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AdS Black Holes with a Bouncing Interior

by Sean A. Hartnoll, Navonil Neogi

This is not the latest submitted version.

Submission summary

Authors (as Contributors): Sean Hartnoll
Submission information
Arxiv Link: (pdf)
Date submitted: 2022-10-05 13:12
Submitted by: Hartnoll, Sean
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
Approach: Theoretical


We construct planar black hole solutions of AdS gravity minimally coupled to a scalar field with an even, super-exponential potential. We show that the evolution of the black hole interior exhibits an infinite sequence of Kasner epochs, as the scalar field rolls back and forth in its potential. We obtain an analytic expression for the `bounces' between each Kasner epoch and also give an explicit formula for the times and strengths of the bounces at late interior times, thereby fully characterizing the interior evolution. In this way we show that the interior geometry approaches the Schwarzschild singularity at late times, even as the scalar field is driven higher up its potential with each bounce.

Current status:
Has been resubmitted

Submission & Refereeing History

Reports on this Submission

Anonymous Report 2 on 2022-11-2 (Invited Report)


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.

  • validity: top
  • significance: high
  • originality: high
  • clarity: high
  • formatting: perfect
  • grammar: perfect

Anonymous Report 1 on 2022-10-27 (Invited Report)


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.

Requested changes

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 $|k_n| |\Delta v_n|$?

- 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 $V''' V' \approx (V'')^2$.

  • validity: high
  • significance: high
  • originality: top
  • clarity: good
  • formatting: perfect
  • grammar: perfect

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