De Sitter space

In mathematical physics, n-dimensional (often denoted dSn) is a maximally symmetric Lorentzian manifold with constant positive scalar curvature. It is analogue of an n-sphere, with a Lorentzian metric in place of the Riemannian metric of the latter.

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 in which the cosmological constant <math>\Lambda</math> is positive (corresponding to a positive vacuum energy density and negative pressure).

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 Leiden Observatory. Willem de Sitter and Albert Einstein worked closely together in Leiden in the 1920s on the spacetime structure of the universe. De Sitter space was also discovered, independently, and about the same time, by Tullio Levi-Civita.

Definition

A de Sitter space can be defined as a submanifold of a generalized Minkowski space of one higher dimension, including the induced metric. Take Minkowski space R1,n with the standard metric:

The n-dimensional de Sitter space is the submanifold described by the 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 induced metric on the de Sitter space is induced from the ambient Lorentzian metric. It is nondegenerate and has Lorentzian signature. (If one replaces <math>\alpha^2</math> with <math>-\alpha^2</math> 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. See '.)

The de Sitter space can also be defined as the quotient of two indefinite orthogonal groups, which shows that it is a non-Riemannian symmetric space.

Topologically, dSn is , which is simply connected if .

Properties

The isometry group of de Sitter space is the Lorentz group . The metric therefore then has 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

<math display="block">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 Sitter space is an Einstein manifold since the Ricci tensor is proportional to the metric:

<math display="block">R_{\mu\nu} = R^\lambda{}_{\mu\lambda\nu} = \frac{n - 1}{\alpha^2}g_{\mu\nu}</math>

This means de Sitter space is a vacuum solution of Einstein's equation with cosmological constant given by

<math display="block">\Lambda = \frac{(n - 1)(n - 2)}{2\alpha^2}.</math>

The scalar curvature of de Sitter space is given by

dS slicing

Let

<math display="block">\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 display="block">ds^2 = d\chi^2 + \sin^2\left(\frac{1}{\alpha}\chi\right) ds_{dS,\alpha,n-1}^2,</math>

where

<math display="block">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 Sitter space with radius of curvature <math>\alpha</math> in open slicing coordinates. The hyperbolic metric is given by:

<math display="block">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.

References

External links

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.