De Sitter space

Properties

The isometry group of de Sitter space is the Lorentz group O(1, n). The metric therefore then has n(n + 1)/2 independent Killing vector fields and is maximally symmetric. Every maximally symmetric space has constant curvature. The Riemann curvature tensor of de Sitter space is given by

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

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

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

The scalar curvature of de Sitter space is given by

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

For the case n = 4, we have Λ = 3/α² and R = 4Λ = 12/α².

Coordinates

Static coordinates

We can introduce static coordinates ${\displaystyle (t,r,\dots )}$ for de Sitter as follows:

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

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

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

Note that there is a cosmological horizon at ${\displaystyle r=\alpha }$.

Flat slicing

Let

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

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

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

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

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

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

Open slicing

Let

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

where $\sum _i{z}_i^2=1$ forming a ${\displaystyle S^{n-2}}$ 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

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

where

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

is the standard hyperbolic metric.

Closed slicing

Let

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

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

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

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:

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

These coordinates, also known as "global coordinates" cover the maximal extension of de Sitter space, and can therefore be used to find its Penrose diagram.

dS slicing

Let

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

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

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

where

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

is the metric of an ${\displaystyle n-1}$ dimensional de Sitter space with radius of curvature ${\displaystyle \alpha }$ in open slicing coordinates. The hyperbolic metric is given by:

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

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