de Sitter space

In mathematics and physics, n-dimensional de Sitter space, denoted $dS_{n}$, is the Lorentzian analog of an n-sphere (with its canonical Riemannian metric). It is a maximally symmetric, Lorentzian manifold with constant positive curvature, and is simply-connected for n at least 3.

In the language of general relativity, de Sitter space is the maximally symmetric, vacuum solution of Einstein's field equation with a positive (repulsive) cosmological constant $\Lambda$. When n = 4, it is also a cosmological model for the physical universe; see de Sitter universe.

De Sitter space was discovered by Willem de Sitter, and independently by Tullio Levi-Civita (1917).

More recently it has been considered as the setting for special relativity rather than using Minkowski space and such a formulation is called de Sitter relativity.

Definition

De Sitter space can be defined as a submanifold of Minkowski space in 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

$-x_{0}^{2}+\sum _{i=1}^{n}x_{i}^{2}=\alpha ^{2}$

where $\alpha$ is some positive constant with dimensions of length. The metric on de Sitter space is the metric induced from the ambient Minkowski metric. One can check that 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 that if n ≥ 3 then de Sitter space is simply-connected). Given the standard embedding of the unit (n−1)-sphere in Rⁿ with coordinates y_i one can introduce a new coordinate t so that

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

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

Plugging in the subscripted x's into the induced 4D metric, embedding the de Sitter space in the five-dimensional Minkowski space R^1,4, and being careful to use the Leibniz rule in differentials in $\sum _{i=1}^{n}dx_{i}^{2}$, we find resulting cross terms there vanish on the sphere and one of the remaining squares of sum hypertrig collapse with $-dx_{0}^{2}$ to produce $-dt^{2}$, so the metric in these coordinates (t plus some set of coordinates on Sⁿ⁻¹) is given by

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

where $d\Omega _{n-1}^{2}$ is the standard round metric on the (n−1)-sphere, as concurs reference 3.

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 vectors and is maximally symmetric. Every maximally symmetric space has constant curvature. The Riemann curvature tensor of de Sitter is given by

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

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 $r=\alpha$.

See also

• Anti de Sitter space
• de Sitter universe
• Hyperboloid

References

• Qingming Cheng (2001) [1994], "De Sitter space", Encyclopedia of Mathematics, EMS Press
• 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
• Nomizu, K. (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, 50: 217--228
• Susskind, L.; Lindesay, J. (2005), An Introduction to Black Holes, Information and the String Theory Revolution:The Holographic Universe, p. 119(11.5.25)
• Levi-Civita, Tullio (1917), "Realtá fisica di alconi spazî normali del Bianchi", Rendiconti, Reale Accademia Dei Lincei, 26: 519–31