Local conditions for the generalized covariant entropy bound

A set of sufficient conditions for the generalized covariant entropy bound given by Strominger and Thompson is as follows: Suppose that the entropy of matter can be described by an entropy current $s^a$. Let $k^a$ be any null vector along $L$ and $s\equiv -k^a{s}_a$. Then the generalized bound can be derived from the following conditions: (i) $s^{\prime }\le 2\pi T_{ab}{k}^a{k}^b$, where $s^{\prime }=k^a{\nabla }_{a}s$ and $T_{ab}$ is the stress-energy tensor; (ii) on the initial 2-surface $B$, $s(0)\le -\frac{1}{4}\theta (0)$, where $\theta$ is the expansion of $k^a$. We prove that condition (ii) alone can be used to divide a spacetime into two regions: The generalized entropy bound holds for all light sheets residing in the region where $s<-\frac{1}{4}\theta$ and fails for those in the region where $s>-\frac{1}{4}\theta$. We check the validity of these conditions in FRW flat universe and a scalar field spacetime. Some apparent violations of the entropy bounds in the two spacetimes are discussed. These holographic bounds are important in the formulation of the holographic principle.

Figure 1

A spacetime diagram for the scalar field spacetime. Below the solid line $t=G(r)$, ${\partial }^a\varphi$ is timelike. Between $t=G(r)$ and the timelike singularity $r=2$, no observers associated with the scalar field can be naturally defined.

Figure 2

A spacetime diagram depicting the important regions under discussion. Condition (i) is satisfied below the thick solid line, while condition (A) is valid above the thin solid line. The dotted line represents the apparent horizon.

Figure 3

(a) A light sheet crossing the $s=-\frac{1}{4}\theta$ surface. AH represents the apparent horizon. (b) Change of ratio on the light sheet. The generalized bound is violated in the $s>-\frac{1}{4}\theta$ zone and is satisfied until the light sheet runs certain distance in the $s<-\frac{1}{4}\theta$ zone.

Figure 4

Plots of the proper length and thermal wavelength $\lambda$ for the light sheet in Fig. 3a. This figure shows that the proper length of the light sheet is significantly larger than the thermal wavelength $\lambda$.

Figure 5

A spacetime diagram showing the region where condition (A) $s<-\frac{1}{4}\theta$ holds for past-directed outgoing light sheets.

Figure 6

(a) A past-outgoing light sheet located between the apparent horizon and the past singularity. (b) Plot of $\mathrm{ratio}(\mathrm{r})$ when the light sheet in part (a) of this figure rolls down from the apparent horizon to the singularity.

Figure 7

Plots of the proper length and thermal wavelength $\lambda$ for the light -sheet in Fig. 6a. We consider the light sheet as in Fig. 6a, but we let it start near the apparent horizon and stop it at $r=43.9$ where the singularity is located. Since the comoving frame is not static, this proper length changes with time. But along the light sheet, the time $t$ is a function of $r$. So the proper length can be plotted as a function of $r$. This figures shows that the proper length of the light sheet is significantly larger than the thermal wavelength $\lambda$.