德西特空間

數學與物理學中，一個"n"維德西特空間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formula_1對應正真空能量密度和負壓。已隱藏部分未翻譯內容，歡迎參與[ 翻譯]。In , "n"-dimensional de Sitter space (often abbreviated to dS"n") 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 formula_1 (corresponding to a positive vacuum energy density and negative pressure). There is cosmological evidence that the universe itself is , i.e. it will evolve like the de Sitter universe in the far future when dominates.de Sitter space and are named after (1872–1934), 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 R1,"n" with the standard :formula_3de Sitter space is the submanifold described by the of one sheetformula_4where formula_5 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 formula_6 with formula_7 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 O(1, "n") / O(1, "n" − 1) of two s, which shows that it is a non-Riemannian ., de Sitter space is R × "S""n"−1 (so that if "n" ≥ 3 then de Sitter space is ).Properties.The of de Sitter space is the 勞侖茲群 O(1, "n"). The metric therefore then has "n"("n" + 1)/2 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 byformula_8(using the sign convention formula_9 for the Riemann curvature tensor). de Sitter space is an since the is proportional to the metric:formula_10This means de Sitter space is a vacuum solution of Einstein's equation with cosmological constant given byformula_11The of de Sitter space is given byformula_12For the case "n" = 4, we have Λ = 3/"α"2 and "R" = 4Λ = 12/"α"2.Coordinates.Static coordinates.We can introduce formula_13 for de Sitter as follows:formula_14where formula_15 gives the standard embedding the ("n" − 2)-sphere in R"n"−1. In these coordinates the de Sitter metric takes the form:formula_16Note that there is a at formula_17.Flat slicing.Letformula_18where formula_19. Then in the formula_20 coordinates metric reads:formula_21where formula_22 is the flat metric on formula_23's.Setting formula_24, we obtain the conformally flat metric:formula_25Open slicing.Letformula_26where formula_27 forming a formula_28 with the standard metric formula_29. Then the metric of the de Sitter space readsformula_30whereformula_31is the standard hyperbolic metric.Closed slicing.Letformula_32where formula_15s describe a formula_34. Then the metric reads:formula_35Changing the time variable to the conformal time via formula_36 we obtain a metric conformally equivalent to Einstein static universe:formula_37These 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.Letformula_38where formula_15s describe a formula_40. Then the metric reads:formula_41whereformula_42is the metric of an formula_43 dimensional de Sitter space with radius of curvature formula_5 in open slicing coordinates. The hyperbolic metric is given by:formula_45This is the analytic continuation of the open slicing coordinates under formula_46 and also switching formula_47 and formula_48 because they change their timelike/spacelike nature.