Black hole thermodynamical entropy (1202.2154v2)

Abstract: As early as 1902, Gibbs pointed out that systems whose partition function diverges, e.g. gravitation, lie outside the validity of the Boltzmann-Gibbs (BG) theory. Consistently, since the pioneering Bekenstein-Hawking results, physically meaningful evidence (e.g., the holographic principle) has accumulated that the BG entropy $S_{BG}$ of a $(3+1)$ black hole is proportional to its area $L^2$ ($L$ being a characteristic linear length), and not to its volume $L^3$. Similarly it exists the \emph{area law}, so named because, for a wide class of strongly quantum-entangled $d$-dimensional systems, $S_{BG}$ is proportional to $\ln L$ if $d=1$, and to $L^{d-1}$ if $d>1$, instead of being proportional to $L^d$ ($d \ge 1$). These results violate the extensivity of the thermodynamical entropy of a $d$-dimensional system. This thermodynamical inconsistency disappears if we realize that the thermodynamical entropy of such nonstandard systems is \emph{not} to be identified with the BG {\it additive} entropy but with appropriately generalized {\it nonadditive} entropies. Indeed, the celebrated usefulness of the BG entropy is founded on hypothesis such as relatively weak probabilistic correlations (and their connections to ergodicity, which by no means can be assumed as a general rule of nature). Here we introduce a generalized entropy which, for the Schwarzschild black hole and the area law, can solve the thermodynamic puzzle.

• The paper addresses the anomaly where black hole entropy is non-extensive, unlike traditional thermodynamics, and proposes using generalized entropy frameworks to restore extensivity.
• Authors introduce generalized entropies like S_q and S_delta within nonextensive statistical mechanics to account for complex systems with long-range interactions and reconcile thermodynamic scaling behavior.
• Specific parameter values for these generalized entropies are shown to recover extensivity for black holes, suggesting their potential application in unifying thermodynamics with quantum gravity.

An Expert Analysis of "Black hole thermodynamical entropy"

The paper "Black hole thermodynamical entropy" authored by Constantino Tsallis and Leonardo J.L. Cirto addresses a fundamental issue in black hole thermodynamics. Drawing upon the foundational works of Bekenstein and Hawking, the authors delve into the intriguing notion that black hole entropy does not behave extensively as described by traditional thermodynamic laws. This essay provides a comprehensive overview of the paper, elucidating the theoretical implications and highlighting the novel approach introduced by the authors to address the discrepancies of black hole entropy through generalized entropy frameworks.

Theoretical Insights and Methodological Approach

The paper begins by exploring the well-established anomaly in black hole entropy, which starkly contrasts with the thermodynamic extensivity principle traditionally described by the Boltzmann-Gibbs (BG) entropy framework. According to BG entropy, the entropy of a system should be proportional to its volume. However, in the context of black holes, accumulating evidence, particularly from the holographic principle, indicates that entropy is proportional to the area of the black hole's event horizon, rather than its volume. This is manifestly non-extensive since the entropy scales with the surface area (L²) instead of the volume (L³) for a standard (3+1)-dimensional black hole. The implications are profound, as they suggest an unconventional scaling relation and a potential breakdown of standard thermodynamics in systems exhibiting strong correlation or entanglement.

To resolve this thermodynamic inconsistency, the authors propose a deviation from the BG entropy model by employing generalized nonadditive entropy frameworks. Specifically, they introduce a generalized entropy form, denoted as $S_q$ or $S_\delta$, which offers a pathway to reconcile the extensively anomalous behavior of black hole thermodynamical entropy. The proposed $S_q$ entropy is part of the broader concept of nonextensive statistical mechanics, a framework rooted in accounting for complex systems characterized by long-range interactions and correlations.

Numerical Results and Examination

Throughout the paper, a detailed mathematical exposition is provided for the proposed generalized entropic forms. These extended models, notably the Tsallis entropy and a newly defined $S_\delta$ entropy, adaptively scale with the system's parameters, addressing the profound changes in scaling behavior observed in quantum entangled systems and black holes. The authors successfully connect these entropies to well-documented quantum and gravitational property variations, underscoring the viability of nonadditive entropies in restoring thermodynamic extensivity.

The critical consideration addressed is how $S_\delta$ or the general form $S_{q,\delta}$ recovers extensivity for specific values of $(q,\delta)$ in complex systems. For instance, the extensivity is achieved for $\delta = 3/2$ in a (3+1)-dimensional Schwarzschild black hole scenario, leading to an intriguing expression connecting the thermodynamical entropy with the Bekenstein-Hawking entropy through non-linear power functions. This observation aligns with theoretical expectations and grounds the proposed entropy models as applicable in reconciling thermodynamics with quantum gravitational phenomena.

Implications and Future Prospects

The implications of this research extend to both theoretical understanding and practical applications in fields of quantum mechanics, general relativity, and statistical mechanics. By addressing the thermodynamic extensivity through a nonadditive framework, the authors open avenues for revisiting thermodynamic laws under extreme conditions and in systems where conventional BG statistical mechanics reach their limit.

Future research may explore the application of these nonadditive frameworks in other domains of physics, especially where systems exhibit similar long-range interactions and strong entanglement properties. The potential for these generalizations to become instrumental tools in unifying diverse physical theories or providing insights into the entropy of more complex geometric configurations within the space-time continuum is a promising prospect for the field of theoretical physics.

In conclusion, Tsallis and Cirto contribute a significant theoretical framework addressing longstanding anomalies in black hole entropy. Their approach beyond BG entropy paves the way for deeper exploration into the nonadditive entropy models, bringing us closer to understanding the entropic nature of the universe in scenarios beyond classic thermodynamic boundaries.