Quantum extremal surface and black hole evaporation inquiry

Question

Recently there has been progress made in the black hole information paradox by using the tools of AdS-CFT correspondance. Specifically, the Page curve for an evaporating black hole has been successfully reproduced with the use of the so-called quantum extremal surfaces. The entaglement entropy of the black hole is given by:

$$S_{gen}(X)=\frac{A(X)}{4}+S_{quantum fields}(\Sigma_{X}) $$

where $X$ is the quantum extremal surface and $\Sigma_{X}$ is the volume bounded by $\Sigma$ and a cutoff surface far from the black hole. Initially, the QES is at the singularity so there is no area contribution. Thus the entaglement entropy grows as radiation is emitted and crosses the cutoff surface. At the page time, the QES lies just behind the horizon. Then, it is claimed that the surface term dominates the entropy and since that is decreasing, the Page curve is reproduced.

My question is: why does the surface term dominate at the Page time? At https://arxiv.org/abs/2006.06872 the authors say that the entropy of the quantum fields is small because it does not capture many Hawking quanta. Is this because most of them have escaped the cutoff surface?

Thank you.

Answer 1

First of all you are looking at the calculation of the generalised entropy of the black hole radiation inside the cutoff region. The paper then goes to do the calculation of the region beyond the cutoff, employing the island trick, and showing that the two results match.

So let's focus on the region inside the cutoff.

Here the generalised entropy gets mainly two kinds of contributions: vanishing surface and non-vanishing surface contributions. We are interested in taking the minimum of these two and it will turn out that the vanishing surface is the minimum before the Page time, while the non-vanishing surface is the minimum after the Page time.

The vanishing surface term should be clear: it starts zero since the quantum extremal surface X can be shrunk to radius zero and the quantum state is pure, so the fine grained entropy is zero. Then it increases monotonically, due to the entanglement of the interior Hawking quanta on the region $\Sigma_X$ bounded by X and the cutoff surface, which is crossing the horizon, and those that escaped.

The non-vanishing surface term is where your doubt lies. Here the area term is non zero, since the surface X is close to the horizon, which is close to maximal, since the fine grained entropy is always smaller than the coarse grained entropy given by the usual Bekenstein area entropy. Now the region $\Sigma_X$ bounded by X and the cutoff surface is much smaller though, since the horizon is not too far from the cutoff. Therefore the entanglement term of the region $\Sigma_X$ is negligible. The area term (relative to the non-vanishing surface contribution) is therefore dominant at all times, not just at Page time, but is monotonically decreasing, since the area of the black hole is decreasing due to the evaporation.

To prove the stability of the non-vanishing surface the authors further show how if we were to move X in the ingoing null direction the entanglement term would first decrease and then increase when the entanglement with outgoing quanta outside the black hole region kicks in.