德西特空間

數學與物理學中，一個n維德西特空間（英語：de Sitter space，標作dS_n）為一最大對稱的勞侖茲流形，具有正常數的純量曲率。

提示：此條目的主題不是反德西特空間。

德西特空間

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${\displaystyle \Lambda }$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 dS_n) 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） ${\displaystyle \Lambda }$ (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 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（英語：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（英語：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 group）s, 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 基靈向量場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]

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

(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（英語：Einstein manifold） since the Ricci tensor（英語：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（英語：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（英語：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（英語：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 潘洛斯圖.^[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

• 反德西特空間
• de Sitter universe（英語：de Sitter universe）
• AdS/CFT對偶
• de Sitter–Schwarzschild metric（英語：de Sitter–Schwarzschild metric）

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. Zee 2013，第626頁
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.

延伸閱讀

• Qingming Cheng, De Sitter space, Hazewinkel, Michiel (編), 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

外部鏈接

• Simplified Guide to de Sitter and anti-de Sitter Spaces （頁面存檔備份，存於互聯網檔案館） 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.