德西特空間

德西特空間

數學與物理學中，一個n維德西特空間（Template:Lang-en，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$\mathrm{\Lambda }$對應正真空能量密度和負壓。

In Template:Tsl, n-dimensional de Sitter space (often abbreviated to dS_n) is a maximally symmetric Template:Tsl with constant positive Template:Tsl. It is the Lorentzian analogue of an Template:Tsl (with its canonical Template:Tsl).

The main application of de Sitter space is its use in Template:Tsl, where it serves as one of the simplest mathematical models of the universe consistent with the observed Template:Tsl. More specifically, de Sitter space is the maximally symmetric Template:Tsl of Template:Tsl with a positive Template:Tsl $\mathrm{\Lambda }$ (corresponding to a positive vacuum energy density and negative pressure). There is cosmological evidence that the universe itself is Template:Tsl, i.e. it will evolve like the de Sitter universe in the far future when Template:Tsl dominates.

de Sitter space and Template:Tsl are named after Template:Tsl (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 Template:Tsl of a generalized 閔考斯基時空 of one higher Template:Tsl. Take Minkowski space R^1,n with the standard Template:Tsl: $$d{s}^2=-d{x}_0^2+\sum _{i=1}^{n}d{x}_i^2.$$

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

de Sitter space can also be defined as the Template:Tsl Template:Nowrap of two Template:Tsls, which shows that it is a non-Riemannian Template:Tsl.

Template:Tsl, de Sitter space is Template:Nowrap (so that if Template:Nowrap then de Sitter space is Template:Tsl).

Properties

The Template:Tsl of de Sitter space is the 勞侖茲群 Template:Nowrap. The metric therefore then has Template:Nowrap 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

$R_{\rho \sigma \mu \nu }=\frac{1}{{\alpha }^2}(g_{\rho \mu }{g}_{\sigma \nu }-g_{\rho \nu }{g}_{\sigma \mu })$

(using the sign convention $R^{\rho }{}_{\sigma \mu \nu }={\partial }_{\mu }{\mathrm{\Gamma }}_{\nu \sigma }^{\rho }-{\partial }_{\nu }{\mathrm{\Gamma }}_{\mu \sigma }^{\rho }+{\mathrm{\Gamma }}_{\mu \lambda }^{\rho }{\mathrm{\Gamma }}_{\nu \sigma }^{\lambda }-{\mathrm{\Gamma }}_{\nu \lambda }^{\rho }{\mathrm{\Gamma }}_{\mu \sigma }^{\lambda }$ for the Riemann curvature tensor). de Sitter space is an Template:Tsl since the Template:Tsl 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

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

The Template:Tsl of de Sitter space is given by

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

For the case Template:Nowrap, we have Template:Nowrap and Template:Nowrap.

Static coordinates

We can introduce Template:Tsl $(t,r,\dots )$ for de Sitter as follows:

$\begin{array}{cc}{x}_0& =\sqrt{{\alpha }^2-r^2}\sinh \left(\frac{1}{\alpha }t\right)\\ x_1& =\sqrt{{\alpha }^2-r^2}\cosh \left(\frac{1}{\alpha }t\right)\\ x_i& =r{z}_{i}2\le i\le n.\end{array}$

where $z_i$ gives the standard embedding the Template:Nowrap-sphere in Rⁿ⁻¹. In these coordinates the de Sitter metric takes the form:

$d{s}^2=-(1-\frac{r^2}{{\alpha }^2})d{t}^2+{(1-\frac{r^2}{{\alpha }^2})}^{-1}d{r}^2+r^{2}d{\mathrm{\Omega }}_{n-2}^2.$

Note that there is a Template:Tsl at $r=\alpha$.

Flat slicing

Let

$\begin{array}{cc}{x}_0& =\alpha \sinh \left(\frac{1}{\alpha }t\right)+\frac{1}{2\alpha }{r}^2{e}^{\frac{1}{\alpha }t},\\ x_1& =\alpha \cosh \left(\frac{1}{\alpha }t\right)-\frac{1}{2\alpha }{r}^2{e}^{\frac{1}{\alpha }t},\\ x_i& =e^{\frac{1}{\alpha }t}{y}_i,2\le i\le n\end{array}$

where $r^2=\sum _i{y}_i^2$. Then in the $(t,y_i)$ coordinates metric reads:

$d{s}^2=-d{t}^2+e^{2\frac{1}{\alpha }t}d{y}^2$

where $d{y}^2=\sum _{i}d{y}_i^2$ is the flat metric on $y_i$'s.

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

$d{s}^2=\frac{{\alpha }^2}{({\zeta }_{\mathrm{\infty }}-\zeta {)}^2}(d{y}^2-d{\zeta }^2)$

Open slicing

Let

$\begin{array}{cc}{x}_0& =\alpha \sinh \left(\frac{1}{\alpha }t\right)\cosh \xi ,\\ x_1& =\alpha \cosh \left(\frac{1}{\alpha }t\right),\\ x_i& =\alpha z_i\sinh \left(\frac{1}{\alpha }t\right)\sinh \xi ,2\le i\le n\end{array}$

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

$d{s}^2=-d{t}^2+{\alpha }^2{\sinh }^2\left(\frac{1}{\alpha }t\right)d{H}_{n-1}^2,$

where

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

is the standard hyperbolic metric.

Closed slicing

Let

$\begin{array}{cc}{x}_0& =\alpha \sinh \left(\frac{1}{\alpha }t\right),\\ x_i& =\alpha \cosh \left(\frac{1}{\alpha }t\right)z_i,1\le i\le n\end{array}$

where $z_i$s describe a $S^{n-1}$. Then the metric reads:

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

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

$d{s}^2=\frac{{\alpha }^2}{{\cos }^2\eta }(-d{\eta }^2+d{\mathrm{\Omega }}_{n-1}^2).$

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.

dS slicing

Let

$\begin{array}{cc}{x}_0& =\alpha \sin \left(\frac{1}{\alpha }\chi \right)\sinh \left(\frac{1}{\alpha }t\right)\cosh \xi ,\\ x_1& =\alpha \cos \left(\frac{1}{\alpha }\chi \right),\\ x_2& =\alpha \sin \left(\frac{1}{\alpha }\chi \right)\cosh \left(\frac{1}{\alpha }t\right),\\ x_i& =\alpha z_i\sin \left(\frac{1}{\alpha }\chi \right)\sinh \left(\frac{1}{\alpha }t\right)\sinh \xi ,3\le i\le n\end{array}$

where $z_i$s describe a $S^{n-3}$. Then the metric reads:

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

where

$d{s}_{dS,\alpha ,n-1}^2=-d{t}^2+{\alpha }^2{\sinh }^2\left(\frac{1}{\alpha }t\right)d{H}_{n-2}^2$

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:

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

This is the analytic continuation of the open slicing coordinates under $(t,\xi ,\theta ,{\varphi }_1,{\varphi }_2,\dots ,{\varphi }_{n-3})\to (i\chi ,\xi ,it,\theta ,{\varphi }_1,\dots ,{\varphi }_{n-4})$ and also switching $x_0$ and $x_2$ because they change their timelike/spacelike nature.

See also

• 反德西特空間
• Template:Tsl
• AdS/CFT对偶
• Template:Tsl

參考資料

• Template:Cite book

延伸閱讀

• Template:Springer
• Template:Citation
• Template:Citation
• Template:Citation

外部鏈接

• Simplified Guide to de Sitter and anti-de Sitter Spaces Template:Wayback 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.