The Impact of Generalized Uncertainty Principle on the Bekenstein Bound in Non-Gaussian Statistical Frameworks

Core Concepts

The standard Bekenstein bound, based on the Heisenberg uncertainty principle, is violated when considering non-Gaussian statistics (Barrow, Tsallis, and Kaniadakis) for black hole entropy. However, incorporating the Generalized Uncertainty Principle (GUP) can restore the bound's validity by establishing a connection between the GUP parameter and the indices of these non-Gaussian statistics.

Abstract

Bibliographic Information:

Shokri, M. (2024). Bekenstein bound on black hole entropy in non-Gaussian statistics. arXiv preprint arXiv:2411.00694v1.

Research Objective:

This research paper investigates the validity of the Bekenstein bound on black hole entropy when considering non-Gaussian statistical frameworks, specifically Barrow, Tsallis, and Kaniadakis statistics, in the context of both the Heisenberg Uncertainty Principle (HUP) and the Generalized Uncertainty Principle (GUP).

Methodology:

The author first demonstrates the violation of the standard Bekenstein bound (based on HUP) when applying non-Gaussian statistics to describe the entropy of a Schwarzschild black hole. Subsequently, the GUP is introduced, and its impact on the Bekenstein bound is analyzed within each non-Gaussian framework. Mathematical derivations and graphical representations are used to illustrate the relationship between the GUP parameter and the indices of the respective statistics.

Key Findings:

• The standard Bekenstein bound, incorporating HUP, fails to hold when black hole entropy is described using Barrow, Tsallis, or Kaniadakis statistics.
• Incorporating the GUP into the Bekenstein bound leads to a generalized bound that can be satisfied within these non-Gaussian frameworks.
• The satisfaction of the generalized Bekenstein bound is contingent upon a specific connection between the GUP parameter (β) and the indices of the respective non-Gaussian statistics (∆ for Barrow, q for Tsallis, and κ for Kaniadakis).

Main Conclusions:

The research concludes that while the standard Bekenstein bound is challenged by non-Gaussian statistics in the context of black hole entropy, the GUP offers a potential resolution. By establishing a relationship between the GUP parameter and the indices of these statistics, the generalized Bekenstein bound can be satisfied, suggesting a possible interplay between quantum gravitational effects and non-extensive statistical behavior in black hole thermodynamics.

Significance:

This study contributes to the ongoing discourse on black hole thermodynamics, particularly concerning the interplay between gravity and quantum mechanics. It highlights the limitations of the standard Bekenstein bound in non-Gaussian statistical frameworks and proposes a potential solution by incorporating the GUP, thereby advancing our understanding of entropy bounds in quantum gravity.

Limitations and Future Research:

The research focuses specifically on the Schwarzschild black hole model and three specific non-Gaussian statistics. Further investigation could explore the applicability of these findings to other black hole solutions and alternative statistical frameworks. Additionally, exploring the physical implications and observational consequences of the proposed connection between GUP and non-Gaussian statistics could be a promising avenue for future research.

Stats

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Key Insights Distilled From

Bekenstein bound on black hole entropy in non-Gaussian statistics

by Mehdi Shokri at arxiv.org 11-04-2024

https://arxiv.org/pdf/2411.00694.pdf

Deeper Inquiries

How would the inclusion of quantum field theory considerations, beyond the semi-classical approximations used in this study, affect the interplay between the Bekenstein bound, GUP, and non-Gaussian statistics?

Answer:
Delving into the realm of quantum field theory (QFT) introduces a new level of complexity to the interplay between the Bekenstein bound, GUP, and non-Gaussian statistics. Here's a breakdown of potential implications:

Entanglement Entropy: QFT highlights the significance of entanglement entropy, particularly in the context of black hole horizons. Entanglement between degrees of freedom inside and outside the horizon contributes to the black hole entropy. Incorporating entanglement entropy, potentially within the framework of non-Gaussian statistics, could modify the Bekenstein bound.

Quantum Fluctuations: QFT predicts quantum fluctuations of spacetime itself at the Planck scale. These fluctuations could alter the horizon geometry, leading to corrections in the black hole entropy calculation. Such modifications might necessitate a re-evaluation of the Bekenstein bound, especially when considering GUP effects that also emerge at the Planck scale.

Beyond Semi-classical Gravity: The study primarily operates within the semi-classical regime, where gravity is treated classically, and matter fields are quantized. A full QFT treatment of gravity, such as loop quantum gravity or string theory, might reveal a different relationship between entropy and area, potentially impacting the Bekenstein bound.

Modified Dispersion Relations: GUP scenarios often imply modified dispersion relations for particles. These modifications could affect the thermodynamics of the black hole, leading to corrections in the entropy and potentially influencing the validity of the Bekenstein bound in non-Gaussian frameworks.
In essence, incorporating QFT considerations beyond semi-classical approximations could significantly impact the interplay between the Bekenstein bound, GUP, and non-Gaussian statistics. It might lead to modifications of the bound itself or require a more nuanced interpretation of its implications for black hole thermodynamics.

Could the violation of the standard Bekenstein bound in non-Gaussian frameworks point towards a deeper inconsistency in our understanding of black hole thermodynamics, potentially requiring a paradigm shift beyond GUP modifications?

Answer:
The violation of the standard Bekenstein bound in non-Gaussian frameworks indeed raises intriguing questions about our understanding of black hole thermodynamics. While not necessarily signaling a fundamental inconsistency, it suggests several possibilities:

Incompleteness of Semi-classical Gravity: The violation might stem from the limitations of the semi-classical approach. Treating gravity classically while quantizing matter fields might not capture the full picture of black hole thermodynamics. A complete theory of quantum gravity could potentially resolve the apparent violation.

Beyond Area Law: The Bekenstein-Hawking entropy formula, relating entropy to horizon area, might be an approximation that breaks down in certain regimes, particularly when considering non-Gaussian statistics and GUP effects. A more fundamental entropy formula, perhaps incorporating additional geometric or topological factors, might be necessary.

Non-Extensive Nature of Gravity: Non-Gaussian statistics, like Tsallis entropy, often arise in systems with long-range interactions or non-extensive behavior. The violation of the standard Bekenstein bound could hint at the non-extensive nature of gravity itself, suggesting that the entropy of a black hole might not scale linearly with its size in all situations.

Rethinking the Bound: The violation might compel us to re-examine the assumptions underlying the Bekenstein bound. The bound was derived using heuristic arguments and might not hold universally, especially in scenarios involving modified gravity or quantum gravitational effects.

New Physics at the Planck Scale: The interplay between GUP and non-Gaussian statistics, leading to the violation, could point towards new physics at the Planck scale. This new physics might involve modifications to spacetime structure, quantum fluctuations of the horizon, or non-trivial entanglement entropy contributions.
While a paradigm shift beyond GUP modifications might not be immediately necessary, the violation of the standard Bekenstein bound in non-Gaussian frameworks encourages exploring these possibilities. It underscores the need for a deeper understanding of quantum gravity and its implications for black hole thermodynamics.

If the universe itself exhibits some level of fractal structure, as suggested by certain cosmological models, could the insights from Barrow entropy and its connection to GUP offer a new perspective on the overall entropy budget of the cosmos?

Answer:
The possibility of a fractal universe, coupled with insights from Barrow entropy and its connection to GUP, opens up fascinating avenues for understanding the cosmic entropy budget:

Beyond Standard Cosmology: Standard cosmological models often assume a homogeneous and isotropic universe at large scales. However, a fractal structure implies a more complex, inhomogeneous distribution of matter and energy, potentially altering our understanding of the universe's entropy content.

Modified Entropy-Area Relation: Barrow entropy, inspired by fractal geometries, suggests a modification to the standard entropy-area relationship for black holes. If the universe itself exhibits fractal properties, a similar modification might apply to cosmological horizons, impacting the calculation of the universe's total entropy.

Quantum Gravitational Effects at Large Scales: The connection between Barrow entropy and GUP implies that quantum gravitational effects, typically confined to the Planck scale, could manifest at cosmological scales in a fractal universe. These effects could influence the expansion rate, structure formation, and ultimately, the entropy evolution of the cosmos.

New Sources of Entropy: Fractal structures are characterized by intricate patterns repeating at different scales. This self-similarity could imply the existence of hidden degrees of freedom and new sources of entropy in the universe, contributing to the overall entropy budget.

Implications for the Entropic Arrow of Time: The concept of an entropic arrow of time relies on the universe's entropy increasing over time. A fractal universe, with its potentially modified entropy budget and quantum gravitational influences, might lead to a more nuanced understanding of the entropic arrow of time and its cosmological implications.
In conclusion, if the universe possesses a fractal structure, Barrow entropy and its connection to GUP offer a fresh perspective on the cosmic entropy budget. It suggests a departure from standard cosmological models and hints at the intriguing possibility of quantum gravitational effects playing a role in the universe's large-scale structure and entropy evolution.