DeSitter

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Four-dimensional deSitter spacetime satisfies the Einstein equations in vacuum with a positive cosmological constant: G_{\mu\nu} + \Lambda g_{\mu\nu} = 0. deSitter spacetime is the surface

-(T)^2 + (X)^2 + (Y)^2 + (Z)^2 + (W)^2 = \ell^2

where \Lambda \equiv 3/\ell^2, in the five-dimensional flat spacetime

ds^2 = - (dT)^2 + (dX)^2 + (dY)^2 + (dZ)^2 + (dW)^2

The surface can be covered with coordinates t, \chi, \theta and \phi, defined by

\begin{array}{rcl} T & = & \ell \sinh(t/\ell) \\ X & = & \ell \cosh(t/\ell) \cos\chi \\ Y & = & \ell \cosh(t/\ell) \sin\chi \cos\theta \\ Z & = & \ell \cosh(t/\ell) \sin\chi \sin\theta \cos\phi\\ W & = & \ell \cosh(t/\ell) \sin\chi \sin\theta \sin\phi \end{array}

In these coordinates the deSitter metric is

ds^2 = - dt^2 + \ell^2 \cosh^2(t/\ell)\, d\chi^2 + \ell^2 \cosh^2(t/\ell)\, \sin^2\chi\, (d\theta^2 + \sin^2\theta \, d\phi^2)

[edit] Static coordinates

Let

\begin{array}{rcl} x_0 & = & \sqrt{\ell^2 - r^2} \sinh(t/\ell) \\ x_1 & = & \sqrt{\ell^2 - r^2} \cosh(t/\ell) \\ x_2 & = & r \, \sin\theta\, \sin\phi \\ x_3 & = & r\, \sin\theta \, \cos\phi \\ x_4 & = & r \,\cos\theta \end{array}

The deSitter metric is

ds^2 = -(1 - r^2/\ell^2) dt^2 + \frac{1}{(1 - r^2/\ell^2)} dr^2 + r^2 (d\theta^2 + \sin^2\theta \, d\phi^2 )

These coordinates cover only the strip -\ell \leq x_4 \leq \ell of the full manifold. Let r = \ell \sin\chi and the metric becomes

ds^2 = -\cos^2\chi\, dt^2 + \ell^2 dr^2 + \ell^2\sin^2\chi (d\theta^2 + \sin^2\theta \, d\phi^2 )

[edit] Conformally flat coordinates

Let

\begin{array}{rcl} x_0 & = & \ell \sinh(t/\ell) + e^{t/\ell} X_i X_i /(2\ell)\\ x_1 & = & \ell \cosh(t/\ell) - e^{t/\ell} X_i X_i /(2\ell)\\ x_2 & = & e^{t/\ell} X_1 \\ x_3 & = & e^{t/\ell} X_2 \\ x_4 & = & e^{t/\ell} X_3 \end{array}

where X^i X^i \equiv (X_1)^2 + (X_2)^2 + (X_3)^2. The metric is

ds^2 = -dt^2 + e^{2t/\ell} (dX_1^2 + dX_2^2 + dX_3^2)

Only the region x_0 + x_1 \geq 0 is covered by these coordinates.

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