curvature <math>\alpha</math> in open slicing coordinates. The hyperbolic metric is given by:
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| :<math>dH_{n-2}^2 = d\xi^2 + \sinh^2\xi d\Omega_{n-3}^2.</math>
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| This is the analytic continuation of the open slicing coordinates under <math>(t,\xi,\theta,\phi_1,\phi_2,\cdots,\phi_{n-3}) \to (i\chi,\xi,it,\theta,\phi_1,\cdots,\phi_{n-4})</math> and also switching <math>x_0</math> and <math>x_2</math> because they change their timelike/spacelike nature.
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| ==See also==
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| * [[Anti-de Sitter space]]
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| * [[de Sitter universe]]
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| * [[AdS/CFT correspondence]]
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| * [[Hyperboloid]]
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| * [[De Sitter–Schwarzschild metric]]
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| ==References==
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| *{{springer|id=d/d110040|title=De Sitter space|author=Qingming Cheng}}
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| *{{Citation|last=de Sitter|first=W.|year=1917|title=On the relativity of inertia: Remarks concerning Einstein's latest hypothesis|journal=Proc. Kon. Ned. Acad. Wet.|volume=19|pages=1217–1225}}
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| *{{Citation|last=de Sitter|first=W.|year=1917|title=On the curvature of space|journal=Proc. Kon. Ned. Acad. Wet.|volume=20|pages=229–243}}
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| * {{Citation|first=Tullio|last=Levi-Civita|authorlink=Tullio Levi-Civita|title=Realtà fisica di alcuni spazî normali del Bianchi|journal=Rendiconti, Reale Accademia Dei Lincei|volume=26|year=1917|pages=519–31}}
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| * {{Citation|last=Nomizu|first=Katsumi|authorlink=Katsumi Nomizu|title=The Lorentz-Poincaré metric on the upper half-space and its extension|journal=[[Hokkaido Mathematical Journal]]|volume=11|year=1982|pages=253–261|issue=3}}
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| * {{Citation|last=Coxeter|first=H. S. M.|authorlink=Harold Scott MacDonald Coxeter|title=A geometrical background for de Sitter's world|journal=[[American Mathematical Monthly]]|volume=50|year=1943|pages=217–228|doi=10.2307/2303924|issue=4|publisher=Mathematical Association of America|jstor=2303924}}
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| * {{Citation|last1=Susskind|first1=L.|last2=Lindesay|first2=J.|title=An Introduction to Black Holes, Information and the String Theory Revolution:The Holographic Universe|year=2005|page=119(11.5.25)}}
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| {{DEFAULTSORT:De Sitter Space}}
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| [[Category:Exact solutions in general relativity]]
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| [[Category:Riemannian geometry]]
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