⚛️ general relativity

Entropy-geometry correspondence as effective nonlocal gravity

This paper establishes an operator formulation of the entropy-geometry correspondence for static, spherically symmetric gravity, demonstrating how various generalized entropy models reconstruct effective nonlocal gravitational theories with scale-dependent Newton couplings that thermodynamically reproduce the area law while recovering standard Schwarzschild geometry in appropriate limits.

Original authors: Kimet Jusufi, Ankit Anand

Published 2026-08-10

📖 9 min read🧠 Deep dive

CC BY 4.0

Original authors: Kimet Jusufi, Ankit Anand

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, cosmic movie screen. For over a century, physicists have used a script called General Relativity to explain how gravity works on this screen. In this story, gravity isn't a force pulling things together; it's the screen itself bending and warping when heavy objects like stars or black holes sit on it. But there's a twist: when we look at the tiniest, most extreme spots on this screen—like the center of a black hole—the script starts to glitch. The math breaks down, suggesting that our current understanding of the universe is missing a few key pages.

Enter the concept of "entropy." In everyday life, entropy is a measure of messiness or disorder. Think of a clean bedroom that slowly gets messy over time; that's increasing entropy. In the world of black holes, however, entropy is a bit more magical. It turns out that the amount of "messiness" or information a black hole holds is directly tied to the size of its surface area, not its volume. This is the famous "area law." But here is the big question that keeps physicists up at night: Is this area law just a rule that black holes follow, or is it actually the source code that tells gravity how to behave? What if the "messiness" of the universe is the real architect of gravity, and the bending of space is just a side effect?

This paper by Kimet Jusufi and Ankit Anand takes a bold step to answer that question. Instead of starting with a known shape of space and calculating the entropy, they do the reverse: they start with different ideas about how entropy might behave and ask, "If the universe's messiness works this way, what does the shape of space look like?" They discover that every different type of entropy formula acts like a unique filter or lens. When you look at a black hole through these different lenses, the gravity around it changes. Specifically, the paper shows that these entropy corrections are mathematically identical to a theory called "nonlocal gravity." In this view, gravity doesn't just act on a single point; it gets "dressed up" or smeared out over a region, like a fuzzy cloud of influence rather than a sharp pinprick. The authors prove that if you accept these new entropy formulas, you must also accept that gravity itself changes its strength depending on how close you are to the source, effectively rewriting the rules of the cosmic movie screen to fix the glitches at the center of black holes.

The Great Cosmic Reverse-Engineer

Imagine you have a mysterious machine that makes a specific sound. Usually, scientists try to figure out what's inside the machine by listening to the sound. But in this paper, the authors decide to do the opposite. They say, "Let's imagine the machine makes a different sound, and then let's figure out what the machine must look like inside to produce that new noise."

In the world of physics, the "sound" is the entropy of a black hole. Entropy is a measure of how much information or "disorder" is hidden inside. For a long time, everyone agreed on one specific sound: the Bekenstein-Hawking entropy, which says the entropy is exactly proportional to the black hole's surface area. This led to the standard picture of gravity, where space bends in a very specific way (the Schwarzschild solution).

But recent theories suggest the "sound" might be different. Maybe the entropy grows faster, slower, or in a wobbly, non-straight line due to quantum effects. The authors of this paper asked: If the entropy behaves differently, what does gravity look like?

They built a mathematical bridge that connects the "sound" (entropy) directly to the "shape" of the machine (spacetime geometry). They found that every time you change the entropy formula, you aren't just changing the black hole's temperature; you are fundamentally changing how gravity works.

The Magic of "Fuzzy" Gravity

The most exciting discovery in this paper is that these new gravity shapes can be described as nonlocal.

Think of a standard light bulb. It shines from a single, sharp point. If you stand right next to it, it's blindingly bright. If you step back, it gets dim quickly. This is how standard gravity works: it comes from a sharp point (the center of the black hole).

Now, imagine you put a frosted glass globe around that light bulb. The light doesn't come from a single point anymore; it's "smeared" out. It's softer, and the way it fades as you walk away is different. The authors show that when you use these new, generalized entropy formulas, gravity acts exactly like that frosted globe. The point source of gravity gets "dressed" in a cloud of influence.

They call this a nonlocal form factor. In simple terms, it means gravity doesn't just happen at one spot; it's a relationship that stretches out. The paper shows that different entropy formulas create different "frosting" patterns:

Some create a fractional pattern, where the gravity fades in a weird, mathematical way that isn't a simple curve.
Others create an exponential pattern, where the gravity is heavily screened or hidden at very small distances, like a thick fog.
Some create a logarithmic pattern, where the point source is completely replaced by a smooth, extended cloud.

The "Running" Gravity Constant

One of the coolest metaphors the paper uses is the idea of a running Newton coupling.

In standard physics, Newton's constant ( $G$ ) is like a fixed rulebook number. It never changes. $G$ is $G$ , no matter where you are in the universe. But this paper suggests that if you accept these new entropy ideas, $G$ isn't a fixed number at all. It's more like a volume knob that changes depending on how close you are to the black hole.

They call this a scale-dependent coupling.

If you are far away, the gravity might look normal (like the standard volume).
If you get very close, the "volume" of gravity might turn down or turn up, depending on which entropy formula you use.

The authors prove that this "running" gravity is exactly what you need to make the math work. If you calculate the entropy using this new, changing gravity, it matches the input entropy perfectly. It's a self-consistent loop: the entropy changes the gravity, and the changing gravity explains the entropy.

What the Paper Rules Out

The authors didn't just find a new way to play with math; they also tested two different ways to interpret their results and ruled one of them out.

The Two Stories:

Story A: The universe is made of normal Einstein gravity, but the black hole is surrounded by a weird, invisible "matter cloud" that changes the shape of space.
Story B: The "matter cloud" isn't matter at all. Instead, the law of gravity itself has changed. The "fuzziness" is a property of spacetime, not a pile of stuff sitting on top of it.

The Verdict:
The paper argues strongly for Story B. They show that for some of the entropy formulas (like the Rényi and Tsallis-Cirto types), Story A leads to a physical impossibility. It would require the "matter cloud" to have negative entropy, which is like saying a messy room is somehow "more organized" than a clean one. That doesn't make sense in the real world.

However, in Story B, where the gravity law itself changes, everything works perfectly. The math stays consistent, and the entropy matches the area law exactly if you use the new, changing gravity constant. So, the paper concludes that these entropy corrections aren't just about adding new stuff to the universe; they are about rewriting the rules of gravity itself.

The Toolkit of Entropy

The authors didn't just stop at one idea. They took a whole toolbox of different entropy formulas that physicists have been discussing for years and applied their method to each one. They found that each formula creates a unique "signature" of nonlocal gravity:

Rényi Entropy: Creates a gravity that gets stronger at large distances, like a long-range infrared dressing.
Tsallis-Cirto & Barrow Entropy: These create "fractional" gravity. Imagine gravity behaving like a fractal pattern, where the rules change depending on the scale you look at.
Kaniadakis Entropy: This creates a gravity that is "screened" by an exponential cloud, effectively hiding the singularity at the center.
Logarithmic Entropy: This completely removes the sharp point source, replacing it with a smooth, extended distribution.
Exponential & LQG Entropy: These keep a point source but dress it in a specific, localized cloud that changes how gravity behaves right at the edge.

Why This Matters

This work is a bit like finding a universal translator between two languages that were thought to be unrelated. On one side, we have thermodynamics (the study of heat and entropy). On the other, we have quantum gravity (the study of how gravity works at the smallest scales).

For a long time, these two fields spoke different languages. This paper shows that they are actually speaking the same language, just with different accents. If you know the entropy of a black hole, you automatically know the structure of the nonlocal gravity that surrounds it.

The authors are careful to note that this is a "static" picture—it describes a black hole that isn't spinning or moving. They don't claim to have solved the entire mystery of quantum gravity yet. They haven't built a full, 4D movie of the universe with these rules; they've just built a perfect, working model for a single, stationary black hole. But it's a massive step forward. It suggests that the "messiness" of the universe (entropy) isn't just a side effect of gravity; it might be the very thing that creates the shape of gravity.

In the end, the paper suggests that if we want to understand the deepest secrets of the universe, we shouldn't just look at the geometry of space. We should listen to the entropy. Because it turns out, the universe's "messiness" might be the architect of its own structure.

Technical Summary: Entropy-Geometry Correspondence as Effective Nonlocal Gravity

Problem Statement
While the Bekenstein-Hawking entropy formula ( $S_{BH} = A/4G$ ) established a profound link between gravitation, geometry, and statistical mechanics, numerous quantum-gravitational and statistical frameworks predict corrections to this area law. These include logarithmic, nonextensive (Rényi, Tsallis-Cirto, Kaniadakis), fractal (Barrow), and loop quantum gravity (LQG) inspired entropies. Historically, these generalized entropies have been treated as derived quantities calculated from a pre-existing spacetime geometry. Consequently, the geometric significance of the entropy function itself—specifically, how a generalized entropy dictates the structure of the exterior spacetime and the nature of the gravitational interaction—remains underexplored. Furthermore, when a generalized entropy is used to reconstruct a static, spherically symmetric metric, the resulting geometry implies an effective anisotropic matter source. The physical origin of this source is ambiguous: is it a modification of the gravitational sector (nonlocal gravity) or a dressing of the matter source within standard Einstein gravity? The paper addresses this ambiguity and establishes a rigorous operator formulation linking generalized entropy to effective nonlocal gravity.

Methodology
The authors develop an operator formulation of the entropy-geometry correspondence for static, spherically symmetric spacetimes. The methodology proceeds through the following steps:

Reconstruction of Geometry: Starting from a generalized entropy function $S(r_+)$ defined at the horizon radius $r_+$ , the authors utilize a thermodynamic identity extended to the radial domain to reconstruct the metric function $f_S(r)$ . This yields the relation $f_S(r) = 1 - 4\pi M / S'(r)$ , where $M$ is the mass parameter.
Identification of Effective Sources: By comparing the reconstructed metric to the standard Schwarzschild form, the authors identify an effective cumulative mass profile $u_S(r) = 2\pi r / S'(r)$ . This profile allows the definition of an effective energy density $\rho_S(r)$ via Einstein's equations, which acts as an anisotropic fluid source ( $p_r = -\rho$ ).
Nonlocal Operator Formulation: The effective density is interpreted as the result of a nonlocal operator acting on a point source. The authors define an entropy-dependent form factor $A_S^{-2}(-\nabla^2)$ such that the effective density is $\rho_S(r) = M A_S^{-2}(-\nabla^2)\delta^{(3)}(r)$ .
Momentum Space Mapping: Using spherical symmetry, the coordinate-space kernel is transformed into momentum space to determine the nonlocal form factor $A_S^{-2}(k)$ . This establishes a direct map from the entropy function to a momentum-dependent gravitational dressing.
Running Coupling Interpretation: The cumulative profile $u_S(r)$ is reinterpreted as a scale-dependent Newton coupling $G_S(r) = G_N u_S(r)$ . This allows the field equations to be written in a compact form involving a running coupling operator $G_S(\square)$ .
Thermodynamic Consistency Check: The authors analyze whether the reconstructed geometry carries the input entropy. They compare two interpretations: (a) the geometry is sourced by dressed matter in unmodified Einstein gravity, and (b) the geometry arises from a modified gravitational sector with a running coupling. They verify which reading satisfies the first law of thermodynamics ( $dS = dA/4G_S(r_+)$ ) and physical entropy positivity constraints.

Key Contributions and Results

Operator Correspondence: The paper establishes a one-to-one correspondence $S(r) \to u_S(r) \to A_S^{-2}(-\nabla^2)$ , demonstrating that every generalized entropy uniquely determines an effective nonlocal operator, its coordinate-space source kernel, and the resulting spacetime geometry within the static spherical sector.
Running Newton Coupling: The authors derive that the entropy derivative directly determines the radial running of the gravitational coupling: $G_S(r)/G_N = 2\pi r / S'(r)$ . They show that the generalized entropy is exactly reproduced by the area law evaluated with this running coupling ( $S(r_+) = \int dA/4G_S(r)$ ), provided the gravitational sector is modified.
Resolution of Interpretational Degeneracy: The paper resolves the ambiguity between "modified gravity" and "dressed matter." It demonstrates that for specific entropies (Rényi and Tsallis-Cirto with $\delta < 1$ ), the "dressed matter" interpretation is thermodynamically inconsistent because it would require negative matter entropy to satisfy the first law. Thus, the entropy correction must be attributed to the gravitational sector itself.
Classification of Entropy Models: The formalism is applied to a wide range of entropy models, revealing distinct classes of nonlocal gravitational modifications:
- Bekenstein-Hawking: Corresponds to the identity operator and constant $G_N$ (standard Schwarzschild).
- Rényi: Generates an infrared inverse-Laplacian operator ( $1/k^2$ ), leading to a linear term in the metric and a non-asymptotically flat geometry.
- Tsallis-Cirto & Barrow: Naturally generate fractional Laplace operators ( $(-\nabla^2)^\delta$ and $(-\nabla^2)^{\Delta/2}$ ), corresponding to scale-free fractional modifications of gravity.
- Kaniadakis: Produces a non-polynomial, exponentially decaying form factor, resulting in a screened effective matter distribution.
- Logarithmically Corrected: Completely replaces the point source with a smooth extended distribution, removing the Dirac delta contribution for positive correction parameters.
- Exponentially Corrected & LQG: Retain a renormalized point source but dress it with an exponentially localized cloud, leading to exponential screening of the gravitational interaction at large distances.
Horizon Conditions: The paper identifies the conditions $S'(r) > 0$ and $S''(r) > 0$ as necessary for the reconstructed horizon to be an event horizon with positive temperature. Violations of these conditions (e.g., in Tsallis-Cirto with $\delta < 1/2$ ) lead to degenerate horizons or cosmological-type horizons rather than black holes.

Significance
The paper claims to provide a direct bridge between generalized black-hole thermodynamics and effective nonlocal gravity. Its primary significance lies in showing that generalized entropy is not merely a thermodynamic correction but encodes the underlying structure of the gravitational interaction itself. By reconstructing the nonlocal operator and the running Newton coupling directly from the entropy function, the work offers a unified language to compare diverse entropy proposals. It demonstrates that different entropy corrections correspond to qualitatively distinct classes of nonlocal gravitational dynamics, ranging from infrared modifications to fractional and non-polynomial form factors.

Crucially, the work argues that the thermodynamic consistency of the first law favors the interpretation where the entropy correction modifies the gravitational sector (running coupling) rather than the matter sector. This distinction is vital for understanding the physical viability of various quantum-gravity inspired entropy models. The authors note that while the correspondence is exact within the static, spherically symmetric sector, it serves as a symmetry-reduced representation, and a full covariant nonlocal action requires further investigation regarding tensorial consistency and causal prescriptions.