Anti-deSitter

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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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