From information bounds to infrared gravity: implications of Sharma-Mittal entropy

This paper demonstrates that the two-parameter Sharma-Mittal entropy framework unifies black hole thermodynamics and information theory to derive modified gravitational laws that naturally reproduce a MOND-like regime, offering a potential alternative explanation for dark matter phenomena.

The Gist

Imagine the universe as a giant, cosmic computer. For a long time, scientists thought the "hard drive" of this computer (space and time) worked in a very standard, predictable way. But recently, researchers have started wondering if the rules of information and heat might actually be the source of gravity itself.

This paper by Benkrane, Luciano, and Sheykhi explores a new, more flexible set of rules for how information is stored in the universe, specifically around black holes. They use a mathematical tool called Sharma–Mittal (SM) entropy.

Here is a breakdown of their findings using simple analogies:

1. The "Universal Remote" of Entropy

Think of the standard way we measure information (called Boltzmann-Gibbs entropy) as a basic TV remote with fixed buttons. Scientists previously found two "upgraded" remotes: the Rényi remote and the Tsallis remote. These work differently when dealing with huge, complex systems like black holes.

The authors introduce the Sharma–Mittal (SM) entropy as a "Universal Remote." It has two adjustable dials (parameters).

• If you turn one dial, it acts exactly like the Rényi remote.
• If you turn the other, it acts like the Tsallis remote.
• If you set them just right, it acts like the standard remote.

The paper asks: "What happens to gravity if we use this Universal Remote instead of the old, fixed one?"

2. The Black Hole "Weight Limit" (The Bekenstein Bound)

There is a famous rule in physics called the Bekenstein bound. Imagine it as a strict weight limit on an elevator. No matter how much information (people) you try to pack into a black hole (the elevator), there is a maximum amount of "stuff" it can hold based on its size and energy.

The researchers checked if their Universal Remote respects this weight limit.

• The Result: Yes, it does, but only if the two dials on the remote are set within a specific "safe zone." If you turn the dials too far in certain directions, the math breaks the rules of physics. This tells us that the universe is picky; it won't allow just any random way of counting information.

3. Erasing a Bit of Information Costs Energy

There is a principle called Landauer's Principle. It says that if you want to delete a single "bit" of information (like pressing the "delete" key on a computer), you must waste a tiny amount of energy as heat. You can't delete things for free.

The authors applied this to black holes. They asked: "If a black hole deletes one bit of information, how much does its mass change?"

• The Result: In the old standard model, the mass change is predictable. With the SM Universal Remote, the mass change depends on the settings of the two dials.
  • In some settings, the black hole loses mass slower than expected.
  • In other settings, it loses mass faster.
  • This suggests that the "cost" of deleting information isn't the same for every black hole; it depends on the hidden "texture" of the universe's information storage.

4. Gravity as an "Emergent" Force

The paper uses a theory by physicist Erik Verlinde, which suggests gravity isn't a fundamental force (like magnetism) but something that emerges from information, much like how "wetness" emerges from the movement of many water molecules.

When they used the SM entropy to calculate gravity:

• Close up: Gravity looks exactly like Newton's classic laws (what we learn in school).
• Far away: Gravity starts to behave differently. The "Universal Remote" predicts that at very large distances (like across a galaxy), gravity gets stronger or weaker than Newton predicted, depending on the dial settings.

5. The "Dark Matter" Mystery and MOND

This is the most exciting part. Astronomers see galaxies spinning so fast that they should fly apart. To explain this, they usually invent "Dark Matter" (invisible stuff holding them together).

However, there is another theory called MOND (Modified Newtonian Dynamics). It suggests that gravity just changes its behavior at very large distances, so you don't need invisible matter; the rules of gravity are just different out there.

• The Discovery: The authors found that if they set their Universal Remote's dials to a specific ratio (one dial is 1.5 times the other), the math naturally produces the exact behavior of MOND.
• The Implication: This suggests that the weird way galaxies spin might not be caused by invisible "dark matter" at all. Instead, it might be because the universe's information storage (entropy) has a specific "texture" that makes gravity act differently on a galactic scale.

Summary

The paper argues that if we view the universe through the lens of this flexible "Universal Remote" for information (Sharma–Mittal entropy):

1. It respects the fundamental limits of black holes.
2. It changes how black holes lose mass when they "delete" information.
3. It naturally explains why galaxies spin the way they do without needing to invent "Dark Matter," by simply adjusting how gravity works at the largest scales.

It's a proposal that the "glue" holding the universe together might be a result of how information is stored and processed, rather than a mysterious invisible substance.

Technical Summary

Technical Summary: From Information Bounds to Infrared Gravity: Implications of Sharma–Mittal Entropy

Problem Statement
The thermodynamic description of gravitating systems remains a subject of debate, particularly regarding the non-extensive nature of Bekenstein–Hawking entropy and the unknown microphysical structure of spacetime. While generalized entropy formalisms, such as Rényi and Tsallis entropies, have been proposed to address these issues and explore modified gravity scenarios, a more flexible framework is needed to interpolate between different non-extensive regimes. This work investigates the Sharma–Mittal (SM) entropy, a two-parameter generalization encompassing both Rényi and Tsallis frameworks, to determine its thermodynamic and gravitational implications. Specifically, the authors examine whether the SM entropy is compatible with fundamental information bounds, how it modifies black hole evaporation via the Landauer principle, and whether it can induce infrared (IR) modifications to gravity that reproduce Modified Newtonian Dynamics (MOND) without invoking dark matter.

Methodology
The study proceeds through a systematic application of the SM entropy formalism to three distinct physical contexts:

1. Generalized Entropy Framework: The authors define the SM entropy ($S_{SM}$) using two deformation parameters, $r$ and $\delta$ (or $R = 1-r$ and $\delta$). They establish its gravitational realization by substituting the Bekenstein–Hawking entropy ($S_{BH}$) into the SM functional form, creating a direct link between the statistical framework and black hole physics.
2. Bekenstein Bound Compatibility: The authors test the consistency of the gravitational realization of SM entropy against the Bekenstein bound ($S \leq 2\pi k_B RE/\hbar c$). They derive a consistency condition involving the deformation parameters and analyze the admissible regions in the parameter space $(x, y)$, where $x = R S_{SM}$ and $y = \delta S_{SM}$.
3. Landauer Principle and Mass Loss: By incorporating Landauer's principle (which posits a minimum energy cost for erasing one bit of information), the authors derive a modified mass-loss relation ($\Delta M$) for a Schwarzschild black hole. They analyze the asymptotic behavior of this relation in both small-mass ($4\pi\delta M^2 \ll 1$) and large-mass ($4\pi\delta M^2 \gg 1$) regimes.
4. Emergent Gravity and MOND: Within Verlinde's entropic gravity framework, the authors derive modified gravitational force and acceleration laws induced by the SM entropy. They analyze the asymptotic behavior of the acceleration at large distances to determine if the framework can reproduce the deep-MOND regime ($a \propto 1/r$). They further relate the resulting acceleration scale to the SM deformation parameters.

Key Contributions and Results

• Bekenstein Bound Constraints: The study establishes that the SM framework is not arbitrary; it must satisfy a nontrivial consistency condition to remain compatible with the Bekenstein bound. Numerical analysis reveals that the deformation parameters $R$ and $\delta$ are subject to correlated constraints. The limiting cases ($R \to 0$ and $R \to \delta$) consistently recover the known results for Rényi entropy and standard Bekenstein–Hawking entropy, respectively, validating the internal consistency of the formalism.
• Modified Mass-Loss Relation: The incorporation of the Landauer principle yields a modified mass-loss relation for black holes erasing one bit of information. The deviation from the standard semiclassical result is controlled by the ratio $R/\delta$.
  • In the small-mass regime, the leading correction depends on the mismatch $(\delta - R)$.
  • In the large-mass regime, the asymptotic behavior is governed by the ratio $R/\delta$. A critical value of $R/\delta = 1/2$ separates regimes where the mass variation increases or decreases with black hole mass. This suggests the energetic cost of information erasure is not universal in the large-mass limit but depends on the specific generalized entropy parameters.
• Modified Gravitational Laws: The authors derive a modified entropic force and acceleration law. The deviation from Newtonian gravity is fully controlled by the parameters $R$ and $\delta$.
  • The modification can be interpreted as an effective mass density surrounding the gravitating source. For $R > \delta$, this density is positive (attractive), while for $R < \delta$, it is negative (repulsive).
  • The gravitational potential is expressed in terms of the Gauss hypergeometric function, reducing to the standard Newtonian potential when $R = \delta$.
• Emergence of MOND-like Behavior: A significant finding is that the SM entropy naturally reproduces a MOND-like regime at large distances under the specific condition $R/\delta = 3/2$. Under this condition, the acceleration scales as $a \propto 1/r$, matching the deep-MOND behavior.
  • This allows for a direct connection between the SM parameters and the MOND acceleration scale $a_0$. The authors derive a relation where $a_0$ depends on the gravitating mass $M$ and the deformation parameter $\delta$.
  • For typical astrophysical masses, the required deformation parameter $\delta$ is extremely small, but the correction term $\delta \pi r^2 / \ell_p^2$ becomes relevant at galactic scales due to the hierarchy between astrophysical and Planck length scales.

Significance and Claims
The paper claims that the Sharma–Mittal entropy framework offers a unified link between black hole thermodynamics, information theory, and infrared modifications of gravity. By demonstrating that a specific relation between the SM parameters ($R/\delta = 3/2$) naturally yields MOND-like dynamics, the authors suggest that observed IR modifications in galactic dynamics may admit a thermodynamic interpretation rooted in the non-extensive nature of entropy.

The findings highlight that generalized entropy formalisms can provide alternative explanations for phenomena traditionally attributed to dark matter, without altering the underlying gravitational dynamics but rather by modifying the thermodynamic description of spacetime degrees of freedom. The work emphasizes that the SM framework is not merely a statistical extension but imposes physical constraints on the deformation parameters through consistency with fundamental bounds (Bekenstein) and information principles (Landauer). The authors conclude that while the SM framework shows promise, further investigation is required to constrain these parameters via astrophysical observations and to derive them from fundamental quantum gravity approaches.