Dynamic black holes may follow thermodynamic laws after all, Penn State physicists propose

For half a century, the laws of black hole mechanics formulated by Stephen Hawking and collaborators have stood as one of the most elegant bridges between general relativity and thermodynamics. But those laws carried a serious limitation: they only applied to black holes sitting quietly at equilibrium, unchanging over time. Real black holes, however, are anything but quiet. They form from collapsing stars, swallow matter, merge with other black holes, and slowly evaporate through Hawking radiation. A new study led by physicists at Penn State University now proposes a way to extend those celebrated thermodynamic laws to dynamic, out-of-equilibrium black holes, potentially reshaping how we understand entropy in extreme gravitational environments.

The research, published on June 24, 2026 in Physical Review Letters and highlighted as an editor’s suggestion, was carried out by Abhay Ashtekar, Atherton University Professor and Evan Pugh Professor of Physics Emeritus at Penn State, together with graduate students Daniel E. Paraizo and Jonathan Shu.

The problem with event horizons

The original framework of black hole thermodynamics, established by Bardeen, Carter, and Hawking in the early 1970s, drew direct analogies between black hole mechanics and classical thermodynamics. Hawking showed that the area of a black hole’s event horizon is proportional to its entropy and that its temperature is related to its surface gravity. These results opened a remarkable conceptual window: objects governed by Einstein’s curved spacetime equations appeared to obey the same statistical rules as a gas in a box.

However, event horizons have a peculiar property that undermines this analogy in dynamic situations. The event horizon is a global, teleological construct. It is defined not by what is happening at the black hole right now, but by the entire future history of the spacetime. In practical terms, this means the event horizon can begin to grow in regions of flat, empty spacetime before any matter has even reached the black hole. Its properties depend on events that may or may not happen in the future.

«In dynamic situations, event horizons can form and grow in what we call flat regions of space-time, where nothing is happening,» explained Jonathan Shu, co-author of the study. «This makes them teleological. Their properties cannot be determined just by the local physics of the black hole but instead rely on prediction of events that may or may not happen in the future.»

This teleological character makes the area of the event horizon a problematic candidate for physical entropy in dynamic black holes. Physical quantities, by their very nature, should be determined by the local, present state of a system, not by hypothetical future developments.

Dynamical horizons as the solution

Ashtekar and his collaborators propose replacing event horizons with so-called dynamical horizons, a quasi-local framework that Ashtekar himself helped develop over two decades of prior research. Unlike event horizons, dynamical horizons are defined using the local properties of the black hole at each instant in time. They are surfaces of marginally trapped light rays, surfaces where outgoing light just barely fails to escape, determined entirely by the geometry of spacetime right there and then.

Using mathematical structures known as dynamical horizon segments (DHSs), the team extended the first law of black hole mechanics to encompass black holes that are arbitrarily far from equilibrium. In the classical Bardeen-Carter-Hawking formulation, the first law relates infinitesimal changes between nearby equilibrium states without specifying any physical process that drives the transition. The new formulation, by contrast, deals with finite changes that occur due to actual physical processes at the horizon, such as fluxes of energy crossing the black hole boundary.

The extension to the second law follows naturally. The generalized second law on dynamical horizon segments, building on earlier work by Ashtekar and Badri Krishnan, shows that the area of the dynamical horizon never decreases when matter satisfying the null energy condition falls in. Moreover, this result is now a quantitative statement: it directly relates the change in the horizon area to the energy fluxes entering the black hole, providing a concrete accounting of entropy production rather than just stating that entropy cannot decrease.

Black hole entropy redefined

One of the most profound implications of this framework is a conceptual shift in how black hole entropy should be understood. Instead of identifying entropy with the area of the event horizon (the Bekenstein-Hawking formula S = A/4G), the new formulation identifies it with the area of marginally trapped surfaces inside quasi-local horizons. In stationary eras, the two coincide. But in dynamic situations, the dynamical horizon gives a physically more meaningful measure of entropy because it responds only to local physics.

This distinction matters enormously for two of the most important phenomena in black hole physics. First, for evaporating black holes in quantum theory, where the Hawking radiation gradually carries energy away from the black hole. Second, for black hole mergers, the violent events detected by the LIGO-Virgo-KAGRA collaboration through gravitational waves. In both cases, the system is profoundly far from equilibrium, and a proper thermodynamic description requires exactly the kind of quasi-local framework that Ashtekar, Paraizo, and Shu have developed.

«We can apply these generalized laws to better understand evaporating black holes in quantum theory and black hole mergers, like those detected by the LIGO-Virgo-KAGRA collaboration using gravitational waves,» said Ashtekar.

A framework two decades in the making

The current result is not an isolated breakthrough but rather the culmination of a research program spanning more than twenty years. Ashtekar and Krishnan introduced the concept of dynamical horizons and established their basic properties in the early 2000s. Since then, a broader community of researchers has explored related ideas, including the Hollands-Wald-Zhang entropy formula proposed in 2024, which independently arrived at the conclusion that entropy should be associated with the apparent horizon rather than the event horizon, at least for perturbative departures from equilibrium.

What makes the Penn State result distinctive is its generality. The new first law applies to black holes that can be arbitrarily far from equilibrium, not just small perturbations around a stationary background. This non-perturbative extension brings the thermodynamic analogy much closer to the full physical reality of astrophysical black holes.

The research was supported by the Penn State Atherton Professorship Program and the Penn State Eberly College of Science.

Why it matters

If the identification of entropy with the area of dynamical horizons holds up under further theoretical scrutiny, it would represent a significant conceptual advance. It would mean that the thermodynamic character of black holes is not merely an analogy limited to idealized, unchanging configurations, but a robust physical feature that persists through the most violent processes in the universe: mergers, accretion, and quantum evaporation. For the broader quest to reconcile general relativity with quantum mechanics, understanding entropy in dynamic gravitational systems remains one of the central open problems, and the quasi-local horizon framework provides a concrete path forward.

The paper «Thermodynamics of Black Holes, Far from Equilibrium» by Abhay Ashtekar, Daniel E. Paraizo, and Jonathan Shu is published in Physical Review Letters, volume 136, article 251405 (2026). A detailed companion paper (arXiv: 2604.00170) provides the full mathematical treatment.

© 2026 SKYCR.ORG | Homer Dávila Gutiérrez, FRAS. All rights reserved. Total or partial reproduction prohibited without express authorization. Original source: Penn State University / Physical Review Letters DOI: 10.1103/3c1r-v8f1.