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

Sara Motalebi

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} $r_{\rm crit}=(β/γ)^{1/4}\sqrt{\ell_{P}L}$, which marks a phase transition where quantum gravitational ($β$) and AdS curvature ($γ$) 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 $\langle T_{μν}\rangle=0$ under the self-duality condition $\partial_z g_{μν}|_{z=r_{\rm crit}}=0$; (ii) a topological transition manifested by the complexification of the central charge, $c_{\rm eff}=c\left(1+\frac{i}{2}\sqrtκ\ell_{P}^{2}\right)$ with $κ=(β/γ)^{1/4}\sqrt{βγ\ell_P/L}$; 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 $ΔS_{\rm recovery}>0$. These effects establish a consistency condition $L>\sqrtβ\ell_{P}$ 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.

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

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} , which marks a phase transition where quantum gravitational () and AdS curvature () 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 under the self-duality condition ; (ii) a topological transition manifested by the complexification of the central charge, with ; 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 . These effects establish a consistency condition for a valid AdS/CFT correspondence and identify as the thermodynamic critical point where black holes transition to stringy remnants and information is topologically scrambled.

Paper Structure

This paper contains 17 sections, 1 theorem, 60 equations.

Table of Contents

  1. Introduction
  2. First-Principles Derivation of Critical Radius
  3. Physical Foundations of AdS-Modified GUP
  4. GUP-Modified Klein-Gordon Equation
  5. Physical Significance of Critical Radius
  6. Vanishing Holographic Stress Tensor
  7. EGUP-Modified Gravitational Framework
  8. Holographic RG Flow and Critical Behavior
  9. Proof of Vanishing Stress Tensor
  10. Physical Verification and Interpretation
  11. Derivation of Complex Central Charge
  12. Complex Central Charge from Holographic Anomalies
  13. Information Recovery Framework
  14. A Potential Topological Information Mechanism at Critical Radius
  15. A Scenario for Modified Page Curve Dynamics
  16. ...and 2 more sections

Key Result

Theorem 1

At $z=r_{\text{crit}}$, the expansion coefficient $g^{(d)}_{\mu\nu} = 0$, implying vanishing boundary stress tensor $\langle T_{\mu\nu}\rangle = 0$.

Theorems & Definitions (2)

  • Theorem 1
  • proof