De Sitter space

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

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).

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]

Definition

de Sitter space can be defined as a submanifold of a generalized Minkowski space of one higher dimension. Take Minkowski space R^1,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 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]

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 $R^{\rho }{}_{\sigma \mu \nu }=\partial _{\mu }\Gamma _{\nu \sigma }^{\rho }-\partial _{\nu }\Gamma _{\mu \sigma }^{\rho }+\Gamma _{\mu \lambda }^{\rho }\Gamma _{\nu \sigma }^{\lambda }-\Gamma _{\nu \lambda }^{\rho }\Gamma _{\mu \sigma }^{\lambda }$ for the Riemann curvature tensor). de Sitter space is an Einstein manifold since the Ricci tensor is proportional to the metric:

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[4]

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{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 $z_{i}$ 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{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 $\left(t,y_{i}\right)$ coordinates metric reads:

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 $y_{i}$ '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{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 $S^{n-2}$ with the standard metric ${\textstyle \sum _{i}dz_{i}^{2}=d\Omega _{n-2}^{2}}$ . Then the metric of the de Sitter space reads

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{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 $z_{i}$ 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 ${\textstyle \tan \left({\frac {1}{2}}\eta \right)=\tanh \left({\frac {1}{2\alpha }}t\right)}$ we obtain a metric conformally equivalent to Einstein static universe:

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]

dS slicing

Let

{\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 $z_{i}$ 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.

References

de Sitter, W. (1917), "On the relativity of inertia: Remarks concerning Einstein's latest hypothesis" (PDF), Proc. Kon. Ned. Acad. Wet., 19: 1217–1225, Bibcode:1917KNAB...19.1217D
de Sitter, W. (1917), "On the curvature of space" (PDF), Proc. Kon. Ned. Acad. Wet., 20: 229–243
Levi-Civita, Tullio (1917), "Realtà fisica di alcuni spazî normali del Bianchi", Rendiconti, Reale Accademia dei Lincei, 26: 519–31
Zee 2013, p. 626
Hawking & Ellis. The large scale structure of space–time. Cambridge Univ. Press.

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

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[1] Sitter, W. (1917), "On the relativity of inertia: Remarks concerning Einstein's latest hypothesis" (PDF), Proc. Kon. Ned. Acad. Wet., 19: 1217–1225, Bibcode:1917KNAB...19.1217D

[2] Sitter, W. (1917), "On the curvature of space" (PDF), Proc. Kon. Ned. Acad. Wet., 20: 229–243

[3] Levi-Civita, Tullio (1917), "Realtà fisica di alcuni spazî normali del Bianchi", Rendiconti, Reale Accademia dei Lincei, 26: 519–31

[zee-p626-4] Zee 2013, p. 626

[5] Hawking & Ellis. The large scale structure of space–time. Cambridge Univ. Press.