Anti de Sitter Space

de Sitter space is important because it may describe the early and late time behavior of the real universe. An ti de Sitter space is important for an entirely different reason. It is the background in which the holographic principle is best understood. But for our present purpose all we need are the properties of its PB diagram.

Fig. 1.1. Static surface of large area in AdS space.

AdS space is the vacuum of theories with a negative cosmological constant. The space-time in appropriate coordinates is static but unlike de Sitter space the static coordinates cover the entire space. The vacuum of AdS is a genuine zero temperature state.

The PB diagram consists of an infinite strip bounded on the left by the spatial origin and on the right by a boundary. The PB diagram consists of a single region in which both negative expansion light sheets point toward the origin. Let us consider a static surface of large area A far from the spatial origin. The surface is denoted by the dotted vertical line L in Figure 1.1. We will think of L as an infrared cutoff. Consider an arbitrary point p on L. Evidently Bousso's rules bound the entropy on past and future light sheets bounded by p. Therefore the entropy on any space-like surface bounded by p and including the origin is also holographically bounded. In other words the entire region to the left of L can be foliated with space-like surfaces such that the maximum entropy on each surface is A/4G.

AdS space is an example of a special class of geometries which have time-like Killing vectors and which can be foliated by space-like surfaces that satisfy the Holographic bound. These two properties imply a very far reaching conclusion. All physics taking place in such backgrounds (in the interior of the infrared cutoff L) must be described in terms of a Hamiltonian that acts in a Hilbert space of dimensionality

N_states = exp(A/4G)        (1.1)

The holographic description of AdS space is the subject of the next chapter