De Sitter space explained

In mathematical physics, n-dimensional de Sitter space (often denoted dS_n) 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).

Λ (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 our 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 R^1,n with the standard metric:$$ds^2\; =\; -dx\_0^2\; +\; \backslash sum\_^n\; dx\_i^2.$$

The n-dimensional de Sitter space is the submanifold described by the hyperboloid of one sheet$$-x\_0^2\; +\; \backslash sum\_^n\; x\_i^2\; =\; \backslash alpha^2,$$where

\alpha is some nonzero constant with its dimension being that of length. The induced metric on the de Sitter space induced from the ambient Minkowski metric. It is nondegenerate and has Lorentzian signature. (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. 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, dS_n 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 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 R ^{\rho} {} _{\sigma\mu\nu} = \partial _{\mu} \Gamma ^{\rho} _{\nu\sigma} - \partial _{\nu} \Gamma ^{\rho} _{\mu\sigma} + \Gamma ^{\rho} _{\mu λ} \Gamma ^{λ} _{\nu\sigma} - \Gamma ^{\rho} _{\nu λ} \Gamma ^{λ} _{\mu\sigma} 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 ^{λ {}} _{\mu λ \nu} = (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

Λ = ((n -1 )(n -2 ))/(2 \alpha ^{2}) .

The scalar curvature of de Sitter space is given by

R = (n(n -1 ))/(\alpha ^{2}) = (2 n)/(n -2) Λ .

For the case, we have and .

Coordinates

Static coordinates

(t, r, \ldots) for de Sitter as follows:\begin{align} x _{0} & = \sqrt{\alpha ^{2} - r ^{2 }\sinh\left( (1)/(\alpha)} t\right) \\ x _{1} & = \sqrt{\alpha ^{2} - r ^{2 }\cosh\left( (1)/(\alpha)} t\right) \\ x _{i} & = r z _{i} 2 \le i \le n. \end{align}

where

z _{i} gives the standard embedding the -sphere in Rⁿ⁻¹. In these coordinates the de Sitter metric takes the form:ds ^{2} = - \left( 1 - (r ^{2})/(\alpha ^{2}) \right)dt ^{2} + \left( 1 - (r ^{2})/(\alpha ^{2}) \right) ^{-1} dr ^{2} + r ^{2} d\Omega ^{2 .} _{n -2}

Note that there is a cosmological horizon at

r = \alpha .

Flat slicing

Let

\begin{align} x _{0} & = \alpha \sinh\left( (1)/(\alpha) t\right) + (1)/(2 \alpha) r ^{2} e ^{(1)/(t \alpha)} , \\ x _{1} & = \alpha \cosh\left( (1)/(\alpha) t\right) - (1)/(2 \alpha) r ^{2} e ^{(1)/(t \alpha)} , \\ x _{i} & = e ^{(1)/(t \alpha)} y _{i,} 2 \leq i \leq n \end{align} where $r^2\; =\; \backslash sum\_i\; y\_i^2$. Then in the \left(t, y _{i\right)} coordinates metric reads:ds ^{2} = - dt ^{2} + e ^{2 (1)/(t \alpha)} dy ^{2} where $dy^2\; =\; \backslash sum\_i\; dy\_i^2$ is the flat metric on y _{i} 's.

Setting

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

Open slicing

Let

\begin{align} x _{0} & = \alpha \sinh\left( (1)/(\alpha) t\right) \cosh\xi, \\ x _{1} & = \alpha \cosh\left( (1)/(\alpha) t\right), \\ x _{i} & = \alpha z _{i} \sinh\left( (1)/(\alpha) t\right) \sinh\xi, 2 \leq i \leq n \end{align} where $\backslash sum\_i\; z\_i^2\; =\; 1$ forming a S ^{n-2} with the standard metric $\backslash sum\_i\; dz\_i^2\; =\; d\backslash Omega\_^2$. Then the metric of the de Sitter space readsds ^{2} = - dt ^{2} + \alpha ^{2} \sinh ^{2 \left( (1)/(\alpha)} t\right) dH ^{2 ,} _{n -1} wheredH ^{2} _{n -1} = d\xi ^{2} + \sinh ^{2(\xi)} d\Omega ^{2} _{n -2} is the standard hyperbolic metric.

Closed slicing

Let

\begin{align} x _{0} & = \alpha \sinh\left( (1)/(\alpha) t\right), \\ x _{i} & = \alpha \cosh\left( (1)/(\alpha) t\right) z _{i,} 1 \leq i \leq n \end{align} where z _{i} s describe a S ^{n-1} . Then the metric reads:ds ^{2} = - dt ^{2} + \alpha ^{2} \cosh ^{2 \left( (1)/(\alpha)} t\right) d\Omega ^{2 .} _{n -1}

Changing the time variable to the conformal time via $\backslash tan\backslash left(\backslash frac\backslash eta\backslash right)\; =\; \backslash tanh\backslash left(\backslash fract\backslash right)$ we obtain a metric conformally equivalent to Einstein static universe:

ds ^{2} = (\alpha ^{2})/(\cos ^{2η}) \left( - d η ^{2} + d\Omega ^{2 \right).} _{n -1}

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.^[1]

dS slicing

Let

\begin{align} x _{0} & = \alpha \sin\left( (1)/(\alpha) \chi\right) \sinh\left( (1)/(\alpha) t\right) \cosh\xi, \\ x _{1} & = \alpha \cos\left( (1)/(\alpha) \chi\right), \\ x _{2} & = \alpha \sin\left( (1)/(\alpha) \chi\right) \cosh\left( (1)/(\alpha) t\right), \\ x _{i} & = \alpha z _{i} \sin\left( (1)/(\alpha) \chi\right) \sinh\left( (1)/(\alpha) t\right) \sinh\xi, 3 \leq i \leq n \end{align} where z _{i} s describe a S ^{n-3} . Then the metric reads:ds ^{2} = d\chi ^{2} + \sin ^{2 \left( (1)/(\alpha)} \chi\right) ds ^{2 ,} _{dS,\alpha,n -1} whereds ^{2} _{dS,\alpha,n -1} = - dt ^{2} + \alpha ^{2} \sinh ^{2 \left( (1)/(\alpha)} t\right) dH ^{2} _{n -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 ^{2} _{n -2} = d\xi ^{2} + \sinh ^{2(\xi)} d\Omega ^{2 .} _{n -3}

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.

See also

• Anti-de Sitter space
• De Sitter universe
• AdS/CFT correspondence
• De Sitter–Schwarzschild metric

References

• Book: Zee, Anthony. Anthony Zee. Einstein Gravity in a Nutshell. 2013 . Princeton University Press. 9780691145587.

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.

Notes and References

1. Book: Hawking & Ellis . The large scale structure of space–time . Cambridge Univ. Press.