The EGUP-Induced Critical Radius: A New Holographic Scale for Quantum Gravity

TL;DR

This paper introduces a new quantum gravity scale called the critical radius in AdS space, where holographic duality breaks down and topological effects emerge, providing insights into black hole information paradox resolution.

Contribution

The work unifies GUP and EUP principles in AdS space to identify a critical radius marking phase transitions and holographic breakdowns, offering a novel perspective on quantum gravity and information recovery.

Findings

Identification of the critical radius as a phase transition point.

Demonstration of holographic duality breakdown at the critical radius.

Proposal of topological mechanisms for information recovery.

Abstract

We present a unified framework incorporating both the Generalized and Extended Uncertainty Principles (GUP/EUP) in Anti-de Sitter space. This reveals a fundamental quantum gravity scale, the \textit{critical radius} $r_{\rm crit}=(\beta/\gamma)^{1/4}\sqrt{\ell_{P}L}$, which marks a phase transition where quantum gravitational ($\beta$) and AdS curvature ($\gamma$) effects equilibrate. At this scale, we demonstrate three interconnected phenomena: (i) a breakdown of the standard holographic duality, signaled by the exact vanishing of the boundary stress tensor $\langle T_{\mu\nu}\rangle=0$ under the self-duality condition $\partial_z g_{\mu\nu}|_{z=r_{\rm crit}}=0$; (ii) a topological transition manifested by the complexification of the central charge, $c_{\rm eff}=c\left(1+\frac{i}{2}\sqrt{\kappa}\ell_{P}^{2}\right)$ with $\kappa=(\beta/\gamma)^{1/4}\sqrt{\beta\gamma\ell_P/L}$; and (iii) a mechanism as a scenario for information paradox resolution, where information is recovered via topological storage in Chern-Simons states, modifying the Page curve with $\Delta S_{\rm recovery}>0$. These effects establish a consistency condition $L>\sqrt{\beta}\ell_{P}$ for a valid AdS/CFT correspondence and identify $r_{\rm crit}$ as the thermodynamic critical point where black holes transition to stringy remnants and information is topologically scrambled.