De Sitter space

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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^{[further explanation needed]} 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 ${\displaystyle \Lambda }$ (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),^[1][2] 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.^[3]

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: $${\displaystyle ds^{2}=-dx_{0}^{2}+\sum _{i=1}^{n}dx_{i}^{2}.}$$

The n-dimensional de Sitter space is the submanifold described by the hyperboloid of one sheet $${\displaystyle -x_{0}^{2}+\sum _{i=1}^{n}x_{i}^{2}=\alpha ^{2},}$$ where ${\displaystyle \alpha }$ 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 ${\displaystyle \alpha ^{2}}$ with ${\displaystyle -\alpha ^{2}}$ 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 Minkowski space § Geometry.)

The 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, dS_n is R × Sⁿ⁻¹ (which is simply connected if n ≥ 3).

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 is given by^[4]

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

(using the sign convention ${\displaystyle R^{\rho }{}_{\sigma \mu \nu }=\partial _{\mu }\Gamma _{\nu \sigma }^{\rho }-\partial _{\nu }\Gamma _{\mu \sigma }^{\rho }+\Gamma _{\mu \lambda }^{\rho }\Gamma _{\nu \sigma }^{\lambda }-\Gamma _{\nu \lambda }^{\rho }\Gamma _{\mu \sigma }^{\lambda }}$ 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

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

The scalar curvature of de Sitter space is given by^[4]

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

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

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

${\displaystyle {\begin{aligned}x_{0}&={\sqrt {\alpha ^{2}-r^{2}}}\sinh \left({\frac {1}{\alpha }}t\right)\\x_{1}&={\sqrt {\alpha ^{2}-r^{2}}}\cosh \left({\frac {1}{\alpha }}t\right)\\x_{i}&=rz_{i}\qquad \qquad \qquad \qquad \qquad 2\leq i\leq n.\end{aligned}}}$

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:

${\displaystyle ds^{2}=-\left(1-{\frac {r^{2}}{\alpha ^{2}}}\right)dt^{2}+\left(1-{\frac {r^{2}}{\alpha ^{2}}}\right)^{-1}dr^{2}+r^{2}d\Omega _{n-2}^{2}.}$

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

Let

${\displaystyle {\begin{aligned}x_{0}&=\alpha \sinh \left({\frac {1}{\alpha }}t\right)+{\frac {1}{2\alpha }}r^{2}e^{{\frac {1}{\alpha }}t},\\x_{1}&=\alpha \cosh \left({\frac {1}{\alpha }}t\right)-{\frac {1}{2\alpha }}r^{2}e^{{\frac {1}{\alpha }}t},\\x_{i}&=e^{{\frac {1}{\alpha }}t}y_{i},\qquad 2\leq i\leq n\end{aligned}}}$

where ${\textstyle r^{2}=\sum _{i}y_{i}^{2}}$. Then in the ${\displaystyle \left(t,y_{i}\right)}$ coordinates metric reads:

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

where ${\textstyle dy^{2}=\sum _{i}dy_{i}^{2}}$ is the flat metric on ${\displaystyle y_{i}}$'s.

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

${\displaystyle ds^{2}={\frac {\alpha ^{2}}{(\zeta _{\infty }-\zeta )^{2}}}\left(dy^{2}-d\zeta ^{2}\right)}$

Let

${\displaystyle {\begin{aligned}x_{0}&=\alpha \sinh \left({\frac {1}{\alpha }}t\right)\cosh \xi ,\\x_{1}&=\alpha \cosh \left({\frac {1}{\alpha }}t\right),\\x_{i}&=\alpha z_{i}\sinh \left({\frac {1}{\alpha }}t\right)\sinh \xi ,\qquad 2\leq i\leq n\end{aligned}}}$

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

${\displaystyle ds^{2}=-dt^{2}+\alpha ^{2}\sinh ^{2}\left({\frac {1}{\alpha }}t\right)dH_{n-1}^{2},}$

where

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

is the standard hyperbolic metric.

Let

${\displaystyle {\begin{aligned}x_{0}&=\alpha \sinh \left({\frac {1}{\alpha }}t\right),\\x_{i}&=\alpha \cosh \left({\frac {1}{\alpha }}t\right)z_{i},\qquad 1\leq i\leq n\end{aligned}}}$

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

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

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

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

These coordinates, also known as "global coordinates" cover the maximal extension of de Sitter space, and can therefore be used to find its Penrose diagram.^[5]

Let

${\displaystyle {\begin{aligned}x_{0}&=\alpha \sin \left({\frac {1}{\alpha }}\chi \right)\sinh \left({\frac {1}{\alpha }}t\right)\cosh \xi ,\\x_{1}&=\alpha \cos \left({\frac {1}{\alpha }}\chi \right),\\x_{2}&=\alpha \sin \left({\frac {1}{\alpha }}\chi \right)\cosh \left({\frac {1}{\alpha }}t\right),\\x_{i}&=\alpha z_{i}\sin \left({\frac {1}{\alpha }}\chi \right)\sinh \left({\frac {1}{\alpha }}t\right)\sinh \xi ,\qquad 3\leq i\leq n\end{aligned}}}$

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

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

where

${\displaystyle ds_{dS,\alpha ,n-1}^{2}=-dt^{2}+\alpha ^{2}\sinh ^{2}\left({\frac {1}{\alpha }}t\right)dH_{n-2}^{2}}$

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

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

This is the analytic continuation of the open slicing coordinates under ${\displaystyle \left(t,\xi ,\theta ,\phi _{1},\phi _{2},\ldots ,\phi _{n-3}\right)\to \left(i\chi ,\xi ,it,\theta ,\phi _{1},\ldots ,\phi _{n-4}\right)}$ and also switching ${\displaystyle x_{0}}$ and ${\displaystyle x_{2}}$ because they change their timelike/spacelike nature.

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

• Zee, Anthony (2013). Einstein Gravity in a Nutshell. Princeton University Press. ISBN 9780691145587.

• Qingming Cheng (2001) [1994], "De Sitter space", Encyclopedia of Mathematics, EMS Press
• Nomizu, Katsumi (1982), "The Lorentz–Poincaré metric on the upper half-space and its extension", Hokkaido Mathematical Journal, 11 (3): 253–261, doi:10.14492/hokmj/1381757803
• Coxeter, H. S. M. (1943), "A geometrical background for de Sitter's world", American Mathematical Monthly, 50 (4), Mathematical Association of America: 217–228, doi:10.2307/2303924, JSTOR 2303924
• Susskind, L.; Lindesay, J. (2005), An Introduction to Black Holes, Information and the String Theory Revolution:The Holographic Universe, p. 119(11.5.25)

• 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.