A fussy thermometer shows different temperatures depending on who is looking at it. A black hole behaves much the same way: its thermal properties, entropy and temperature, depend on how we describe the information hidden inside. In a new study, physicists propose measuring entropy via the KL divergence — a quantity that tells how much one probability distribution differs from another.
The authors start with a Bernoulli ensemble that depends on the black hole mass and compare it to an "unbiased" reference. It turns out that this information-based approach exactly reproduces the surface gravity temperature and Wald entropy, but not in standard general relativity, rather in a slightly deformed version of it. This deformation acts at large distances and, curiously, generates a positive cosmological constant on its own. Its smallness is naturally explained by a huge parameter N — the number of microscopic states.
The idea extends to rotating, charged Kerr–Newman black holes using the concept of irreducible mass. This move preserves the connection between entropy and horizon area and the correct semiclassical limit. At the same time, geometric extremality (the limiting case of maximal rotation or charge) no longer coincides with the universal point of statistical freezing — the latter turns out to be independent of angular momentum and charge. It's as if the fussy thermometer has finally been calibrated: now it shows not "extreme" values but the true thermal death of the system.
as written by the authors arXiv:2608.05801v1
We investigate gravitational and Kerr-Newman extensions of an information-theoretic black-hole entropy that satisfies the Nernst formulation of the third law of thermodynamics. The entropy is identified with the Kullback-Leibler divergence between a mass-dependent Bernoulli ensemble and an unbiased reference ensemble, and therefore measures relative information rather than the logarithm of the number of black-hole microstates. We show perturbatively that its temperature and entropy can be reproduced by an infrared deformation of General Relativity. To linear order in the deformation couplings, the surface-gravity temperature and the Wald entropy agree with the information-theoretic results. In an explicit realization, the matching generates a positive effective cosmological term whose smallness is related to the large microscopic parameter $$N$$. We also propose an extension to Kerr-Newman black holes based on the irreducible mass. This construction preserves the connection with the horizon area and the semiclassical limit, while distinguishing geometrical extremality from the universal statistical-freezing endpoint, which is independent of angular momentum and charge.
The irreducible mass of a black hole is the part of its total mass that cannot be extracted by any process, even if you stop the rotation and neutralize the charge. It is directly related to the horizon area and entropy.