⚛️ high-energy theory

Information--Theoretic Black Hole Entropy II: Infrared Gravity and Charged/Rotating Extensions

This paper extends an information-theoretic black hole entropy, defined via Kullback-Leibler divergence and satisfying the Nernst third law, to charged and rotating Kerr-Newman cases while demonstrating that its thermodynamic properties can be reproduced by an infrared deformation of General Relativity that generates a small positive effective cosmological term.

Original authors: Alex Kehagias

Published 2026-08-07

📖 8 min read🧠 Deep dive

CC BY 4.0

Original authors: Alex Kehagias

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 library. In this library, every black hole is a special book. For decades, scientists have known that the "size" of the cover of this book (its surface area) tells us how much information is hidden inside. This is the famous "area law" of black hole physics. But there's a catch: the old rules of this library had a glitch. They suggested that if a black hole got infinitely heavy and cold, the amount of information inside would explode to infinity. That doesn't make sense in the real world. In everyday thermodynamics—the science of heat and energy—there's a rule called the "Third Law." It says that as things get colder and colder, approaching absolute zero, their disorder (entropy) should settle down to a specific, finite number, not go wild. It's like a noisy party that eventually quiets down to a single, steady hum rather than getting louder and louder forever.

This paper tackles that glitch. The author is asking: "What if the rules of gravity themselves are slightly different than we thought, just enough to fix this cold-black-hole problem?" They are exploring a new way to think about black hole entropy not as a count of hidden rooms, but as a measure of "missing information" compared to a perfectly random guess. They want to know if we can tweak the laws of gravity to make the math work, and if these new rules apply to spinning or electrically charged black holes, not just the simple, still ones.

The Story of the "Perfectly Quiet" Black Hole

In the world of black holes, there's a classic puzzle. According to the standard rules of General Relativity, a black hole's temperature drops as it gets heavier. If you keep adding mass, the black hole gets colder and colder. But here's the weird part: as it gets colder, the standard math says its "entropy" (a measure of how many ways the black hole could be arranged internally) gets bigger and bigger, eventually becoming infinite. It's like a library that keeps adding more and more books as it gets quieter, until the shelves collapse under the weight of infinite possibilities. This breaks the "Third Law" of thermodynamics, which demands that at the coldest possible temperature, the system should settle into a calm, finite state.

In a previous paper, the author proposed a fix. They suggested that black hole entropy isn't just a raw count of microstates, but a measure of relative information. Imagine you have a bag of coins. If the coins are perfectly fair (50% heads, 50% tails), you have maximum uncertainty. But if the coins are biased (say, 90% heads), you have less uncertainty because you can predict the outcome better. The author treats a black hole like a biased coin. The "bias" depends on the black hole's mass. As the black hole gets heavier and colder, this bias approaches a limit, and the "missing information" stops growing. Instead of exploding to infinity, the entropy hits a ceiling, a maximum value, satisfying the Third Law.

Rewriting the Rules of Gravity

The big question this paper asks is: What kind of gravity theory would actually produce this behavior? If the standard rules of Einstein don't work, what do we need to change?

The author suggests that gravity isn't just the smooth, simple curve we see in textbooks. They propose that at very large scales (infrared gravity), the rules get a little "wobbly." They introduce a new, universal mass scale called $M_0$ . Think of $M_0$ as a cosmic speed limit for mass. You can't have a black hole heavier than this. As a black hole approaches this limit, the laws of gravity subtly shift to ensure the temperature drops to zero and the entropy stops growing.

To prove this works, the author built a mathematical model. They took the standard equations of gravity and added some "extra terms"—like adding a few new ingredients to a recipe. These ingredients involve complex shapes of space and time (curvature invariants). When they ran the math, they found that these extra terms naturally created the exact temperature and entropy curve they were looking for. It's as if they found a hidden knob on the gravity machine that, when turned just right, makes the black hole obey the Third Law.

Interestingly, this new gravity theory also predicts a tiny, positive push on the universe, known as a cosmological constant. This is the force that makes the universe expand. The author found that the size of this push is directly linked to the size of the black hole's entropy limit. If the universe is huge (which it is), then the "cosmological constant" is tiny, which matches what we observe. It's a neat connection: the reason the universe is expanding so slowly might be tied to the reason black holes have a maximum amount of information.

Spinning and Charged Black Holes

So far, we've only talked about simple, still black holes. But real black holes often spin like tops and carry electric charge. Does the new rule work for them too?

The author says yes, but with a twist. For a spinning or charged black hole, you can't just use the total mass. Why? Because some of that mass is "spinning energy" or "electric energy" that you could, in theory, siphon off without changing the black hole's core size. It's like a spinning top: you can slow it down, but the wood it's made of stays the same.

To fix this, the author uses a concept called irreducible mass. This is the mass of the black hole that cannot be removed, no matter how much energy you extract. It's the "core" of the black hole, tied directly to the size of its event horizon (the point of no return). The author proposes that the new entropy formula should use this "irreducible mass" instead of the total mass.

When they do this, the math works beautifully. The entropy still hits a maximum limit as the black hole gets cold, regardless of how fast it spins or how much charge it holds. This means the "Third Law" holds true for all types of black holes, not just the simple ones.

The Difference Between "Geometric" and "Statistical" Cold

One of the most fascinating discoveries in this paper is a distinction between two ways a black hole can get cold.

Geometric Extremality: This happens when a black hole spins so fast or has so much charge that its inner and outer horizons merge. In standard physics, this is a "zero temperature" state. But the author argues this is just a geometric trick. The black hole looks cold because of how its shape is arranged, but the microscopic "coins" inside are still flipping around. It's like a room that looks empty because the furniture is arranged in a perfect circle, but the people inside are still talking.
Statistical Freezing: This is the true "Third Law" endpoint. This happens when the black hole reaches that universal mass limit $M_0$ . Here, the "coins" inside actually stop flipping. The system reaches a state of maximum order (or rather, a fixed state of relative information). This is the true "frozen" state.

The author emphasizes that these two states are different. A black hole can be "geometrically cold" (extremal) but still have microscopic activity. It only becomes "statistically cold" when it hits the universal mass limit. This distinction helps clear up confusion about what happens at the very edge of black hole physics.

The Big Picture

This paper doesn't claim to have solved the mystery of quantum gravity. Instead, it offers a compelling "existence proof." It shows that it is possible to tweak the laws of gravity in a specific way (adding infrared deformations) to make black hole thermodynamics behave nicely, obeying the Third Law and matching our observations of the universe's expansion.

The author suggests that the universe might have a built-in "maximum mass" scale, $M_0$ , which acts as a cosmic speed limit. This scale isn't arbitrary; it's linked to the cosmological constant, the force driving the universe's expansion. If the universe is huge, this scale is huge, and the corrections to gravity are tiny—just enough to fix the black hole entropy glitch without breaking the rest of physics.

In short, the paper paints a picture where black holes are not just simple pits of gravity, but complex information systems that obey the same rules of heat and cold as a cup of coffee or a spinning top. By rethinking entropy as a measure of "missing information" rather than just a count of states, and by tweaking gravity to include a universal mass limit, the author provides a consistent story that connects the smallest black holes to the largest scales of the cosmos. It's a reminder that even in the most extreme corners of the universe, the rules of information and thermodynamics still hold the key.

Technical Summary: Information–Theoretic Black Hole Entropy II: Infrared Gravity and Charged/Rotating Extensions

Problem Statement
The standard Bekenstein-Hawking entropy ( $S_{BH} \propto M^2$ ) for Schwarzschild black holes satisfies the first and second laws of thermodynamics but violates the Nernst formulation of the third law. As the Hawking temperature $T \to 0$ (formally occurring as mass $M \to \infty$ ), the entropy diverges rather than approaching a universal constant. Furthermore, the microscopic origin of this entropy remains unclear, as it is traditionally expressed via geometric horizon area rather than a probability distribution over microstates. While string theory reproduces the area law for specific supersymmetric cases, a universal statistical interpretation for generic black holes is lacking. This paper, a companion to a previous work [14], addresses two specific questions arising from a proposed information-theoretic entropy function $S_{bh}(M)$ that satisfies the third law:

What modification to the gravitational theory (specifically in the infrared regime) reproduces this entropy and its associated temperature?
How does this entropy formula generalize to charged and rotating (Kerr-Newman) black holes?

Methodology
The author employs a perturbative approach to construct an effective gravitational theory and extend the statistical framework to rotating/charged systems.

Infrared Deformation of General Relativity:
The author proposes an effective action containing higher-curvature and inverse-curvature operators suppressed by a universal mass scale $M_0$ . The action takes the schematic form:
$I = \frac{1}{16\pi G_N} \int d^4x \sqrt{-g} \left( R - c_0 M_0^{-2} + c_1 M_0^{-4} I_1 + \dots \right)$
where $I_1$ is a specific curvature invariant constructed from the Riemann tensor. The author seeks static, spherically symmetric solutions perturbatively in the couplings $c_i$ . They calculate the horizon radius $r_H$ and the Wald entropy ( $S_W$ ) using the Noether-charge formula. The couplings are fixed by demanding that the perturbative expansion of $S_W$ matches the information-theoretic entropy derived in the companion paper.
Extension to Kerr-Newman Black Holes:
For charged and rotating black holes, the author argues that the ADM mass $M$ is not the appropriate variable for the entropy function because it includes extractable energy (rotational and electromagnetic). Instead, they propose replacing $M$ with the irreducible mass ( $M_{irr}$ ), which is directly tied to the horizon area and invariant under reversible extraction processes (Christodoulou-Ruffini mass formula). The generalized entropy is defined as $S_{bh}(M_{irr})$ .
Thermodynamic Analysis:
Using the first law of black hole mechanics ( $dM = T dS + \Omega dJ + \Phi dQ$ ), the author derives the temperature for the generalized system. They analyze the behavior of the temperature and entropy in two distinct zero-temperature limits: the geometric extremal limit (where inner and outer horizons coincide) and the statistical limit where $M_{irr} \to M_0$ .

Key Contributions and Results

Infrared Gravity Realization: The paper demonstrates that the information-theoretic entropy $S_{bh}(M)$ and its corresponding temperature can be reproduced by an infrared-deformed theory of gravity. To linear order in the deformation couplings, the Wald entropy and surface-gravity temperature of the modified theory agree exactly with the information-theoretic results. The required deformation involves non-analytic operators (inverse powers of curvature invariants) rather than standard higher-curvature terms, which typically yield corrections scaling as inverse powers of mass.
Explicit Coupling Fixing: By matching the perturbative entropy expansion, the author fixes the coefficients of the effective action:
$c_0 = \frac{1}{8G_N^2}, \quad c_1 = -\frac{1}{40G_N^4}$
This matching generates a positive effective cosmological constant $\Lambda_{eff} = c_0 / (2M_0^2)$ .
Generalization to Kerr-Newman: The entropy formula is successfully extended to charged and rotating black holes by substituting the ADM mass with the irreducible mass $M_{irr}$ . This preserves the connection to the horizon area and the semiclassical limit while ensuring the entropy depends only on the non-extractable "core" of the black hole.
Distinction Between Extremality and the Third Law: A crucial result is the distinction between two zero-temperature limits:
1. Geometric Extremality: Occurs when the horizon degenerates ( $\Delta \to 0$ ). Here, the Hawking temperature vanishes due to a mechanical factor, but the underlying statistical ensemble (Bernoulli distribution) retains non-zero fluctuations and entropy.
2. Statistical Freezing ( $M_{irr} \to M_0$ ): This is the endpoint required by the Nernst third law. Here, the statistical temperature vanishes, and the entropy approaches a universal constant $S_0 = N \ln 2$ , independent of charge or angular momentum.
  The paper concludes that geometric extremality does not imply the statistical freezing of microstates; the latter is a distinct limit governed by the universal scale $M_0$ .
Cosmological Constant Connection: The universal scale $M_0$ is identified with the observed cosmological constant $\Lambda_{obs}$ . The smallness of $\Lambda_{obs}$ is mapped to the largeness of the microscopic parameter $N$ (the number of bits in the ensemble), with $\Lambda_{eff} \sim M_P^2 / N$ . This suggests $M_0$ is a fundamental constant of nature rather than a contingent bound related to the mass of the universe.

Significance and Claims
The paper claims to provide a consistent framework where the Bekenstein-Hawking area law emerges as the leading approximation to a relative entropy (Kullback-Leibler divergence) between a mass-biased ensemble and an unbiased reference.

Thermodynamic Consistency: The construction resolves the conflict between black hole thermodynamics and the Nernst third law by introducing a universal infrared scale $M_0$ .
Gravitational Origin: It establishes that such thermodynamics can arise from a specific class of infrared-deformed gravitational actions, providing an existence proof for a theory where the third law is satisfied.
Statistical Interpretation: The entropy is reinterpreted not as the logarithm of the number of microstates, but as a measure of relative information (entropy deficit). Consequently, the finite entropy at geometric extremality is not interpreted as a ground state degeneracy but as a non-frozen statistical state.
Modesty: The author explicitly states that their construction is an effective description. They do not claim to have identified the unique UV-complete theory of quantum gravity, noting that additional physical input is required to select the specific curvature invariants. Furthermore, they distinguish their KL-entropy approach from supersymmetric BPS state counting, acknowledging that their Nernst interpretation applies to the thermodynamic family of Bernoulli distributions rather than protected BPS sectors.

In summary, the paper links the third law of black hole thermodynamics to an infrared modification of gravity and a universal cosmological scale, offering a statistical-mechanical interpretation of black hole entropy that remains consistent with the third law and distinguishes between geometric and statistical zero-temperature limits.