# Anti-deSitter

### From GRwiki

Four-dimensional anti-deSitter spacetime is the surface

in the five-dimensional flat spacetime

Anti-deSitter spacetime satisfies the Einstein equations in vacuum with a negative cosmological constant: where .

## Static Coordinates

The surface can be covered with coordinates , , , and defined by

In these coordinates the anti-deSitter metric is

where is the metric for the unit two-sphere.

Related coordinates are defined by setting ; this yields

## Synchronous Coordinates

Let

The anti-deSitter metric becomes

These coordinates cover the strip .

Let . Then

where . This is the form of a Friedman-Robertson-Walker cosmology with negative spatial curvature and scale factor .

## Other coordinates

Let

The anti-deSitter metric becomes

These coordinates cover only part of the manifold.