Blackish Holes with Stringy Backreaction

TL;DR

This work analyzes how a Dirichlet wall placed near a black hole horizon can encode non-trivial quantum level correlations in a probe scalar spectrum by revealing a two-scale hierarchy tied to radial localization and near-horizon dynamics. Using a WKB treatment in BTZ and in D$p$-brane/HV-Lifshitz geometries, it shows that the Dip-time of the spectral form factor scales universally as $t_{\rm dip}\sim (\log\epsilon_0)^2$, with further Parametric enhancements when backreaction from string sources is included. The authors introduce a dressed effective framework with string backreaction controlled by a dimensionless parameter $Q$, demonstrating that $t_{\rm dip}$ can be significantly increased by UV/IR data, bridging near-horizon quantum dynamics and UV completions. The results illuminate a path toward more realistic effective descriptions of black-hole microphysics and ECOs, with implications for holographic duals in the Veneziano limit and potential connections to OTOCs and non-equilibrium chaotic dynamics.

Abstract

Recent studies have demonstrated that an $\textit{ad hoc}$ Dirichlet boundary condition, placed outside but close to an event horizon, for probe degrees of freedom in an otherwise black hole geometry is capable of capturing non-trivial level-correlations of the corresponding spectrum of the probe sector. Much of the interesting physics stems from a hierarchy of scales that is present in the quantum spectrum, in terms of two quantum numbers that characterize it. In this work, we establish an explicit connection with the hierarchy of these scales with a $\textit{radial localization}$ or the absence of it of the probe scalar WKB-wavefunction. Subsequently, this scale separation can be traced back to the hierarchy between the local red-shift and the classical light-traversing time in a geometry that produces a Rindler-throat. The classical null ray takes a logarithmically divergent time to reach the Dirichlet wall, and interestingly, we explicitly demonstrate that the scalar quantum spectrum arising from the Rindler throat yields a Dip-time of the corresponding spectral form factor, which scales with a universal power of the light traversing time. Armed with these, we further consider a $\textit{dressed effective model}$ where the Dirichlet boundary condition is inserted in a ten-dimensional supergravity geometry, where classical string sources back-react. We demonstrate that, as a result of this backreaction, the quantum-dynamical time-scales, $\textit{e.g.}$ the Dip time of the corresponding spectral form factor can be further enhanced with factors of the string length, thereby making the Dirichlet wall configuration better mimic the true black hole. In the dual field theory, the geometry corresponds to thermal states of a large $N$ gauge theory in the Veneziano limit, where both the number of colour and the flavour degrees of freedom are large.

Figures (7)

• Figure 1: Left: The behavior of the effective WKB potential \ref{['wkbpot']} as a function of the angular momentum quantum number $m$. The potential approaches zero as the horizon is reached at $z \rightarrow \infty$, while the presence of the brick wall at $z = z_B$ leads to an infinite potential barrier, indicated by the red vertical dashed line. Right: The forbidden and allowed regions of the WKB problem, labeled as Region I and Region II respectively, for fixed $m$ and $\omega^2$.
• Figure 2: Modes obtained by solving \ref{['quant11']} for $z_B = 10$. The left panel shows modes as a function of $m$ for fixed $n = 1$, whereas the right panel shows modes as a function of $n$ for fixed $m = 20$. Note the very slow dependence of the modes on $m$ compared to the linear dependence on the quantum number $n$. These two markedly different growth rates of the modes are the main reason for the emergence of two distinct scales in the system.
• Figure 3: Behavior of the WKB wavefunction \ref{['reg2F']}. Left: dependence on $n$ for fixed $m$, showing the expected increase in the number of nodes, with one node at the brick wall ($z = z_B$). Right: dependence on $m$ for fixed $n$, showing much slower variation due to the quasi-degeneracy of the spectrum along the $m$-direction.
• Figure 4: Dependence of $|\psi"_{II}|$ on the quantum numbers. Left: variation with $n$ for fixed $m$. Right: variation with $m$ for fixed $n$.
• Figure 5: Spectral form factor (SFF) for the modes along the $m$-direction with cutoff $m_{\text{cut}} = 500$. The SFF clearly exhibits the dip–ramp–plateau structure. $z_B=10$ and $n$ is fixed at $n = 1$.