Egup-induced Critical Radius Reveals Holographic Scale with Complexification at and Modifies Page Curve by 0.1

The interplay between gravity and quantum mechanics remains a central challenge in modern physics, and recent work by Sara Motalebi from Tarbiat Modares University, and colleagues, sheds new light on this fundamental connection. The team investigates how modifications to the principles governing uncertainty at extremely small scales impact our understanding of gravity within Anti-de Sitter space, a theoretical framework used to model the universe. Their research reveals a critical radius where gravitational and curvature effects balance, leading to a breakdown of standard holographic duality, a complexification of key mathematical properties, and a potential mechanism for resolving the long-standing information paradox associated with black holes. These interconnected findings establish a crucial consistency condition for the holographic principle and pinpoint a thermodynamic critical point where black holes transition into stringy remnants, with information topologically encoded and preserved.

Scientists investigate how incorporating a minimum length scale, predicted by many quantum gravity theories, alters our understanding of black holes and potentially resolves the long-standing information paradox. The team utilizes non-Hermitian quantum mechanics to describe modified black hole solutions arising from the GUP, emphasizing the importance of understanding how physical quantities change with energy scale through renormalization group flow. The study demonstrates that the GUP modifies black hole thermodynamics, leading to corrections in entropy and temperature that become significant at small scales and affect the black hole’s stability and evaporation process.

By employing a non-Hermitian approach, researchers can effectively handle the complex potentials and energies that emerge in these scenarios. The holographic description, leveraging the AdS/CFT correspondence, allows them to relate the black hole’s properties to those of a dual conformal field theory, providing a powerful tool for understanding quantum gravity and the nature of spacetime. This combined approach offers a potential pathway to resolving the information paradox, suggesting that information is not lost but encoded in the boundary theory. This work contributes to the ongoing effort to develop a consistent theory of quantum gravity, reconciling general relativity with quantum mechanics. It provides new insights into black hole behavior at the quantum level and reinforces the importance of the holographic principle as a powerful tool for understanding quantum gravity and the nature of spacetime. The use of non-Hermitian quantum mechanics and pseudo-Hermitian representations demonstrates the power of mathematical techniques in tackling complex physical problems.

Critical Radius Defines Black Hole Momentum Scale

Scientists have developed a comprehensive framework to investigate the interplay between Generalized and Extended Uncertainty Principles within Anti-de Sitter (AdS) space. This work reveals a fundamental gravitational scale, termed the “critical radius”, where gravitational and AdS curvature effects achieve equilibrium. This radius triggers a cascade of interconnected phenomena crucial to understanding black hole thermodynamics and resolving the information paradox. The team analyzed the Klein-Gordon equation near black hole horizons, utilizing approximation techniques to determine the momentum distribution and demonstrating that integration over the near-horizon region yields a momentum scale proportional to ħ/rs, where rs represents the Schwarzschild radius.

Rigorous validation involved thermodynamic extremization and analysis of entropy divergence, establishing a robust scaling relationship independent of the chosen method. Researchers constructed a GUP-corrected Bekenstein-Hawking entropy formula, incorporating logarithmic and higher-order correction terms, and demonstrated that the heat capacity diverges precisely at the critical radius, confirming its status as a thermodynamic critical point. This divergence signifies a transition where black holes evolve from quantum-dominated states to those governed by AdS curvature, with the negative sign of the EUP correction arising fundamentally from the influence of AdS curvature. Further investigation involved deriving a complete bulk action incorporating both GUP and EUP corrections by extending the Einstein-Hilbert action with terms capturing quantum and curvature modifications, ensuring mathematical consistency.

Varying this action yielded modified Einstein equations, revealing correction tensors that satisfy the Bianchi identity, guaranteeing mathematical consistency. Analyzing the holographic Renormalization Group (RG) flow using a specific coordinate system demonstrated that at the critical radius, correction terms cancel, reducing the Hamiltonian to its conformal fixed point form and signifying a stable Planck-scale remnant where information is topologically scrambled and evaporation terminates. This innovative approach provides a pathway to resolving the information paradox by storing information in Chern-Simons states, modifying the Page curve and establishing a consistency condition for a valid AdS/CFT correspondence.

Holographic Breakdown, Topological Transition, Information Recovery

This work presents a unified framework incorporating both Generalized and Extended Uncertainty Principles in Anti-de Sitter space, revealing a fundamental gravity scale termed the critical radius. At this scale, gravitational and AdS curvature effects equilibrate, leading to three interconnected phenomena. First, the standard holographic duality breaks down, signaled by the exact vanishing of the boundary stress tensor under specific conditions. Second, a topological transition occurs, manifested by the complexification of the central charge, which shifts from its standard value to a complex number with an imaginary component of 2√κ.

Third, a mechanism for information paradox resolution emerges, where information is recovered via topological storage in Chern-Simons states, modifying the Page curve with a correction term of 1/120. Measurements confirm that the finite, cutoff-independent term resulting from calculations is 1/120, a value consistent with established regularization techniques but derived from physical principles related to the high-energy density of states. The imaginary component in the central charge does not violate unitarity, but instead encodes a topological information storage mechanism, preserving conformal symmetry through modular invariance. This topological protection is achieved through a Chern-Simons holonomy of πi/2√κ, providing protected states.

The consistency condition ensures the physical validity of κ and prevents unitarity violation while decoupling ultraviolet and infrared scales. Researchers demonstrate a three-stage topological mechanism for information preservation at the critical radius, encoding information in Chern-Simons states with a Hilbert space dimension potentially matching string state degeneracy. Non-unitary dynamics facilitate information transfer to radiation, and holographic decoding unitarizes the process. The resulting entropy correction quantifies information retrieval from topological memory, with the complex central charge deforming the radiation density matrix and introducing coherence through phase correlations. This framework offers a potential resolution to the tension between apparent unitarity violation and actual information recovery, with the imaginary entropy potentially quantifying information capacity.

Critical Radius Defines Holographic Breakdown and Recovery

This research establishes a critical radius as a fundamental scale governing quantum gravitational phenomena within Anti-de Sitter (AdS) space. Scientists demonstrate that this radius marks a point where gravitational and AdS curvature effects balance, leading to a breakdown of the standard holographic duality. Specifically, the boundary stress tensor vanishes at this scale, signalling a restructuring of the relationship between gravity and quantum fields. Furthermore, the analysis reveals a topological transition at the critical radius, evidenced by the complexification of the central charge, and a mechanism for information recovery.

This suggests that information potentially lost within black holes may be preserved through topological storage in Chern-Simons states, modifying the Page curve and offering a pathway towards resolving the information paradox. The universality of the critical radius is supported by its consistent emergence from multiple independent approaches, including modified field equations, thermodynamic extremization, uncertainty principle balance, and heat capacity divergence, confirming its role as a Planck-scale threshold. Researchers acknowledge that a detailed mapping between their framework and recent approaches to the black hole/string transition remains a future research direction.