When two black holes orbit each other, they will eventually spiral inward and collide in one of the most violent phenomena in the universe. The event is so energetic that it significantly distorts the universe around it. It emits gravitational waves - ripples in the fabric of spacetime - that are strong enough to be detected with precision instruments on Earth even when they originate billions of light-years away. These gravitational waves carry information about the event that physicists use to predict the size of the merger's resulting new larger black hole - referred to as a remnant. But accurate predictions involve complex equations originally developed by Einstein as part of his theory of general relativity that require supercomputers to solve. Now, a team of researchers, led by physicists at Penn State, have shown that there may be simpler way, which also points towards obtaining a deeper understanding of the physics contained in those complex equations.
A paper describing the research published today (July 7) in the journal Physical Review Letters, where it is being highlighted as an "editor's suggestion."
"The final black hole after a merger is ringing like a struck bell, and it radiates away more gravitational waves until it settles into a calm, stable state described by just two numbers - its final mass and spin," said Monica Rincon-Ramirez, a postdoctoral scholar in physics in the Penn State Eberly College of Science and the first author of the paper. "The question we asked is: Can we predict what that final state looks like using arguments from thermodynamics?"
Thermodynamics is the branch of physics that studies how quantities such as energy, heat and entropy determine the macroscopic behavior of systems containing many interacting particles - from gases in engines and the atmosphere to everyday activities such as cooking. General relativity, on the other hand, describes gravity through the geometry of spacetime, and is thus primarily needed to describe astronomical observations. Before Stephen Hawking showed that black holes could radiate energy, they were generally believed to fall outside the realm of thermodynamics.
Another recent study from Penn State overcame a limitation of Hawking's formulation of black hole mechanics, making them applicable to dynamic black holes that form, merge and evaporate.
"The concepts and laws of thermodynamics apply to systems with many particles, like gases," said Nathan K. Johnson-McDaniel, a postdoctoral researcher at the University of Mississippi who earned a doctorate in physics at Penn State in 2011 and is an author of the paper. "Usually, we are interested in predicting the coarse-grained properties of these gases and not what every molecule is doing. Black holes, on the other hand, are described by the deterministic equations of general relativity and seemingly have no relationship with the gases. But starting in the 1970s, leading physicists found an interesting parallel between the properties of black holes and those of gases. We wanted to extend this analogy to binary black hole systems."
The new work suggests that once the energy and angular momentum - a measure of the system's rotational motion - carried away by gravitational waves are properly accounted for, the final black hole appears to be the state that maximizes entropy, the measure of randomness in a system, tracking the natural tendency of the universe to go from a state of order to a state of chaos.
"Entropy is essentially a measure of disorder, or more precisely, of how many ways something can be arranged," said Vaishak Prasad, a postdoctoral researcher in astronomy and astrophysics at Penn State and an author of the paper. "A messy room has high entropy - there are countless ways things can be strewn about. A perfectly tidy room has low entropy - there are only a few arrangements that count as 'tidy.' Nature tends to drift toward high entropy states simply because there are more of them. Our results suggest that black hole mergers do something similar."
The team developed what they call the "maximum entropy conjecture for black hole mergers," which is strikingly similar to ordinary thermodynamics.
"When two hot gases are brought into contact, one does not need to track every microscopic interaction of the molecules in the gases to determine the final state of the combined gas," said Eugenio Bianchi, professor of physics at Penn State and an author of the paper. "Maximizing entropy, while accounting for other physical laws, predicts the outcome."
The team's new conjecture suggests that a related principle may govern black hole mergers.
"The central finding emerged from studying how the merging black holes' evolving mass and angular momentum map onto those of a sequence of hypothetical rotating black hole remnants," Rincon-Ramirez said. "Remarkably, we observe that the entropy of this sequence reaches a maximum at values strikingly close to the mass and angular momentum of the actual final remnant predicted independently by numerical relativity simulations. The agreement is within a few percent."
When two black holes collide and merge to form a single black hole, the remnant black hole left behind seems to "forget" almost everything about the collision, except its mass and spin, the researchers explained.
"We found that the most natural way to describe what it does remember can be explained using thermodynamic concepts," said B.S. Sathyaprakash, Elsbach Professor of Physics and professor of astronomy and astrophysics in the Penn State Eberly College of Science, the leader of the research team. "This work explores a surprising possibility at the intersection of gravity, black hole physics and thermodynamics that goes beyond the established laws of black hole mechanics and thermodynamics and raises a potentially transformative question: Could entropy maximization be a fundamental organizing principle governing black hole interactions more generally?"
In addition to Rincon-Ramirez, Johnson-McDaniel, Prasad, Bianchi and Sathyaprakash, the research team included Ish Gupta, a postdoctoral researcher at the University of California, Berkeley, who earned a doctorate in physics at Penn State in 2025. The U.S. National Science Foundation funded the research.
