Usually, when physicists want to model the merger of two black holes, they use supercomputers. Now, however, physicists in the US have found a possible shortcut. In a study published in Physical Review Letters, they show that, of the final states permitted by energy and angular momentum, the remnant of a non-spinning black hole merger ends up in a state quite close to one that maximizes the system’s entropy – a surprising and puzzling result that reinforces the deep connection between thermodynamics and black holes.

“Our group has long believed that black hole thermodynamics could provide a new perspective on binary black hole mergers,” says team member Monica Rincon-Ramirez, a postdoctoral researcher at Pennsylvania State University. “While black hole thermodynamics is well established for equilibrium systems, much less is understood in the highly dynamical regime of mergers.”

Merging black holes

Black hole mergers occur when two black holes that orbit each other spiral inward with increasing speed, losing both energy and angular momentum by emitting gravitational waves until they eventually converge in a violent clash. After the moment of merger, the remnant oscillates vigorously, “ringing down” into a quiescent, rotating object known as a Kerr black hole that can be described in terms of just two quantities: mass and spin.

Meanwhile, the gravitational waves – ripples in the fabric of spacetime – from this energetic collision propagate through space virtually uninterrupted. These gravitational waves encode information about their source, and when we detect them on Earth (millions of light-years away from their origin), we can use them to predict the remnant’s size and spin. However, doing this requires solving complex equations from Einstein’s general theory of relativity using computationally expensive numerical simulations.

Connecting black holes and thermodynamics

Thermodynamics is the branch of physics concerned with the relationships between energy, heat, work and temperature. For ordinary matter, thermodynamics can describe how energy flows from one system to another in simple terms, even for complex systems with numerous particles and degrees of freedom. For example, rather than meticulously tracking each individual particle, concepts such as entropy – a measure of a system’s disorder – can be used to describe the final state of the entire system.

Physicists have long known that black holes obey laws that closely parallel the laws of thermodynamics, with the event horizon and surface gravity of black holes being analogous to entropy and temperature, respectively. Building on this established connection, Rincon-Ramirez and colleagues at the University of Mississippi, the University of California, Berkeley, and Northwestern University as well as Penn State investigated whether black hole thermodynamics could go beyond describing the properties of isolated, stationary black holes and provide, in addition, a principle for determining the final state of a merger.

The maximum-entropy conjecture

After accounting for the energy and angular momentum carried away by gravitational waves, the researchers tracked the evolving mass and angular momentum of a binary, mapped these values to those of hypothetical Kerr black holes, and calculated the corresponding entropy. They found that the Kerr entropy reaches a maximum at a mass and spin strikingly close to the final mass and spin predicted by numerical relativity simulations. Indeed, the two methods agreed with each other to within a few percent. “The Kerr entropy turned out to be remarkably meaningful for the binary system as a whole,” Rincon-Ramirez says.

Interpreting and presenting these results proved to be a challenge, however. “What we found was completely unexpected,” says Rincon-Ramirez. After deliberating among themselves as to why the final black hole’s parameters should lie so close to those of the maximum-entropy black hole, they ultimately chose to extend the conversation to the scientific community. By presenting their puzzling results and a broad conjecture, which they term the “maximum entropy conjecture for black hole mergers”, they hope to sharpen their findings through rigorous testing and eventually place them on a firm mathematical foundation within general relativity.

In the meantime, the researchers plan to extend their conjecture to more general binary black hole systems. By studying systems such as highly spinning and eccentric binaries, they hope to gain further understanding of the role of thermodynamic quantities in general relativity. “Ultimately, we hope this work will contribute to a more complete thermodynamic framework for highly dynamical spacetimes,” Rincon-Ramirez says.

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“Solar towers” that convert sunlight into heat could play a useful role in the renewable energy transition, but high installation costs and complex requirements limit their attractiveness to commercial operators. A team of researchers from Germany’s Karlsruher Institut für Technologie (KIT) and the German Aerospace Center (DLR) is now taking steps to change that. By creating a freely accessible database of information about how these plants work, they aim to accelerate research on solar thermal energy and drive improvements that could allow the technology to be more widely deployed in the future.

Unlike photovoltaic panels, concentrating solar power (CSP) plants do not convert sunlight directly into electricity. Instead, they generate heat by using an array of movable mirrors called heliostats to funnel sunlight onto a receiver located at the top of a tower. This heat can be used to drive a turbine to produce electricity immediately, but it can also be stored and used to generate electricity at night or on cloudy days. Depending on the plant’s location and setup, the heat could even be used directly for industrial applications and district heating.

The problem, explains study leader Kaleb Phipps, is that operating solar power tower plants safely and efficiently is a complex and expensive task. “CSP technologies have the potential to help the transition to a renewable energy system,” says Phipps, a research fellow at KIT’s Scientific Computing Center. “However, they are still not financially viable and there are operating challenges associated with them.”

Before researchers can develop and reliably test new, improved processes, Phipps explains that they need access to real-world operational CSP data. “We believe that modern data-driven machine learning methods like ours can help solve a lot of these problems,” he says.

849 gigabytes of operational data

To construct their database, which is known as PAINT, researchers collected data from the CSP tower near Jülich, Germany. These data include the exact positions of the tower’s 2014 mirrors, their dimensions, how they are warped and the way they rotate and tilt. The team also obtained fine-grained weather data for the entire 2021-2024 period under study, plus a further 218 000 images to help determine whether the mirrors were efficiently directing sunlight precisely to the top of the tower.

All in all, the PAINT database contains 849 gigabytes of operational data organized using the SpatioTemporal Asset Catalog (STAC) standard. To make these data FAIR (findable, accessible, interoperable and reusable), the researchers developed software to simplify access to the database and then published the data in a way that is freely available to all users.

Their goal, they say, is for other researchers to use this FAIR database to develop “digital twins”, or virtual replicas, of real-world CSP plants. When combined with machine-learning models, such twins can be used to answer questions such as whether the mirrors are aligned for maximal light absorption (one of the most important parameters for efficient operation) in near real time.

Community reaction

“The fact that we are receiving many positive responses regarding the data and that people seem to appreciate it is our biggest success,” Phipps says. Such responses are especially gratifying, he adds, because constructing databases like PAINT can be an arduous task.

“The main difficulties were about collecting the data, cleaning and unifying it into standard formats and parsing and organising all the metadata, which was originally complete chaos,” Phipps recalls. “We therefore put a lot of time and effort into creating standardized and automated workflows so that the process is as transparent as possible and can be reused for future data.”

The team also faced challenges when trying to convince peers that this work counted as “research”, he adds. “The fact that the data itself didn’t directly result in a new algorithm or method meant that lots of people didn’t believe we would be able to publish the results or achieve anything with them,” says Phipps. “Luckily, we were able to prove the doubters wrong.”

While the PAINT database currently contains data from only one CSP, the KIT researchers, who report their work in Nature Energy, are hoping to expand it. “We really need contributions from other CSP power plants around the world to transform the database into a truly international and diverse source for researchers,” Phipps says. “As data from different facilities is added in the future, it will be possible to develop a common standard for open operational data in solar tower research. This would speed up the development and promote a widespread adoption of this technology.”

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