德西特空间

数学与物理学中，一个n维德西特空间（英语：de Sitter space，标作dS_n）为一最大对称的洛伦兹流形，具有正常数的数量曲率。

提示：此条目的主题不是反德西特空间。

德西特空间

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的最大对称真空解：具正宇宙学常数${\displaystyle \Lambda }$对应正真空能量密度和负压。

已隐藏部分未翻译内容，欢迎参与翻译。

In mathematical physics（英语：mathematical physics）, n-dimensional de Sitter space (often abbreviated to dS_n) 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） ${\displaystyle \Lambda }$ (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]

定义

de Sitter space can be defined as a submanifold（英语：submanifold） of a generalized 闵可夫斯基时空 of one higher dimension（英语：dimension）. Take Minkowski space R^1,n with the standard metric: $${\displaystyle ds^{2}=-dx_{0}^{2}+\sum _{i=1}^{n}dx_{i}^{2}.}$$

de Sitter space is the submanifold described by the hyperboloid（英语：hyperboloid） of one sheet $${\displaystyle -x_{0}^{2}+\sum _{i=1}^{n}x_{i}^{2}=\alpha ^{2},}$$ where ${\displaystyle \alpha }$ 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 ${\displaystyle \alpha ^{2}}$ with ${\displaystyle -\alpha ^{2}}$ 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 group）s, 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

The isometry group（英语：isometry group） of de Sitter space is the 洛伦兹群 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 黎曼曲率张量 of de Sitter is given by^[4]

${\displaystyle R_{\rho \sigma \mu \nu }={1 \over \alpha ^{2}}\left(g_{\rho \mu }g_{\sigma \nu }-g_{\rho \nu }g_{\sigma \mu }\right)}$

(using the sign convention ${\displaystyle R^{\rho }{}_{\sigma \mu \nu }=\partial _{\mu }\Gamma _{\nu \sigma }^{\rho }-\partial _{\nu }\Gamma _{\mu \sigma }^{\rho }+\Gamma _{\mu \lambda }^{\rho }\Gamma _{\nu \sigma }^{\lambda }-\Gamma _{\nu \lambda }^{\rho }\Gamma _{\mu \sigma }^{\lambda }}$ 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:

${\displaystyle R_{\mu \nu }=R^{\lambda }{}_{\mu \lambda \nu }={\frac {n-1}{\alpha ^{2}}}g_{\mu \nu }}$

This means de Sitter space is a vacuum solution of Einstein's equation with cosmological constant given by

${\displaystyle \Lambda ={\frac {(n-1)(n-2)}{2\alpha ^{2}}}.}$

The scalar curvature（英语：scalar curvature） of de Sitter space is given by^[4]

${\displaystyle R={\frac {n(n-1)}{\alpha ^{2}}}={\frac {2n}{n-2}}\Lambda .}$

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

Coordinates

Static coordinates

We can introduce static coordinates（英语：static spacetime） ${\displaystyle (t,r,\ldots )}$ for de Sitter as follows:

${\displaystyle {\begin{aligned}x_{0}&={\sqrt {\alpha ^{2}-r^{2}}}\sinh \left({\frac {1}{\alpha }}t\right)\\x_{1}&={\sqrt {\alpha ^{2}-r^{2}}}\cosh \left({\frac {1}{\alpha }}t\right)\\x_{i}&=rz_{i}\qquad \qquad \qquad \qquad \qquad 2\leq i\leq n.\end{aligned}}}$

where ${\displaystyle z_{i}}$ gives the standard embedding the (n − 2)-sphere in Rⁿ⁻¹. In these coordinates the de Sitter metric takes the form:

${\displaystyle ds^{2}=-\left(1-{\frac {r^{2}}{\alpha ^{2}}}\right)dt^{2}+\left(1-{\frac {r^{2}}{\alpha ^{2}}}\right)^{-1}dr^{2}+r^{2}d\Omega _{n-2}^{2}.}$

Note that there is a cosmological horizon（英语：cosmological horizon） at ${\displaystyle r=\alpha }$.

Flat slicing

Let

${\displaystyle {\begin{aligned}x_{0}&=\alpha \sinh \left({\frac {1}{\alpha }}t\right)+{\frac {1}{2\alpha }}r^{2}e^{{\frac {1}{\alpha }}t},\\x_{1}&=\alpha \cosh \left({\frac {1}{\alpha }}t\right)-{\frac {1}{2\alpha }}r^{2}e^{{\frac {1}{\alpha }}t},\\x_{i}&=e^{{\frac {1}{\alpha }}t}y_{i},\qquad 2\leq i\leq n\end{aligned}}}$

where ${\textstyle r^{2}=\sum _{i}y_{i}^{2}}$. Then in the ${\displaystyle \left(t,y_{i}\right)}$ coordinates metric reads:

${\displaystyle ds^{2}=-dt^{2}+e^{2{\frac {1}{\alpha }}t}dy^{2}}$

where ${\textstyle dy^{2}=\sum _{i}dy_{i}^{2}}$ is the flat metric on ${\displaystyle y_{i}}$'s.

Setting ${\displaystyle \zeta =\zeta _{\infty }-\alpha e^{-{\frac {1}{\alpha }}t}}$, we obtain the conformally flat metric:

${\displaystyle ds^{2}={\frac {\alpha ^{2}}{(\zeta _{\infty }-\zeta )^{2}}}\left(dy^{2}-d\zeta ^{2}\right)}$

Open slicing

Let

${\displaystyle {\begin{aligned}x_{0}&=\alpha \sinh \left({\frac {1}{\alpha }}t\right)\cosh \xi ,\\x_{1}&=\alpha \cosh \left({\frac {1}{\alpha }}t\right),\\x_{i}&=\alpha z_{i}\sinh \left({\frac {1}{\alpha }}t\right)\sinh \xi ,\qquad 2\leq i\leq n\end{aligned}}}$

where ${\textstyle \sum _{i}z_{i}^{2}=1}$ forming a ${\displaystyle S^{n-2}}$ with the standard metric ${\textstyle \sum _{i}dz_{i}^{2}=d\Omega _{n-2}^{2}}$. Then the metric of the de Sitter space reads

${\displaystyle ds^{2}=-dt^{2}+\alpha ^{2}\sinh ^{2}\left({\frac {1}{\alpha }}t\right)dH_{n-1}^{2},}$

where

${\displaystyle dH_{n-1}^{2}=d\xi ^{2}+\sinh ^{2}(\xi )d\Omega _{n-2}^{2}}$

is the standard hyperbolic metric.

Closed slicing

Let

${\displaystyle {\begin{aligned}x_{0}&=\alpha \sinh \left({\frac {1}{\alpha }}t\right),\\x_{i}&=\alpha \cosh \left({\frac {1}{\alpha }}t\right)z_{i},\qquad 1\leq i\leq n\end{aligned}}}$

where ${\displaystyle z_{i}}$s describe a ${\displaystyle S^{n-1}}$. Then the metric reads:

${\displaystyle ds^{2}=-dt^{2}+\alpha ^{2}\cosh ^{2}\left({\frac {1}{\alpha }}t\right)d\Omega _{n-1}^{2}.}$

Changing the time variable to the conformal time via ${\textstyle \tan \left({\frac {1}{2}}\eta \right)=\tanh \left({\frac {1}{2\alpha }}t\right)}$ we obtain a metric conformally equivalent to Einstein static universe:

${\displaystyle ds^{2}={\frac {\alpha ^{2}}{\cos ^{2}\eta }}\left(-d\eta ^{2}+d\Omega _{n-1}^{2}\right).}$

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

Let

${\displaystyle {\begin{aligned}x_{0}&=\alpha \sin \left({\frac {1}{\alpha }}\chi \right)\sinh \left({\frac {1}{\alpha }}t\right)\cosh \xi ,\\x_{1}&=\alpha \cos \left({\frac {1}{\alpha }}\chi \right),\\x_{2}&=\alpha \sin \left({\frac {1}{\alpha }}\chi \right)\cosh \left({\frac {1}{\alpha }}t\right),\\x_{i}&=\alpha z_{i}\sin \left({\frac {1}{\alpha }}\chi \right)\sinh \left({\frac {1}{\alpha }}t\right)\sinh \xi ,\qquad 3\leq i\leq n\end{aligned}}}$

where ${\displaystyle z_{i}}$s describe a ${\displaystyle S^{n-3}}$. Then the metric reads:

${\displaystyle ds^{2}=d\chi ^{2}+\sin ^{2}\left({\frac {1}{\alpha }}\chi \right)ds_{dS,\alpha ,n-1}^{2},}$

where

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

is the metric of an ${\displaystyle n-1}$ dimensional de Sitter space with radius of curvature ${\displaystyle \alpha }$ in open slicing coordinates. The hyperbolic metric is given by:

${\displaystyle dH_{n-2}^{2}=d\xi ^{2}+\sinh ^{2}(\xi )d\Omega _{n-3}^{2}.}$

This is the analytic continuation of the open slicing coordinates under ${\displaystyle \left(t,\xi ,\theta ,\phi _{1},\phi _{2},\ldots ,\phi _{n-3}\right)\to \left(i\chi ,\xi ,it,\theta ,\phi _{1},\ldots ,\phi _{n-4}\right)}$ and also switching ${\displaystyle x_{0}}$ and ${\displaystyle x_{2}}$ because they change their timelike/spacelike nature.

See also

• 反德西特空间
• de Sitter universe（英语：de Sitter universe）
• AdS/CFT对偶
• de Sitter–Schwarzschild metric（英语：de Sitter–Schwarzschild metric）

参考资料

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. Zee 2013，第626页
5. Hawking & Ellis. The large scale structure of space–time. Cambridge Univ. Press.

• Zee, Anthony. Einstein Gravity in a Nutshell. Princeton University Press. 2013. ISBN 9780691145587.

延伸阅读

• Qingming Cheng, De Sitter space, Hazewinkel, Michiel (编), 数学百科全书, Springer, 2001, ISBN 978-1-55608-010-4
• Nomizu, Katsumi, The Lorentz–Poincaré metric on the upper half-space and its extension, Hokkaido Mathematical Journal, 1982, 11 (3): 253–261, doi:10.14492/hokmj/1381757803 
• Coxeter, H. S. M., A geometrical background for de Sitter's world, American Mathematical Monthly (Mathematical Association of America), 1943, 50 (4): 217–228, JSTOR 2303924, doi:10.2307/2303924
• Susskind, L.; Lindesay, J., An Introduction to Black Holes, Information and the String Theory Revolution:The Holographic Universe: 119(11.5.25), 2005

外部链接

• Simplified Guide to de Sitter and anti-de Sitter Spaces （页面存档备份，存于互联网档案馆） A pedagogic introduction to de Sitter and anti-de Sitter spaces. The main article is simplified, with almost no math. The appendix is technical and intended for readers with physics or math backgrounds.