Overcounting of interior excitations: A resolution to the bags of gold paradox in AdS

TL;DR

The bags-of-gold paradox asks how a black hole interior can host exponentially many semiclassical excitations beyond the coarse-grained Bekenstein-Hawking entropy. The authors argue that interior bulk states are not independent: gravity induces nonzero overlaps between seemingly orthogonal semiclassical states, allowing an exponentially larger set of interior vectors within the same finite interior Hilbert space. They formalize a kinematic bound showing that, with overlaps of order $e^{-S/2}$, one can pack about $m \approx n \exp\left( \tfrac{n\epsilon^2}{2} \right)$ vectors into an $n$-dimensional space, resolving the paradox via overcounting. The resolution is supported by boundary (CFT) arguments showing no paradox in fine-grained entropy and by toy matrix models that reproduce the growth of interior-like excitations with small overlaps. They further show that naive EFT treatments of bags-of-gold configurations conflict with spectral observables, and that the observed agreement with black-hole physics is restored once overcounting is properly incorporated. Overall, the work highlights a special feature of quantum gravity: interior bulk states are non-orthogonal and overcounting, rather than literal independence, reconciles interior richness with entropy bounds and spectral data.

Abstract

In this work, we investigate how single-sided and eternal black holes in AdS can host an enormous number of semiclassical excitations in their interior, which is seemingly not reflected in the Bekenstein Hawking entropy. In addition to the paradox in the entropy, we argue that the treatment of such excitations using effective field theory also violates black holes' expected spectral properties. We propose that these mysteries are resolved because apparently orthogonal semiclassical bulk excitations have small inner products between them; and consequently, a vast number of semiclassical excitations can be constructed using the Hilbert space which describes black hole's interior. We show that there is no paradox in the dual CFT description and comment upon the initial bulk state, which leads to the paradox. Further, we demonstrate our proposed resolution in the context of small $N$ toy matrix models, where we model the construction of these large number of excitations. We conclude by discussing why this resolution is special to black holes.

Figures (10)

• Figure 1: The left figure displays the maximal volume slices for the eternal AdS Schwarzchild black hole. The volumes of these slices increasingly grow with boundary time $t$ thereby becoming very large at late times. The right figure demonstrates the bags of gold paradox for the eternal black hole on the maximal volume slices. We can accommodate an increasingly vast number of excitations placed far apart from each other on these slices which leads to the paradox.
• Figure 2: Bulk excitations denoted by orange and magenta lines arise from the left past horizon and fall into the left future horizon. These states come out of equilibrium as indicated in equation \ref{['ooe']} around the time $t_1$ and $t_2$ for the orange and magenta excitations. In the bulk this out of equilibrium behaviour is indicated by how far the excitations protrude out on the left. The unitaries control the position of the excitation on the slice, and large ${} {\Biggl{}\Biggr}{\left{}\right} {} -t_2{\lvert}{\rvert}{}$ leads to large spatial separation. Interior excitations at late times are visible only to later slices. Consequently, we can keep accomodating more and more excitations at later and later times which leads to the paradox.
• Figure 3: The initial state of the black hole is created by glueing the Euclidean AdS to the bottom half of the Lorentzian Penrose diagram. Boundary deformations of the Euclidean AdS create our excitations. Just after the initial time, all the bags of gold excitations are in the left exterior.
• Figure 4: We demonstrate the paradox for pure state black holes which are single-sided. The dotted line on the left denotes the UV cutoff for the theory living on the right boundary, which prohibits us from reaching arbitrary close to the left boundary. Since the interior region is similar for both the single-sided and eternal black holes, the physical picture of the paradox and its resolution is similar.
• Figure 5: The spectral form factors for different distributions obeying violations of type 1 (See Table \ref{['table2']}). The figure on the top left consists of random energy levels taken from a uniform probability distribution. The figure on the top right has energy levels picked from a near-uniform probability distribution. Both plots are for 1000 energy levels at $\beta = 2$ over 100 iterations. The bottom figures have uniformly spaced energy levels, and as a consequence, are integrable. The bottom left figure is plotted with 50 energy levels, with $\beta =1$ over 50 iterations, while the bottom right figure is plotted with 250 energy levels, with $\beta =1$ over 50 iterations.