Holography, Brick Wall and a Little Hierarchy Problem

Abstract

We propose a heuristic for the brick wall in AdS/CFT: the location where a boundary mode's local bulk energy reaches a (Planckian) UV cut-off. This accomplishes two things: (a) the brick wall is framed as a breakdown criterion for bulk effective field theory, and (b) the definition is boundary-anchored rather than horizon-anchored, aligning it with holography. Near the horizon, spacetime effectively gets cut-off due to blueshift relative to the boundary, and leads to normal modes. By directly computing these new modes for the BTZ black hole, we show that they are qualitatively unchanged from conventional 't Hooftian brick wall normal modes in the relevant part of the spectrum -- successfully reproducing black hole thermodynamics and exterior smooth-horizon correlators, under similar approximations. However, unlike 't Hooft's (and our own previous) calculations, we also do an $exact$ numerical evaluation of the normal mode partition function. This allows us to identify a "little hierarchy" problem in the brick wall paradigm, irrespective of whether it is horizon-anchored or boundary-anchored: because the modes are not exactly degenerate in the $J$-direction, the coefficient of the area law is slightly subleading, unless the brick wall is slightly trans-Planckian. One way to evade the problem is to increase the number of active species. While this is certainly a possibility in string theory, we argue that a natural resolution is to take into account the degrees of freedom intrinsic to the (stretched) horizon, as suggested by the recent results in arXiv:2601.18775. We argue that this will lead to a dominant contribution from a quantum number associated to the radial direction, while retaining the successes of the $J$-degenerate toy model. We discuss the possible significance of these observations for (a) quantum chaos in black holes, and (b) the fuzzball program.

Figures (10)

• Figure 1: Spectrum from BTZ Exact Phase Equation in (a) the boundary-anchored and (b) the hard-wall cutoff case with $\widetilde{\ell} = 10^{-10}$ and $r_h/L = 1$ for $n=1,2$ and $3$.
• Figure 2: Comparison between the exact boundary-anchored spectrum (Blue) and the hardwall spectrum (yellow) with $\widetilde{\ell} = 10^{-10}$ and $r_h/L = 1$, for (a) $n=1$ and (b) $n=2$.
• Figure 3: (a) Exact spectrum (blue) vs holographic ALLS (yellow). (b) Holographic ALLS in \ref{['BTZALLS']} (blue) and Hardwall ALLS\ref{['BTZhardwall']} (yellow) with $\widetilde{\ell} = 10^{-10}$ and $r_h/L = 1$, for $n = 1$.
• Figure 4: The shaded region satisfies \ref{['omegarplot']}. The zoomed-in region is what contributes dominantly to the thermodynamics (as we argue in the text, this corresponds to $\frac{\omega L^2}{r_h} \lesssim \mathcal{O}(1)$): the pink line represents the radial region bounded by \ref{['horN']} for $\alpha=1$, and we work with $L/\ell_P=100$. Part of the goal of this plot is also to illustrate that the bulk UV cut off is effective only close to the horizon.
• Figure 5: Spectral contribution density $-\rho(\omega)\log(1-e^{-\beta_H\omega})$ for $n=1$ and $\alpha=1$.