數學與物理學中，一個 n 維德西特空間（，標作 dSn）為一最大對稱的勞侖茲流形，具有正常數的純量曲率。 主要應用是在廣義相對論作為最簡單的宇宙數學模型。 「德西特」是以威廉德西特（18721934）為名，他與阿爾伯特愛因斯坦於 1920 年代一同研究宇宙中的時空結構。 以廣義相對論的語言來說，德西特空間為愛因斯坦場方程式的最大對稱真空解：具正宇宙學常數 Lambda 對應正真空能量密度和負壓。 In ndimensional de Sitter space often abbreviated to dSn is a maximally symmetric with constant positive It is the Lorentzian analogue of an with its canonical The main application of de Sitter space is its use in where it serves as one of the simplest mathematical models of the universe consistent with the observed More specifically de Sitter space is the maximally symmetric of with a positive Lambda corresponding to a positive vacuum energy density and negative pressure There is cosmological evidence that the universe itself is ie it will evolve like the de Sitter universe in the far future when dominates de Sitter space and are named after 18721934 professor of astronomy at Leiden University and director of the 萊頓天文臺 Willem de Sitter and Albert Einstein worked closely together in Leiden in the 1920s on the spacetime structure of our universe de Sitter space was also discovered independently and about the same time by 圖利奧列維 齊維塔 定義 de Sitter space can be defined as a of a generalized 閔考斯基時空 of one higher Take Minkowski space R1n with the standard ds2 dx_02 sum_i1n dx_i2 de Sitter space is the submanifold described by the of one sheet x_02 sum_i1n x_i2 alpha2 where alpha is some nonzero constant with its dimension being that of length The on de Sitter space is the metric induced from the ambient Minkowski metric The induced metric is and has Lorentzian signature Note that if one replaces alpha2 with alpha2 in the above definition one obtains a of two sheets The induced metric in this case is and each sheet is a copy of For a detailed proof see de Sitter space can also be defined as the of two s which shows that it is a nonRiemannian de Sitter space is so that if then de Sitter space is Properties The of de Sitter space is the 勞侖茲群 The metric therefore then has independent 基靈向量場 s and is maximally symmetric Every maximally symmetric space has constant curvature The 黎曼曲率張量 of de Sitter is given by R_rhosigmamunu 1 over alpha2left g_rhomu g_sigmanu g_rhonu g_sigmamuright using the sign convention Rrho_sigmamunu partial_muGammarho_nusigma partial_nuGammarho_musigma Gammarho_mulambdaGammalambda_nusigma Gammarho_nulambdaGammalambda_musigma for the Riemann curvature tensor de Sitter space is an since the is proportional to the metric R_munu Rlambda _mulambdanu frac n 1alpha2 g_munu This means de Sitter space is a vacuum solution of Einsteins equation with cosmological constant given by Lambda frac n 1n 22alpha2 The of de Sitter space is given by R frac n n 1alpha2 frac 2nn 2Lambda For the case we have and Coordinates Static coordinates We can introduce t r ldots for de Sitter as follows begin align x_0 sqrt alpha2 r2sinhleft frac 1alpha tright x_1 sqrt alpha2 r2coshleft frac 1alpha tright x_i r z_i qquadqquadqquadqquadqquad 2 le i le n end align where z_i gives the standard embedding the sphere in Rn1 In these coordinates the de Sitter metric takes the form ds2 left 1 frac r2alpha2right dt2 left 1 frac r2alpha2right1 dr2 r2 dOmega_n22 Note that there is a at r alpha Flat slicing Let begin align x_0 alpha sinhleft frac 1alpha tright frac 12alpha r2 efrac 1alpha t x_1 alpha coshleft frac 1alpha tright frac 12alpha r2 efrac 1alpha t x_i efrac 1alpha t y_i qquad 2 leq i leq n end align where r2 sum_i y_i2 Then in the left t y_iright coordinates metric reads ds2 dt2 e2frac 1alpha t dy2 where dy2 sum_i dy_i2 is the flat metric on y_is Setting zeta zeta_infty alpha efrac 1alpha t we obtain the conformally flat metric ds2 frac alpha2zeta_infty zeta2left dy2 dzeta2right Open slicing Let begin align x_0 alpha sinhleft frac 1alpha tright coshxi x_1 alpha coshleft frac 1alpha tright x_i alpha z_i sinhleft frac 1alpha tright sinhxi qquad 2 leq i leq n end align where sum_i z_i2 1 forming a Sn2 with the standard metric sum_i dz_i2 dOmega_n22 Then the metric of the de Sitter space reads ds2 dt2 alpha2 sinh2left frac 1alpha tright dH_n12 where dH_n12 dxi2 sinh2 xi dOmega_n22 is the standard hyperbolic metric Closed slicing Let begin align x_0 alpha sinhleft frac 1alpha tright x_i alpha coshleft frac 1alpha tright z_i qquad 1 leq i leq n end align where z_is describe a Sn1 Then the metric reads ds2 dt2 alpha2 cosh2left frac 1alpha tright dOmega_n12 Changing the time variable to the conformal time via tanleft frac 12etaright tanhleft frac 12alpha tright we obtain a metric conformally equivalent to Einstein static universe ds2 frac alpha2cos2etaleft deta2 dOmega_n12right These coordinates also known as global coordinates cover the maximal extension of de Sitter space and can therefore be used to find its 彭羅斯圖 dS slicing Let begin align x_0 alpha sinleft frac 1alphachiright sinhleft frac 1alpha tright coshxi x_1 alpha cosleft frac 1alphachiright x_2 alpha sinleft frac 1alphachiright coshleft frac 1alpha tright x_i alpha z_i sinleft frac 1alphachiright sinhleft frac 1alpha tright sinhxi qquad 3 leq i leq n end align where z_is describe a Sn3 Then the metric reads ds2 dchi2 sin2left frac 1alphachiright ds_dSalphan12 where ds_dSalphan12 dt2 alpha2 sinh2left frac 1alpha tright dH_n22 is the metric of an n 1 dimensional de Sitter space with radius of curvature alpha in open slicing coordinates The hyperbolic metric is given by dH_n22 dxi2 sinh2 xi dOmega_n32 This is the analytic continuation of the open slicing coordinates under left t xi theta phi_1 phi_2 ldots phi_n3right to left ichi xi it theta phi_1 ldots phi_n4right and also switching x_0 and x_2 because they change their timelikespacelike nature See also 反德西特空間 AdSCFT 對偶 參考資料 延伸閱讀 外部連結 Simplified Guide to de Sitter and antide Sitter Spaces A pedagogic introduction to de Sitter and antide Sitter spaces The main article is simplified with almost no math The appendix is technical and intended for readers with physics or math backgrounds CategoryExact solutions in general relativity CategoryDifferential geometry CategoryMinkowski spacetime