Anti-deSitter

Four-dimensional anti-deSitter spacetime is the surface

$-R^2 = -T^2 - W^2 + X^2 + Y^2 + Z^2$

in the five-dimensional flat spacetime

$ds^2 = - dT^2 - dW^2 + dX^2 + dY^2 + dZ^2$

Anti-deSitter spacetime satisfies the Einstein equations in vacuum with a negative cosmological constant: $G_{\mu\nu} + \Lambda g_{\mu\nu} = 0$ where $\Lambda \equiv -3/R^2$.

Static Coordinates

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

$\begin{array}{rcl} T & = & R\sqrt{1 + r^2/R^2} \cos (t/R) \\ W & = & R\sqrt{1 + r^2/R^2} \sin (t/R) \\ X & = & r \cos\theta \\ Y & = & r\sin\theta \cos\phi \\ Z & = & r\sin\theta \sin\phi \end{array}$

In these coordinates the anti-deSitter metric is

$ds^2 = -\left( 1 + r^2/R^2\right) dt^2 + \left( 1 + r^2/R^2 \right)^{-1} dr^2 + r^2 d\Omega^2$

where $d\Omega^2 = d\theta^2 + \sin^2\theta \, d\phi^2$ is the metric for the unit two-sphere.

Related coordinates are defined by setting $r = R\sinh \chi$; this yields

$ds^2 = -\cosh^2\chi \, dt^2 + R^2 d\chi^2 + R^2 \sinh^2\chi \, d\Omega^2$

Synchronous Coordinates

Let

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

The anti-deSitter metric becomes

$ds^2 = -dt^2 + R^2 \sin^2 (t/R) \left[ d\chi^2 + \sinh^2\chi \, d\Omega^2 \right]$

These coordinates cover the strip $-R \leq T \leq R$.

Let $r = R\sinh\chi$. Then

$ds^2 = -dt^2 + R^2 \sin^2(t/R) \left[ \frac{dr^2}{1 - kr^2} + r^2 d\Omega^2\right]$

where $k = -1$. This is the form of a Friedman-Robertson-Walker cosmology with negative spatial curvature and scale factor $R\sin(t/R)$.

Other coordinates

Let

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

The anti-deSitter metric becomes

$ds^2 = d\xi^2 + R^2 \sinh^2 (\xi/R) \left[ -d\chi^2 + \cosh^2\chi \, d\Omega^2 \right]$

These coordinates cover only part of the manifold.
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