De Sitter space

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In mathematics and physics, n-dimensional de Sitter space, denoted <math>dS_n</math>, is the maximally symmetric, simply-connected, Lorentzian manifold with constant positive curvature. It may be regarded as the Lorentzian analog of an n-sphere (with its canonical Riemannian metric).

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

De Sitter space is named for Willem de Sitter.

Contents 1 Definition 2 Properties 3 Static coordinates 4 See also 5 References

Definition

De Sitter space is most easily defined as a submanifold of Minkowski space in one higher dimension. Take Minkowski space R^1,n with the standard metric:

<math>ds^2 = -dx_0^2 + \sum_{i=1}^n dx_i^2.</math>

De sitter space is the submanifold described by the hyperboloid

<math>-x_0^2 + \sum_{i=1}^n x_i^2 = \alpha^2</math>

where <math>\alpha</math> 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 <math>\alpha^2</math> with <math>-\alpha^2</math> 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.)

Topologically, de Sitter space is R × Sⁿ⁻¹ (we shall assume that n ≥ 3 so that de Sitter space is simply-connected). Given the standard embedding of the (n−1)-sphere in Rⁿ with coordinates y_i one can introduce a new coordinate t so that

<math>x_0 = \alpha\sinh(t/a)\,</math>

<math>x_i = \alpha\cosh(t/a)\,y_i\,</math>

The metric in these coordinates (t plus some set of coordinates on Sⁿ⁻¹) is given by

<math>ds^2 = -dt^2 + \alpha^2\cosh^2(t/\alpha)\,d\Omega_{n-1}^2</math>

where <math>d\Omega_{n-1}^2</math> is the standard round metric on the (n−1)-sphere.

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

<math>R_{\rho\sigma\mu\nu} = {1\over \alpha^2}(g_{\rho\mu}g_{\sigma\nu} - g_{\rho\nu}g_{\sigma\mu})</math>

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

<math>R_{\mu\nu} = \frac{n-1}{\alpha^2}g_{\mu\nu}</math>

This means de Sitter space is a vacuum solution of Einstein's equation with cosmological constant given by

<math>\Lambda = \frac{(n-1)(n-2)}{2\alpha^2}.</math>

The scalar curvature of de Sitter space is given by

<math>R = \frac{n(n-1)}{\alpha^2} = \frac{2n}{n-2}\Lambda.</math>

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

Static coordinates

We can introduce static coordinates <math>(t, r, \ldots)</math> for de Sitter as follows:

<math>x_0 = \sqrt{\alpha^2-r^2}\sinh(t/\alpha)</math>

<math>x_1 = \sqrt{\alpha^2-r^2}\cosh(t/\alpha)</math>

<math>x_i = r z_i \qquad\qquad\qquad\qquad\qquad 2\le i\le n.</math>

where <math>z_i</math> gives the standard embedding the (n−2)-sphere in Rⁿ⁻¹. In these coordinates the de Sitter metric takes the form:

<math>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.</math>

Note that there is a cosmological horizon at <math>r = \alpha</math>.

See also

• Anti de Sitter space
• De Sitter universe

References

Nomizu, K. The Lorentz-Poincaré metric on the upper half-space and its extension. Hokkaido Math. J. 11 (1982), no. 3, 253--261.