de Sitter space

In mathematics and physics, a de Sitter space is the analog in Minkowski space, or spacetime, of a sphere in ordinary, Euclidean space. The n-dimensional de Sitter space, denoted dS_n, is the Lorentzian manifold analog of an n-sphere (with its canonical Riemannian metric); it is maximally symmetric, has constant positive curvature, and is simply connected for n at least 3. 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 in the 1920s in Leiden closely together on the spacetime structure of our universe.

In the language of general relativity, 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). When n = 4 (3 space dimensions plus time), it is a cosmological model for the physical universe; see de Sitter universe.

De Sitter space was also discovered, independently, and about the same time, by Tullio Levi-Civita.

More recently it has been considered as the setting for special relativity rather than using Minkowski space, since a group contraction reduces the isometry group of de Sitter space to the Poincaré group, allowing a unification of the spacetime translation subgroup and Lorentz transformation subgroup of the Poincaré group into a simple group rather than a semi-simple group. This alternate formulation of special relativity is called de Sitter relativity.

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 dimensions 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.)

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

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

De Sitter space is an Einstein manifold since the Ricci tensor is proportional to the metric:

$R_{{\mu \nu }}={\frac {n-1}{\alpha ^{2}}}g_{{\mu \nu }}$

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

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

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

Static coordinates

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

$x_{0}={\sqrt {\alpha ^{2}-r^{2}}}\sinh(t/\alpha )$

$x_{1}={\sqrt {\alpha ^{2}-r^{2}}}\cosh(t/\alpha )$

$x_{i}=rz_{i}\qquad \qquad \qquad \qquad \qquad 2\leq i\leq n.$

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

$x_{0}=\alpha \sinh(t/\alpha )+r^{2}e^{{t/\alpha }}/2\alpha ,$

$x_{1}=\alpha \cosh(t/\alpha )-r^{2}e^{{t/\alpha }}/2\alpha ,$

$x_{i}=e^{{t/\alpha }}y_{i},\qquad 2\leq i\leq n$

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

$ds^{{2}}=-dt^{{2}}+e^{{2t/\alpha }}dy^{{2}}$

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

Open slicing

Let

$x_{0}=\alpha \sinh(t/\alpha )\cosh \xi ,$

$x_{1}=\alpha \cosh(t/\alpha ),$

$x_{i}=\alpha z_{i}\sinh(t/\alpha )\sinh \xi ,\qquad 2\leq i\leq n$

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

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

where

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

is the metric of a Euclidean hyperbolic space.

Closed slicing

Let

$x_{0}=\alpha \sinh(t/\alpha ),$

$x_{i}=\alpha \cosh(t/\alpha )z_{i},\qquad 1\leq i\leq n$

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

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

Changing the time variable to the conformal time via $\tan(\eta /2)=\tanh(t/2\alpha )$ we obtain a metric conformally equivalent to Einstein static universe:

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

This serves to find the Penrose diagram of de Sitter space.

dS slicing

Let

$x_{0}=\alpha \sin(\chi /\alpha )\sinh(t/\alpha )\cosh \xi ,$

$x_{1}=\alpha \cos(\chi /\alpha ),$

$x_{2}=\alpha \sin(\chi /\alpha )\cosh(t/\alpha ),$

$x_{i}=\alpha z_{i}\sin(\chi /\alpha )\sinh(t/\alpha )\sinh \xi ,\qquad 3\leq i\leq n$

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

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

where

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

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 $(t,\xi ,\theta ,\phi _{1},\phi _{2},\cdots ,\phi _{{n-3}})\to (i\chi ,\xi ,it,\theta ,\phi _{1},\cdots ,\phi _{{n-4}})$ 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

• Qingming Cheng (2001), "De Sitter space", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• de Sitter, W. (1917), "On the relativity of inertia: Remarks concerning Einstein's latest hypothesis", Proc. Kon. Ned. Acad. Wet., 19: 1217–1225
• de Sitter, W. (1917), "On the curvature of space", 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
• Nomizu, Katsumi (1982), "The Lorentz–Poincaré metric on the upper half-space and its extension", Hokkaido Mathematical Journal, 11 (3): 253–261
• Coxeter, H. S. M. (1943), "A geometrical background for de Sitter's world", American Mathematical Monthly, Mathematical Association of America, 50 (4): 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)