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).

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.

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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).

[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] Flat slicing

Let

x0 = αsinh(t / α) + r2et / α / 2α,
x1 = αcosh(t / α) − r2et / α / 2α,
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,yi) coordinates metric reads:

ds2 = − dt2 + e2t / αdy2

where dy^2=\sum_i dy_i^2 is the flat metric on yi's.

[edit] Open slicing

Let

x0 = αsinh(t / α)coshξ,
x1 = αcosh(t / α),
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 Sn − 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.

[edit] Closed slicing

Let

x0 = αsinh(t / α),
x_i = \alpha \cosh(t/\alpha) z_i, \qquad 1 \leq i \leq n

where zis describe a Sn − 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(η / 2) = tanh(t / 2α) (or equivalently cosη = 1 / cosh(t / α)) 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.

[edit] dS slicing

Let

x0 = αsin(χ / α)sinh(t / α)coshξ,
x1 = αcos(χ / α),
x2 = αsin(χ / α)cosh(t / α),
x_i = \alpha z_i \sin(\chi/\alpha) \sinh(t/\alpha) \sinh\xi, \qquad 3 \leq i \leq n

where zis describe a Sn − 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 α 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 x0 and x2 because they change their timelike/spacelike nature.

[edit] See also

[edit] References

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