查看“︁德西特空間”︁的源代码

{{NoteTA|G1=Math|G2=物理學}}
{{not|反德西特空間}}
[[File:Hyperb1N.png|thumb|德西特空間]]
[[數學]]與[[物理學]]中，一個''n''維'''德西特空間'''（{{lang-en|de Sitter space}}，標作dS<sub>''n''</sub>）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]]<math>\Lambda</math>對應正真空能量密度和負壓。

{{TransH}}

In {{tsl|en|mathematical physics}}, ''n''-dimensional '''de&nbsp;Sitter space''' (often abbreviated to dS<sub>''n''</sub>) is a maximally symmetric {{tsl|en|Lorentzian manifold}} with constant positive {{tsl|en|scalar curvature}}. It is the Lorentzian analogue of an {{tsl|en|n-sphere|''n''-sphere}} (with its canonical {{tsl|en|Riemannian metric}}).

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

de&nbsp;Sitter space and {{tsl|en|anti-de&nbsp;Sitter space}} are named after {{tsl|en|Willem de&nbsp;Sitter}} (1872–1934),<ref>{{Citation|last=de Sitter|first=W.|year=1917|title=On the relativity of inertia: Remarks concerning Einstein's latest hypothesis|journal=Proc. Kon. Ned. Acad. Wet.|volume=19|pages=1217–1225|bibcode=1917KNAB...19.1217D|url=https://www.dwc.knaw.nl/DL/publications/PU00012455.pdf|accessdate=2022-12-01|archive-date=2023-04-07|archive-url=https://web.archive.org/web/20230407093203/https://dwc.knaw.nl/DL/publications/PU00012455.pdf|dead-url=no}}</ref><ref>{{Citation|last=de Sitter|first=W.|year=1917|title=On the curvature of space|journal=Proc. Kon. Ned. Acad. Wet.|volume=20|pages=229–243|url=https://www.dwc.knaw.nl/DL/publications/PU00012216.pdf|accessdate=2022-12-01|archive-date=2023-04-09|archive-url=https://web.archive.org/web/20230409080125/https://dwc.knaw.nl/DL/publications/PU00012216.pdf|dead-url=no}}</ref> professor of astronomy at [[Leiden University]] and director of the [[莱顿天文台]]. Willem de&nbsp;Sitter and [[Albert Einstein]] worked closely together in [[Leiden]] in the 1920s on the spacetime structure of our universe. de&nbsp;Sitter space was also discovered, independently, and about the same time, by [[图利奥·列维-齐维塔]].<ref>{{Citation|first=Tullio|last=Levi-Civita|authorlink=Tullio Levi-Civita|title=Realtà fisica di alcuni spazî normali del Bianchi|journal=Rendiconti, Reale Accademia dei Lincei|volume=26|year=1917|pages=519–31}}</ref>

== 定義 ==
de&nbsp;Sitter space can be defined as a {{tsl|en|submanifold}} of a generalized [[閔考斯基時空]] of one higher {{tsl|en|dimension}}. Take Minkowski space '''R'''<sup>1,''n''</sup> with the standard {{tsl|en|metric tensor|metric}}:
<math display="block">ds^2 = -dx_0^2 + \sum_{i=1}^n dx_i^2.</math>

de&nbsp;Sitter space is the submanifold described by the {{tsl|en|hyperboloid}} of one sheet
<math display="block">-x_0^2 + \sum_{i=1}^n x_i^2 = \alpha^2,</math>
where <math>\alpha</math> is some nonzero constant with its dimension being that of length. The {{tsl|en|metric tensor|metric}} on de&nbsp;Sitter space is the metric induced from the ambient Minkowski metric. The induced metric is {{tsl|en|nondegenerate}} and has Lorentzian signature. (Note that if one replaces <math>\alpha^2</math> with <math>-\alpha^2</math> in the above definition, one obtains a {{tsl|en|hyperboloid}} of two sheets. The induced metric in this case is {{tsl|en|Definite quadratic form|positive-definite}}, and each sheet is a copy of {{tsl|en|hyperbolic space|hyperbolic ''n''-space}}. For a detailed proof, see ''{{section link|Minkowski space#Geometry}}''.)

de&nbsp;Sitter space can also be defined as the {{tsl|en|Homogeneous space|quotient}} {{nowrap|O(1, ''n'') / O(1, ''n'' − 1)}} of two {{tsl|en|indefinite orthogonal group}}s, which shows that it is a non-Riemannian {{tsl|en|symmetric space}}.

{{tsl|en|Topology|Topologically}}, de&nbsp;Sitter space is {{nowrap|'''R''' × ''S''<sup>''n''−1</sup>}} (so that if {{nowrap|''n'' ≥ 3}} then de&nbsp;Sitter space is {{tsl|en|simply connected}}).

==Properties==
The {{tsl|en|isometry group}} of de&nbsp;Sitter space is the [[勞侖茲群]] {{nowrap|O(1, ''n'')}}. The metric therefore then has {{nowrap|''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&nbsp;Sitter is given by<ref name="zee-p626">{{harvnb|Zee|2013|p=626}}</ref>
:<math>R_{\rho\sigma\mu\nu} = {1 \over \alpha^2}\left(g_{\rho\mu}g_{\sigma\nu} - g_{\rho\nu}g_{\sigma\mu}\right)</math>

(using the sign convention <math>
  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}
</math> for the Riemann curvature tensor). de&nbsp;Sitter space is an {{tsl|en|Einstein manifold}} since the {{tsl|en|Ricci tensor}} is proportional to the metric:
:<math>R_{\mu\nu} = R^\lambda{}_{\mu\lambda\nu} = \frac{n - 1}{\alpha^2}g_{\mu\nu}</math>

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

The {{tsl|en|scalar curvature}} of de&nbsp;Sitter space is given by<ref name="zee-p626" />
:<math>R = \frac{n(n - 1)}{\alpha^2} = \frac{2n}{n - 2}\Lambda.</math>

For the case {{nowrap|1=''n'' = 4}}, we have {{nowrap|1=Λ = 3/''α''<sup>2</sup>}} and {{nowrap|1=''R'' = 4Λ = 12/''α''<sup>2</sup>}}.

== Coordinates ==
=== Static coordinates ===
We can introduce {{tsl|en|static spacetime|static coordinates}} <math>(t, r, \ldots)</math> for de&nbsp;Sitter as follows:
:<math>\begin{align}
  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 &= r z_i \qquad\qquad\qquad\qquad\qquad 2 \le i \le n.
\end{align}</math>

where <math>z_i</math> gives the standard embedding the {{nowrap|(''n'' − 2)}}-sphere in '''R'''<sup>''n''−1</sup>. In these coordinates the de&nbsp;Sitter metric takes the form:

:<math>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.</math>

Note that there is a {{tsl|en|cosmological horizon}} at <math>r = \alpha</math>.

=== Flat slicing ===
Let
:<math>\begin{align}
  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{align}</math>

where <math display="inline">r^2 = \sum_i y_i^2</math>.  Then in the <math>\left(t, y_i\right)</math> coordinates metric reads:
:<math>ds^{2} = -dt^{2} + e^{2\frac{1}{\alpha}t} dy^{2}</math>

where <math display="inline">dy^2 = \sum_i dy_i^2</math> is the flat metric on <math>y_i</math>'s.

Setting <math>\zeta = \zeta_{\infty} - \alpha e^{-\frac{1}{\alpha}t}</math>, we obtain the conformally flat metric:
:<math>ds^2 = \frac{\alpha^2}{(\zeta_\infty - \zeta)^2}\left(dy^2 - d\zeta^2\right)</math>

=== Open slicing ===
Let
:<math>\begin{align}
  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{align}</math>

where <math display="inline">\sum_i z_i^2 = 1</math> forming a <math>S^{n-2}</math> with the standard metric <math display="inline">\sum_i dz_i^2 = d\Omega_{n-2}^2</math>.  Then the metric of the de&nbsp;Sitter space reads
:<math>ds^2 = -dt^2 + \alpha^2 \sinh^2\left(\frac{1}{\alpha}t\right) dH_{n-1}^2,</math>

where
:<math>dH_{n-1}^2 = d\xi^2 + \sinh^2(\xi) d\Omega_{n-2}^2</math>

is the standard hyperbolic metric.

=== Closed slicing ===
Let
:<math>\begin{align}
  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{align}</math>

where <math>z_i</math>s describe a <math>S^{n-1}</math>.  Then the metric reads:
:<math>ds^2 = -dt^2 + \alpha^2 \cosh^2\left(\frac{1}{\alpha}t\right) d\Omega_{n-1}^2.</math>

Changing the time variable to the conformal time via <math display="inline">\tan\left(\frac{1}{2}\eta\right) = \tanh\left(\frac{1}{2\alpha}t\right)</math> we obtain a metric conformally equivalent to Einstein static universe:
:<math>ds^2 = \frac{\alpha^2}{\cos^2\eta}\left(-d\eta^2 + d\Omega_{n-1}^2\right).</math>

These coordinates, also known as "global coordinates" cover the maximal extension of de&nbsp;Sitter space, and can therefore be used to find its [[彭罗斯图]].<ref>{{cite book |last1=Hawking & Ellis |title=The large scale structure of space–time |publisher=Cambridge Univ. Press}}</ref>

=== dS slicing ===
Let
:<math>\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}</math>

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

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

is the metric of an <math>n - 1</math> dimensional de&nbsp;Sitter space with radius of curvature <math>\alpha</math> in open slicing coordinates.  The hyperbolic metric is given by:
:<math>dH_{n-2}^2 = d\xi^2 + \sinh^2(\xi) d\Omega_{n-3}^2.</math>

This is the analytic continuation of the open slicing coordinates under <math>\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)</math> and also switching <math>x_0</math> and <math>x_2</math> because they change their timelike/spacelike nature.

==See also==
* [[反德西特空間]]
* {{tsl|en|de Sitter universe}}
* [[AdS/CFT对偶]]
* {{tsl|en|de Sitter–Schwarzschild metric}}

{{TransF}}

== 參考資料 ==
{{reflist|25em}}
*{{cite book|last=Zee|first=Anthony|author-link=Anthony Zee|title=Einstein Gravity in a Nutshell|url=https://archive.org/details/einsteingravityi0000zeea|date=2013 |publisher=Princeton University Press|isbn=9780691145587}}

== 延伸閱讀 ==
*{{springer|id=d/d110040|title=De Sitter space|author=Qingming Cheng}}
*{{Citation|last=Nomizu|first=Katsumi|authorlink=Katsumi Nomizu|title=The Lorentz–Poincaré metric on the upper half-space and its extension|journal=[[Hokkaido Mathematical Journal]]|volume=11|year=1982|pages=253–261|issue=3|doi=10.14492/hokmj/1381757803|doi-access=free}}
*{{Citation|last=Coxeter|first=H. S. M.|authorlink=Harold Scott MacDonald Coxeter|title=A geometrical background for de&nbsp;Sitter's world|journal=[[American Mathematical Monthly]]|volume=50|year=1943|pages=217–228|doi=10.2307/2303924|issue=4|publisher=Mathematical Association of America|jstor=2303924}}
*{{Citation|last1=Susskind|first1=L.|last2=Lindesay|first2=J.|title=An Introduction to Black Holes, Information and the String Theory Revolution:The Holographic Universe|year=2005|page=119(11.5.25)}}

== 外部鏈接 ==
* [http://www.quantumfieldtheory.info/dS_and_AdS_spaces.pdf  Simplified Guide to de&nbsp;Sitter and anti-de&nbsp;Sitter Spaces] {{Wayback|url=http://www.quantumfieldtheory.info/dS_and_AdS_spaces.pdf |date=20230406034257 }} A pedagogic introduction to de&nbsp;Sitter and anti-de&nbsp;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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