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德西特空間

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德西特空間

數學物理學中,一個n德西特空間(英語:de Sitter space,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(1872–1934)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。

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:具正宇宙學常數對應正真空能量密度和負壓。

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In mathematical physics英语mathematical physics, n-dimensional de Sitter space (often abbreviated to dSn) is a maximally symmetric Lorentzian manifold英语Lorentzian manifold with constant positive scalar curvature英语scalar curvature. It is the Lorentzian analogue of an n-sphere英语n-sphere (with its canonical Riemannian metric英语Riemannian metric).

The main application of de Sitter space is its use in general relativity英语general relativity, where it serves as one of the simplest mathematical models of the universe consistent with the observed accelerating expansion of the universe英语accelerating expansion of the universe. More specifically, de Sitter space is the maximally symmetric vacuum solution英语vacuum solution of Einstein's field equations英语Einstein's field equations with a positive cosmological constant英语cosmological constant (corresponding to a positive vacuum energy density and negative pressure). There is cosmological evidence that the universe itself is asymptotically de Sitter英语de Sitter universe, i.e. it will evolve like the de Sitter universe in the far future when dark energy英语dark energy dominates.

de Sitter space and anti-de Sitter space英语anti-de Sitter space are named after Willem de Sitter英语Willem de Sitter (1872–1934),[1][2] 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.[3]

定義

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de Sitter space can be defined as a submanifold英语submanifold of a generalized 閔考斯基時空 of one higher dimension英语dimension. Take Minkowski space R1,n with the standard metric:

de Sitter space is the submanifold described by the hyperboloid英语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英语nondegenerate and has Lorentzian signature. (Note that if one replaces with in the above definition, one obtains a hyperboloid英语hyperboloid of two sheets. The induced metric in this case is positive-definite英语Definite quadratic form, and each sheet is a copy of hyperbolic n-space英语hyperbolic space. For a detailed proof, see Minkowski space § Geometry.)

de Sitter space can also be defined as the quotient英语Homogeneous space O(1, n) / O(1, n − 1) of two indefinite orthogonal group英语indefinite orthogonal groups, which shows that it is a non-Riemannian symmetric space英语symmetric space.

Topologically英语Topology, de Sitter space is R × Sn−1 (so that if n ≥ 3 then de Sitter space is simply connected英语simply connected).

Properties

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The isometry group英语isometry group of de Sitter space is the 勞侖茲群 O(1, n). The metric therefore then has n(n + 1)/2 independent 基灵矢量场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[4]

(using the sign convention for the Riemann curvature tensor). de Sitter space is an Einstein manifold英语Einstein manifold since the Ricci tensor英语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

The scalar curvature英语scalar curvature of de Sitter space is given by[4]

For the case n = 4, we have Λ = 3/α2 and R = 4Λ = 12/α2.

Coordinates

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Static coordinates

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We can introduce static coordinates英语static spacetime for de Sitter as follows:

where gives the standard embedding the (n − 2)-sphere in Rn−1. In these coordinates the de Sitter metric takes the form:

Note that there is a cosmological horizon英语cosmological horizon at .

Flat slicing

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Let

where . Then in the coordinates metric reads:

where is the flat metric on 's.

Setting , we obtain the conformally flat metric:

Open slicing

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Let

where forming a with the standard metric . Then the metric of the de Sitter space reads

where

is the standard hyperbolic metric.

Closed slicing

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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 彭罗斯图.[5]

dS slicing

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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.

See also

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參考資料

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  1. ^ de Sitter, W., On the relativity of inertia: Remarks concerning Einstein's latest hypothesis (PDF), Proc. Kon. Ned. Acad. Wet., 1917, 19: 1217–1225 [2022-12-01], Bibcode:1917KNAB...19.1217D, (原始内容存档 (PDF)于2023-04-07) 
  2. ^ de Sitter, W., On the curvature of space (PDF), Proc. Kon. Ned. Acad. Wet., 1917, 20: 229–243 [2022-12-01], (原始内容存档 (PDF)于2023-04-09) 
  3. ^ Levi-Civita, Tullio, Realtà fisica di alcuni spazî normali del Bianchi, Rendiconti, Reale Accademia dei Lincei, 1917, 26: 519–31 
  4. ^ 4.0 4.1 Zee 2013,第626頁
  5. ^ Hawking & Ellis. The large scale structure of space–time. Cambridge Univ. Press. 

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