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

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)^{۱/۴}\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=۰ under the self-duality condition \partial_z g_{\mu\nu}|_{z=r_{\rm crit}}=۰; (ii) a topological transition manifested by the complexification of the central charge, c_{\rm eff}=c\left(۱+\frac{i}{۲}\sqrt{\kappa}\ell_{P}^{۲}\right) with \kappa=(\beta/\gamma)^{۱/۴}\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}>۰. 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.

کلیدواژه ها:

Generalized Uncertainty Principle ، Extended Uncertainty Principle ، AdS/CFT correspondence ، Black Hole/String Transition