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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。

已隱藏部分未翻譯內容,歡迎參與翻譯

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

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

de Sitter space and anti-de Sitter space英語anti-de Sitter space are named after Willem de Sitter英語Willem de Sitter (1872–1934),[1][2] professor of astronomy at Leiden University and director of the 萊頓天文台. 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 圖利奧·列維-齊維塔.[3]

定義

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

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

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

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

Properties

The isometry group英語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 基靈矢量場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[4]

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

Static coordinates

We can introduce static coordinates英語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英語cosmological horizon at .

Flat slicing

Let

where . Then in the coordinates metric reads:

where is the flat metric on 's.

Setting , we obtain the conformally flat metric:

Open slicing

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

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 彭羅斯圖.[5]

dS slicing

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

參考資料

  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, (原始內容存檔 (PDF)於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], (原始內容存檔 (PDF)於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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