德西特空間

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（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。

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 , n-dimensional 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 , 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 :
ds^2 = -dx_0^2 + \sum_{i=1}^n dx_i^2.

de Sitter space is the submanifold described by the of one sheet
-x_0^2 + \sum_{i=1}^n x_i^2 = \alpha^2,
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 \alpha^2 with -\alpha^2 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 勞侖茲群 . The metric therefore then has independent 基灵矢量场s and is maximally symmetric. Every maximally symmetric space has constant curvature. The 黎曼曲率張量 of de Sitter is given by
:R_{\rho\sigma\mu\nu} = {1 \over \alpha^2}\left(g_{\rho\mu}g_{\sigma\nu} - g_{\rho\nu}g_{\sigma\mu}\right)

(using the sign convention
R^{\rho}{}_{\sigma\mu\nu} =
\partial_{\mu}\Gamma^{\rho}_{\nu\sigma} -
\partial_{\nu}\Gamma^{\rho}_{\mu\sigma} +
\Gamma^{\rho}_{\mu\lambda}\Gamma^{\lambda}_{\nu\sigma} -
\Gamma^{\rho}_{\nu\lambda}\Gamma^{\lambda}_{\mu\sigma}
for the Riemann curvature tensor). de Sitter space is an since the is proportional to the metric:
:R_{\mu\nu} = R^\lambda{}_{\mu\lambda\nu} = \frac{n - 1}{\alpha^2}g_{\mu\nu}

This means de Sitter space is a vacuum solution of Einstein's equation with cosmological constant given by
:\Lambda = \frac{(n - 1)(n - 2)}{2\alpha^2}.

The of de Sitter space is given by

dS slicing
Let
:\begin{align}
x_0 &= \alpha \sin\left(\frac{1}{\alpha}\chi\right) \sinh\left(\frac{1}{\alpha}t\right) \cosh\xi, \\
x_1 &= \alpha \cos\left(\frac{1}{\alpha}\chi\right), \\
x_2 &= \alpha \sin\left(\frac{1}{\alpha}\chi\right) \cosh\left(\frac{1}{\alpha}t\right), \\
x_i &= \alpha z_i \sin\left(\frac{1}{\alpha}\chi\right) \sinh\left(\frac{1}{\alpha}t\right) \sinh\xi, \qquad 3 \leq i \leq n
\end{align}

where z_is describe a S^{n-3}. Then the metric reads:
:ds^2 = d\chi^2 + \sin^2\left(\frac{1}{\alpha}\chi\right) ds_{dS,\alpha,n-1}^2,

where
:ds_{dS,\alpha,n-1}^2 = -dt^2 + \alpha^2 \sinh^2\left(\frac{1}{\alpha}t\right) dH_{n-2}^2

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_{n-2}^2 = d\xi^2 + \sinh^2(\xi) d\Omega_{n-3}^2.

This is the analytic continuation of the open slicing coordinates under \left(t, \xi, \theta, \phi_1, \phi_2, \ldots, \phi_{n-3}\right) \to \left(i\chi, \xi, it, \theta, \phi_1, \ldots, \phi_{n-4}\right) and also switching x_0 and x_2 because they change their timelike/spacelike nature.

参见

反德西特空間

AdS/CFT对偶

參考資料
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