AdS Black Holes with a Bouncing Interior

In this paper, the authors find analytic solutions for spatially homogeneous black hole interiors in AdS gravity coupled to a scalar field. The scalar field oscillates in its potential, producing an infinite sequence of bounces between Kasner-like solutions.

This is an interesting setup because it provides an analytic playground for the study of classical dynamics near a singularity, in a setting that could plausibly have a holographic description. It is well written and thoroughly checked with numerics vs analytics. I recommend this article for publication.

The authors construct a concrete, solvable model of classical interior black hole dynamics. Their model builds on a long history of work in this area. While it has been known for a long time that some classical black hole interiors with exponential potentials consist of an infinite series of Kasner epochs as we flow towards the singularity, the authors construct a simple class of models in which this type of evolution can be analyzed analytically for the case of superexponential potentials. This work is certainly novel and useful.

However, I have some points which I would like the authors to address before I recommend publication. I have also included any observed typos in my report.

1. Below equation (3), note that this potential (or, being more specific, setting $m^2 = -2$) does not necessarily fix $\Delta = 2$. One may also consider the quantization with $\Delta = 1$ for 3 boundary dimensions, which corresponds to a different boundary condition for the scalar (cf. Klebanov-Witten arXiv:9905104). This ultimately does not adversely impact the analysis since the authors set the boundary condition in (4), but they should be clear that they are actually choosing the $\Delta = 2$ quantization explicitly.

2. Equation (16) is unable to pass my own consistency check. Specifically, I start with equation (12), switching from $H$ ot $V$ by substituting $H \to V'/V$, $H' \to (V/V')'$ (inverting (8)) and plugging-in the ansatz $v(\phi) = v_\infty + \delta v(\phi)$. I then expand around $\delta v(\phi) \approx 0$ and truncate at leading order, thereby leaving me with a differential equation relating $\delta v'(\phi)$ and $\delta v''(\phi)$. I would expect (16) to solve the resulting equation, but I find that this is not the case. This means that either I am mistaken somewhere in my approach or (16) is not correct as written. As such, I would appreciate it if the authors could clarify their derivation of (16).

3. I definitely agree that Figure 3 is sufficient to show that there is both a narrow bounce in $V$ and a more extended bounce in $H$. However, I do not think that the current presentation is the simplest way to demonstrate the characterization of the extended bounce regime as that for which $|H| \gg |v|$, since this plot only covers a small $\Delta\rho = \pm 1$ window around the bounce. I think the simplest way to satisfy my complaint here would be to zoom Figure 3 out to a larger domain for $\rho - \rho_H$ which encapsulates more of the pre- and post-bounce behavior. Furthermore, so that the point about $V$ having a very narrow spike is not lost, I believe the authors could do something like in Figure 2 where they embed the plots of $H$ and $V$ over a narrower domain within a plot of the same functions over a wider domain.

4. In the paragraph below (38), the authors arrive at the bounces becoming narrower and the Kasner velocity changing velocity at each bounce in the late-time regime, with the justification being that $|k_n| |\Delta v_n| \to \infty$. However, why this is enough is still unclear to me. I would have thought that the signature of the former phenomenon would be a shortening in the width of the extended region, whereas the signature of the latter would be that $\Delta v_n$ changes sign for each $n$. How can I read off this behavior from the stated limiting behavior of $|k_n| |\Delta v_n|$?

TYPOS:
- pg. 3: Above (1), ``...the Lagrangian density" $\to$ ``...the Lagrangian density is"
- pg. 12: Above (32), the authors write $V''' V' \approx (V')^2$ for large $\phi$ and superexponential potentials. This should read $V''' V' \approx (V'')^2$.