Simple observations concerning black holes and probability

It is argued that black holes and the limit distributions of probability theory share several properties when their entropy and information content are compared. In particular the no-hair theorem, the entropy maximization and holographic bound, and the quantization of entropy of black holes have their respective analogues for stable limit distributions. This observation suggests that the central limit theorem can play a fundamental role in black hole statistical mechanics and in a possibly emergent nature of gravity.

💡 Research Summary

The paper investigates a striking parallel between black‑hole physics and the theory of probability, focusing on how entropy and information are treated in both domains. It begins by recalling three cornerstone concepts of black‑hole thermodynamics: the no‑hair theorem, which states that a black hole is completely characterized by only a few macroscopic parameters (mass, charge, angular momentum); the principle that a black hole maximizes entropy for given macroscopic constraints, leading to the Bekenstein–Hawking entropy proportional to the horizon area; and the holographic bound, which limits the amount of information that can be stored within a region to a quantity proportional to its surface area measured in Planck units.

The author then turns to probability theory, summarizing the central limit theorem (CLT) and its generalization to stable laws. The CLT tells us that the sum of many independent, identically distributed random variables converges to a Gaussian distribution, which is fully specified by just two parameters: mean and variance. More generally, Lévy‑stable distributions arise when the underlying variables have heavy tails; these distributions are characterized by a stability index (the tail exponent) and a scale parameter, again a very small set of numbers. In both cases the microscopic details of the constituent variables are “washed out,” leaving only a universal macroscopic shape.

Four concrete analogies are drawn. First, the no‑hair theorem mirrors the universality of stable limit distributions: regardless of the intricate microstructure, the macroscopic description collapses to a few parameters. Second, entropy maximization in black‑hole thermodynamics corresponds to the fact that, under fixed mean and variance (or fixed stability index and scale), the Gaussian (or Lévy‑stable) distribution maximizes Shannon entropy. Third, the holographic bound is likened to the information‑theoretic limit in probability theory: decreasing the variance (or the scale of a Lévy law) reduces entropy, just as shrinking a black‑hole horizon to the Planck area reduces the number of accessible microstates. Fourth, the quantization of black‑hole entropy—often expressed as the horizon area being an integer multiple of the Planck area—is compared with a possible discretization of the scale parameter in stable laws. The paper argues that, in certain quantum‑gravity models, the scale could take on a discrete spectrum, reproducing the same “area‑quantum” structure seen in black‑hole physics.

These correspondences lead the author to propose that the central limit theorem and its stable‑law extensions may underlie the statistical mechanics of black holes. In this view, a black hole is not a mysterious object with hidden degrees of freedom, but rather the macroscopic manifestation of a huge ensemble of microscopic quantum‑gravitational degrees of freedom whose collective behavior is governed by universal limit theorems. Consequently, gravity itself could be an emergent, statistical phenomenon, with the CLT providing the mathematical backbone for the emergence of spacetime geometry and its thermodynamic properties.

The paper concludes by outlining future research directions: constructing explicit microscopic models of quantum gravity that reproduce stable‑law statistics; establishing a rigorous mapping between the quantized horizon area and the discrete spectrum of stable‑law scale parameters; and re‑examining holographic entropy bounds through the lens of information theory and limit theorems. By highlighting the deep structural similarity between black‑hole entropy and probability‑distribution entropy, the work opens a novel pathway toward understanding gravity as an emergent, statistically driven phenomenon.