Friedman Robertson Walker Geometry

The holographic bound can be generalized to flat F.R.W. geometries, where it is called the Fischler-Susskind (FS) bound[5] and to more general geometries by Bousso[6]. First we will review the F.R.W. case. Consider the general case of d + 1 dimensions. The metric has the form

 (1.1)

where the index m runs over the d spatial directions. The function a(t) is assumed to grow as a power of t.

 (1.2)

Let's also make the usual simplifying cosmological assumptions of homogeneity. In particular we assume that the spatial entropy density (per unit d volume) is homogeneous. Later, we will relax these assumptions.

At time t we consider a spherical region Γ of volume V and area A. The boundary (d − 1)-sphere, ∂Γ, will play the same role as the screen in the previous discussion. The light sheet is now defined by the backward light cone formed by light rays that propagate from ∂Γ into the past (See Figure 1.1).

Fig. 1.1. Holographic surface for calculating entropy bound with a spherical surface as the screen.

As in the previous case the holographic bound applies to the entropy passing through the light sheet. This bound states that the total entropy passing through the light sheet does not exceed A/4G. The key to a proof is again the focusing theorem. We observe that at the screen the area of the outgoing bundle of light rays is increasing as we go to later times. In other words the light sheet has positive expansion into the future and negative expansion into the past. The focusing theorem again tells us that if we map the entropy of black holes passing through the light sheet to the screen, the resulting density satisfies the holographic bound. It is believed that the bound is very general.

It is now easy to see why we concentrate on light sheets instead of spacelike surfaces. Obviously if the spatial entropy density is uniform and we choose Γ big enough, the entropy will exceed the area. However if Γ is larger than the particle horizon at time t the light sheet is not a cone, but rather a truncated cone as in Figure 1.2, which is cut off by the big bang at t = 0. Thus a portion of the entropy present at time t never passed through the light sheet. If we only count that portion of the entropy which did pass through the light sheet, it will scale like the area A. We will return to the question of space-like bounds after discussing Bousso's generalization[6] of the FS bound.

Fig. 1.2. Region of space causally connected to particle horizon.

Next, using the F.R.W. geometry, we will determine if the entropy per horizon-area contained within the particle horizon is increasing or decreasing. Let R_Horizon represent the coordinate size of the particle horizon (Figure 1.3), and d represent the number of spatial dimensions. Let σ be the entropy volume density, so that

This means that the proper size of the horizon is given by a(t)R_Horizon. We want to check whether

 (1.3)

An outgoing light ray (null geodesic) which would generate the particle horizon satisfies dt = a(t)dx, which gives the form for the time dependence

Fig. 1.3. Particle Horizon.

of the size of the particle horizon:

 (1.4)

If we assume the form a(t) = a_o t^p then the particle horizon evolves according to the formula

 (1.5)

Therefore for the entropy bound to continue to be valid, the time dependence must satisfy t(1−p)^d < t^d−1.This bounds the expansion rate coefficient to satisfy

 (1.6)

One sees that if the expansion rate is too slow, then the coordinate volume will grow faster than the area, and the entropy bound will eventually be contradicted.

Suppose that the matter in the F.R.W. cosmology satisfies the equation of state

      P = w u     (1.7)

where P is the pressure and u is the energy density and w is a constant. Given w and the scale factor for the expansion p, one can use the Einstein field equation to calculate a relationship between them:

 (1.8)

We see that the number of spatial dimensions cancels and that the entropy bound is satisfied so long as

      w ≤ 1.         (1.9)

This is an interesting result. Recall that the speed of sound within a medium is given by

Therefore, in the future, the bound will always be satisfied, since the speed of sound is always less than the speed of light. The relation satisfied by the scale factor v²_s = w ≤ 1 is just the usual causality requirement. As one moves forward in time, the entropy bound then becomes more satisfied, not less.

Next, go back in time using the black body radiation background as the dominant entropy. Using a decoupling time t_decoupling ∼ 105 years (when the background radiation fell out of equilibrium with the matter) and extrapolating back using the previously calculated entropy relative to the bound, one gets

(1.10)

The entropy bound S = A/4G is reached when

 (1.11)

This time is comparable to the Planck time (by coincidence??). Therefore the entropy bound is not exceeded after the Planck time.