[Report] Bousso - A Covariant Entropy Conjecture

0. Introduction

 The paper shows developed version of energy-entropy bound, and shows two example. First example is Friedmann-Robertson-Walker (FRW) cosmologies and Oppenheimer–Snyder model. I only focus second one.

In movie, we can see they research black hole collapse before nuclear reaction discovered. This model proved black hole can exists in real.

1. Bekenstein's bound

In quantum statistics. If system is described by Hamiltoniam $\hat{H}$, states is described by density operator $\hat{\rho}$: $$\operatorname{Tr}\hat{\rho}=1.$$ The mean energy is $$\bar{E}=\operatorname{Tr}(\hat{\rho}\hat{H})$$ and the entropy $$S=-\operatorname{Tr}(\hat{\rho}\ln\hat{\rho}).$$ Then we can find which $\hat{\rho}$ makes $S/\bar{E}$ maximal by $$\delta [-\operatorname{Tr}(\hat{\rho}\ln\hat{\rho})/\operatorname{Tr}(\hat{\rho}\hat{H})-\lambda\operatorname{Tr}(\hat{\rho})]=0$$

There exists a universal bound on the entropy $S$ of any thermodynamic system of total energy $M$: $$S\le 2\pi RM.$$ $R$ is defined as teh circumferential radius, $$R=\sqrt{\frac{A}{4\pi}},$$ where $A$ is the area of the smallest sphere circumscribing the system.

For a system contained in a spherical volume, gravitational stability requires that $M\le R/2$. Thus above equation implies $$S\le \frac{A}{4}.$$

2. Self-Gravity

Bekenstein specified conditions for the validity of these bounds. The system must be of constant, finite size and must have limited self-gravity. Self gravity is gravity from itself affect itself. In prove of Bekenstein's bound, any gravity is not considered. So the bound is not clear for system which gravity is dominant. We first generalize entropy bound and compare for self-gravity system.

3. The Conjecture

To make 'covariant' entropty, bound, we have to choose entropy/area and throw energy. 

Generalized version of Bekenstein’s entropy/area bound: Let $A$ be the area of any closed two-dimensional surface $B$, and let $S$ be the entropy on the spatial region $V$ enclosed by $B$. Then $S \le A/4$.

We will use (inside directing) null hypersurface bounded by $B$ to evaluate entropy $S$. 

Covariant Entropy Conjecture. Let $M$ be a four-dimensional space-time on which Einstein's equation is satisfied with the dominant energy condition holding for matter. Let $A$ be the are of a connected two-dimensional spatial surface $B$ contained in $M$. Let $L$ be a hypersurface bounded by $B$ and generated by one of the four null congruences orthogonal to $B$. Let $S$ be the total entropy contained on $L$. If the expansion of the congruence is non-positive (measured in the direction away from $B$) at every point on $L$, then $S\le A/4$.

4. Gravitational Collapse

Let's consider maximal entropy situation in Bekenstein's bound, which is black hole. Then we throw mass shell to black hole. And we assume every dynamics is spherically symmetric. 

The shell fall to black hole after $B$. At $B$, We have to consider three hypersurface: $L_1$, $L_2$ and $L$. (Another one is outgoing.) Let event horizon's radious $r_0$

For $L_1$, this counts the entropy $S_{in}$ that went into the black hole when it born. Entropy is always increasing in isolated system, $$A/4=S_{bh}\ge S_{in}$$

For $L_2$, this is case of FRW cosmology since in Oppenheimer-Snyder model, inside of black hold is FRW model.

For $L$, after matter acrossing, mass of black hole increase. Then radious of event horizon increase. Then $L$ collapses to $r=0$ withing a finite affine time. Then we throw shell to increase $L$ entropy. We want to make shell as wide as possible to store large entropy. 

Unfortunatly, there is maximum wide. If infinitely thin shell of mass $M$ falling to black hole at $r_0$, outside metric is Schwarzschild black hole with mass $\tilde{M}=M+\frac{r_0}{2}$$ When shell across $L$, null generators of $L$ will be moving in a Schwarzschild interior of mass $\tilde{M}$. Then proper time $$\Delta \tau_{dead}=r_0+2\tilde{M}\ln\left(1-\frac{r_0}{2\tilde{M}}\right)\approx \frac{r_0^2}{4\tilde{M}}$$ Let wilde of shell as $\omega$. When half of shell enter $L$, above equation satisfy. Then $$\omgea_{max}\approx r^2_0/2\tilde{M}$$

Final step is calculate maximum entropy of the shell. We make shell at radius $R$ far from black hole. The shell can be split into a large number $n=R^2/\omega^2$ of cubic boxes of volume  $\omgea^3$ and mass $M/n$. By Bekenstein's bound, maximum entropy of each box is $$S_{box}=2\pi\omega\frac{M}{n}$$ We can simply add entropy just think there are independent, but close. $$S_{shell}=2\pi\omega M=\pi r_0^2=\frac{A}{4}$$ 

For last, we can compare with original Bekenstein's bound. Bekenstein bound simply check energy and entropy inside $r_0$, so this can't recognize change of event horizon. Then we can't to further.

5. Percolation

Spherical symmetry can broke easily. If symmetry  was broken, light-ray converges outside of center, which is singularity. We can think this make death time shorter, but it isn't. In collapsing situation, this non-symmetric or high-entropy geometry make path messy. This call path "percolation" to other path. It take more time to die, then bound is larger. We can guess entropy bound also holds.

6. Conclusion

We can generalize Bekenstein's entropy/area bound in covariant way. Checking in collapsing situation is very nontrivial that entropy bound satisfy in dynamical situation. However, in non-spherically symmetric situation, we can only say roughly.

7. Reference

Bousso