德西特空间

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数学与物理学中，一个n维德西特空间（de Sitter space，标作dSn）为一最大对称的劳仑兹流形，具有正常数的纯量曲率。

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

In mathematical physics, n-dimensional de Sitter space (often abbreviated to dSn) 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),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.

## 定义

de Sitter space can be defined as a submanifold of a generalized 闵考斯基时空 of one higher dimension. Take Minkowski space R1,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 劳仑兹群 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

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

(using the sign convention $<pre> R^{\rho}{}_{\sigma\mu\nu} = \partial_{\mu}\Gamma^{\rho}_{\nu\sigma} - \partial_{\nu}\Gamma^{\rho}_{\mu\sigma} + \Gamma^{\rho}_{\mu\lambda}\Gamma^{\lambda}_{\nu\sigma} - \Gamma^{\rho}_{\nu\lambda}\Gamma^{\lambda}_{\mu\sigma} </pre>$ 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

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

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:

$$\begin{align} <pre> x_0 &= \sqrt{\alpha^2 - r^2}\sinh\left(\frac{1}{\alpha}t\right) \\ x_1 &= \sqrt{\alpha^2 - r^2}\cosh\left(\frac{1}{\alpha}t\right) \\ x_i &= r z_i \qquad\qquad\qquad\qquad\qquad 2 \le i \le n. </pre> \end{align}$$

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

$$\begin{align} <pre> x_0 &= \alpha \sinh\left(\frac{1}{\alpha}t\right) + \frac{1}{2\alpha}r^2 e^{\frac{1}{\alpha}t}, \\ x_1 &= \alpha \cosh\left(\frac{1}{\alpha}t\right) - \frac{1}{2\alpha}r^2 e^{\frac{1}{\alpha}t}, \\ x_i &= e^{\frac{1}{\alpha}t}y_i, \qquad 2 \leq i \leq n </pre> \end{align}$$

where $r^2 = \sum_i y_i^2$. Then in the $\left(t, y_i\right)$ coordinates metric reads:

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

where $dy^2 = \sum_i dy_i^2$ 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

$$\begin{align} <pre> x_0 &= \alpha \sinh\left(\frac{1}{\alpha}t\right) \cosh\xi, \\ x_1 &= \alpha \cosh\left(\frac{1}{\alpha}t\right), \\ x_i &= \alpha z_i \sinh\left(\frac{1}{\alpha}t\right) \sinh\xi, \qquad 2 \leq i \leq n </pre> \end{align}$$

where $\sum_i z_i^2 = 1$ forming a $S^{n-2}$ with the standard metric $\sum_i dz_i^2 = d\Omega_{n-2}^2$. 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

$$\begin{align} <pre> x_0 &= \alpha \sinh\left(\frac{1}{\alpha}t\right), \\ x_i &= \alpha \cosh\left(\frac{1}{\alpha}t\right) z_i, \qquad 1 \leq i \leq n </pre> \end{align}$$

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 $\tan\left(\frac{1}{2}\eta\right) = \tanh\left(\frac{1}{2\alpha}t\right)$ 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.

### dS slicing

Let

$$\begin{align} <pre> x_0 &= \alpha \sin\left(\frac{1}{\alpha}\chi\right) \sinh\left(\frac{1}{\alpha}t\right) \cosh\xi, \\ x_1 &= \alpha \cos\left(\frac{1}{\alpha}\chi\right), \\ x_2 &= \alpha \sin\left(\frac{1}{\alpha}\chi\right) \cosh\left(\frac{1}{\alpha}t\right), \\ x_i &= \alpha z_i \sin\left(\frac{1}{\alpha}\chi\right) \sinh\left(\frac{1}{\alpha}t\right) \sinh\xi, \qquad 3 \leq i \leq n </pre> \end{align}$$

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.

## 外部链接

- Simplified Guide to de Sitter and anti-de Sitter Spaces 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.