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You've got a Freund--Rubin in Me: The Aretakis Instability of Extremal Black Branes

Calvin Y. -R. Chen, Áron D. Kovács

TL;DR

This work extends the Aretakis instability to non-dilatonic extremal black branes by tying the late-time horizon behavior of perturbations to the near-horizon AdS$_{p+2}$ scaling dimensions. Using a detailed KK analysis on Freund–Rubin spaces, the authors show that extremal black branes exhibit an Aretakis-type instability driven purely by perturbations of the background fields, with the instability strength controlled by the KK spectrum and the AdS$_{p+2}$ masses. They demonstrate that the scaling dimensions also dictate the smoothness of stationary deformations to the near-horizon geometry, revealing regimes where curvature invariants can become singular and highlighting UV-sensitive cases when these dimensions are integers. Overall, the paper provides a KK-spectrum–driven framework to assess horizon stability and deformation behavior in extremal brane geometries, with implications for holography and higher-dimensional EFT corrections.

Abstract

We investigate how the Aretakis instability affects non-dilatonic extremal black $p$-branes by focusing on their near-horizon geometry. Crucially, the strength of the instability, \textit{i.e.} the number of transverse derivatives needed to see non-decay/blow-up of fields on the horizon at late null time, is given by the scaling dimensions with respect to the near-horizon $\mathrm{AdS}_{p+2}$-factor. This renders the problem of determining the severity of the Aretakis instability equivalent to computing the Kaluza--Klein spectrum of fields on Freund--Rubin spaces. We use this to argue that non-dilatonic extremal black branes suffer from the Aretakis instability even in the absence of additional fields -- we find that this is weaker than for extremal black holes. We also argue that the scaling dimensions determine the smoothness of stationary deformations to the original black brane background -- here, our findings indicate that generically more modes can lead to worse curvature singularities compared to extremal black holes.

You've got a Freund--Rubin in Me: The Aretakis Instability of Extremal Black Branes

TL;DR

This work extends the Aretakis instability to non-dilatonic extremal black branes by tying the late-time horizon behavior of perturbations to the near-horizon AdS scaling dimensions. Using a detailed KK analysis on Freund–Rubin spaces, the authors show that extremal black branes exhibit an Aretakis-type instability driven purely by perturbations of the background fields, with the instability strength controlled by the KK spectrum and the AdS masses. They demonstrate that the scaling dimensions also dictate the smoothness of stationary deformations to the near-horizon geometry, revealing regimes where curvature invariants can become singular and highlighting UV-sensitive cases when these dimensions are integers. Overall, the paper provides a KK-spectrum–driven framework to assess horizon stability and deformation behavior in extremal brane geometries, with implications for holography and higher-dimensional EFT corrections.

Abstract

We investigate how the Aretakis instability affects non-dilatonic extremal black -branes by focusing on their near-horizon geometry. Crucially, the strength of the instability, \textit{i.e.} the number of transverse derivatives needed to see non-decay/blow-up of fields on the horizon at late null time, is given by the scaling dimensions with respect to the near-horizon -factor. This renders the problem of determining the severity of the Aretakis instability equivalent to computing the Kaluza--Klein spectrum of fields on Freund--Rubin spaces. We use this to argue that non-dilatonic extremal black branes suffer from the Aretakis instability even in the absence of additional fields -- we find that this is weaker than for extremal black holes. We also argue that the scaling dimensions determine the smoothness of stationary deformations to the original black brane background -- here, our findings indicate that generically more modes can lead to worse curvature singularities compared to extremal black holes.

Paper Structure

This paper contains 27 sections, 168 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Extremal Black Branes and Freund--Rubin Compactifications
  3. Black Branes
  4. Freund--Rubin Compactifications
  5. Coordinate Charts
  6. The Aretakis Instability
  7. Extremal Reissner-Nordström Black Holes
  8. Symmetry Argument
  9. Extremal Black Branes
  10. Aretakis Scaling
  11. Perturbing Freund--Rubin
  12. Perturbation Equations
  13. Tensor Modes
  14. Vector Modes
  15. Scalar Modes
  16. ...and 12 more sections

Figures (2)

  • Figure 1: Global coordinates on AdS$_{p+2}$. ${\cal H}^\pm$ denotes the future/past Poincaré horizons, $\gamma=\pi$ corresponds to right timelike infinity. The region surrounded by the blue lines represents the exterior of (the near-horizon region of) the black brane, where we solve the wave equation.
  • Figure 2: Plots for effective masses of different modes as a function of $\ell$ in units where $L=1$. On the left, the spectrum of all (tensor, vector, and scalar) modes is shown for $(D,p) = (11,2)$. It is clear that the inequalities \ref{['eq: mass hierarchy']} hold. On the left, we have isolated gravitational scalar and vector modes for $(D,p) = (26,3)$ to illustrate that there is no hierarchy between gravitational scalar and vector modes which holds for all $\ell$.