德西特空間

數學與物理學中，一個n維德西特空間（英語：，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。

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。

已隱藏部分未翻譯内容，歡迎參與翻譯。

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

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

de Sitter space and anti-de Sitter space are named after 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

de Sitter space can be defined as a submanifold of a generalized 閔考斯基時空 of one higher dimension. Take Minkowski space R^1,n with the standard metric:

ds^{2}=-dx_{0}^{2}+\sum _{i=1}^{n}dx_{i}^{2}.

de Sitter space is the submanifold described by the hyperboloid 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 metric on de Sitter space is the metric induced from the ambient Minkowski metric. The induced metric is nondegenerate and has Lorentzian signature. (Note that if one replaces $\alpha ^{2}$ with $-\alpha ^{2}$ in the above definition, one obtains a hyperboloid of two sheets. The induced metric in this case is positive-definite, and each sheet is a copy of hyperbolic n-space. For a detailed proof, see Minkowski space § Geometry.)

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

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

Properties

The 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]

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 $R^{\rho }{}_{\sigma \mu \nu }=\partial _{\mu }\Gamma _{\nu \sigma }^{\rho }-\partial _{\nu }\Gamma _{\mu \sigma }^{\rho }+\Gamma _{\mu \lambda }^{\rho }\Gamma _{\nu \sigma }^{\lambda }-\Gamma _{\nu \lambda }^{\rho }\Gamma _{\mu \sigma }^{\lambda }$ for the Riemann curvature tensor). de Sitter space is an Einstein manifold since the Ricci tensor 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 scalar curvature of de Sitter space is given by[4]

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 $(t,r,\ldots )$ for de Sitter as follows:

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

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 at $r=\alpha$ .

Flat slicing

Let

{\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 $\left(t,y_{i}\right)$ coordinates metric reads:

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 $y_{i}$ 's.

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

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

Open slicing

Let

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

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

where

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

is the standard hyperbolic metric.

Closed slicing

Let

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

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:

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

{\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 $z_{i}$ s 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.

參考資料

de Sitter, W., (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）
de Sitter, W., (PDF), Proc. Kon. Ned. Acad. Wet., 1917, 20: 229–243 [2022-12-01], （原始内容存档 (PDF)于2023-04-09）
Levi-Civita, Tullio, , Rendiconti, Reale Accademia dei Lincei, 1917, 26: 519–31
Zee 2013，第626頁
Hawking & Ellis. . Cambridge Univ. Press.

Zee, Anthony. . Princeton University Press. 2013. ISBN 9780691145587.

延伸閱讀

Qingming Cheng, , Hazewinkel, Michiel (编), , Springer, 2001, ISBN 978-1-55608-010-4
Nomizu, Katsumi, , Hokkaido Mathematical Journal, 1982, 11 (3): 253–261, doi:10.14492/hokmj/1381757803 
Coxeter, H. S. M., , American Mathematical Monthly (Mathematical Association of America), 1943, 50 (4): 217–228, JSTOR 2303924, doi:10.2307/2303924
Susskind, L.; Lindesay, J., : 119(11.5.25), 2005

外部鏈接

Simplified Guide to de Sitter and anti-de Sitter Spaces （页面存档备份，存于） 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.

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[1] Sitter, W., (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] Sitter, W., (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, , Rendiconti, Reale Accademia dei Lincei, 1917, 26: 519–31

[zee-p626-4] Zee 2013，第626頁

[5] Hawking & Ellis. . Cambridge Univ. Press.