De Sitter space

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In mathematics and physics, n-dimensional de Sitter space, denoted dSn, 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 Λ. 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).

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[edit] Definition

De Sitter space can be defined as a submanifold of Minkowski space in one higher dimension. Take Minkowski space R1,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 α 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 α2 with − α2 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 yi 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 R1,4, 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 − dt2, 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.

[edit] 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.

[edit] 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 = r z_i \qquad\qquad\qquad\qquad\qquad 2\le i\le n.

where zi 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 = α.

[edit] See also

[edit] References

  • Qingming Cheng (2001), “De Sitter space”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 
  • de Sitter, W. (1917) On the relativity of inertia: Remarks concerning Einstein's latest hypothesis. Proc. Kon. Ned. Acad. Wet. 19, 1217-1225, and On the curvature of space, Proc. Kon. Ned. Acad. Wet. 20, 229-243
  • Nomizu, K. The Lorentz-Poincaré metric on the upper half-space and its extension. Hokkaido Mathematical Journal 11 (1982), no. 3, 253--261.
  • Coxeter, H. S. M. A geometrical background for de Sitter's world. American Mathematical Monthly 50, (1943), 217--228.
  • Susskind, L. and Lindesay, J. An Introduction to Black Holes, Information and the String Theory Revolution:The Holographic Universe, (2005), p119(11.5.25).
  • Levi-Civitá, T. Realtá fisica di alconi spazî normali del Bianchi. Rendiconti, Reale Accademia Dei Lincei 26, (1917), 519-31.