德西特空间

数学与物理学中，一个"n"维德西特空间（de Sitter space，标作dS"n"）为一最大对称的劳仑兹流形，具有正常数的纯量曲率。
主要应用是在广义相对论作为最简单的宇宙数学模型。
「德西特」是以威廉·德西特（1872–1934）为名，他与阿尔伯特·爱因斯坦于1920年代一同研究宇宙中的时空结构。
以广义相对论的语言来说，德西特空间为爱因斯坦场方程式的最大对称真空解：具正宇宙学常数formula_1对应正真空能量密度和负压。
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 莱顿天文台. 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 图利奥·列维-齐维塔.
定义.
de Sitter space can be defined as a of a generalized 闵考斯基时空 of one higher . Take Minkowski space R1,"n" with the standard :
formula_3
de Sitter space is the submanifold described by the of one sheet
formula_4
where 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 of two s, which shows that it is a non-Riemannian .
, de Sitter space is (so that if then de Sitter space is ).
Properties.
The of de Sitter space is the 劳仑兹群 . The metric therefore then has independent 基灵矢量场s and is maximally symmetric. Every maximally symmetric space has constant curvature. The 黎曼曲率张量 of de Sitter is given by
(using the sign convention formula_9 for the Riemann curvature tensor). de Sitter space is an since the is proportional to the metric:
This means de Sitter space is a vacuum solution of Einstein's equation with cosmological constant given by
The of de Sitter space is given by
For the case , we have and .
Coordinates.
Static coordinates.
We can introduce formula_13 for de Sitter as follows:
where formula_15 gives the standard embedding the -sphere in R"n"−1. In these coordinates the de Sitter metric takes the form:
Note that there is a at formula_17.
Flat slicing.
Let
where formula_19. Then in the formula_20 coordinates metric reads:
where formula_22 is the flat metric on formula_23's.
Setting formula_24, we obtain the conformally flat metric:
Open slicing.
Let
where formula_27 forming a formula_28 with the standard metric formula_29. Then the metric of the de Sitter space reads
where
is the standard hyperbolic metric.
Closed slicing.
Let
where formula_15s describe a formula_34. Then the metric reads:
Changing the time variable to the conformal time via formula_36 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 彭罗斯图.
dS slicing.
Let
where formula_15s describe a formula_40. Then the metric reads:
where
is 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:
This 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.