Black holes, regions in space where gravity is so strong that nothing can escape, have been widely studied over the past decades, due to their unique and intriguing properties. Einstein’s theory of general relativity predicts that black holes obey a set of rules, known as the laws of black hole mechanics. These rules somewhat resemble the laws of thermodynamics, which delineate how energy, heat, and entropy behave in our universe.
The 3rd law of black hole mechanics states that an extremal black hole, or in other words, a black hole that is spinning or charged to its absolute theoretical limit, cannot realistically form in a finite amount of time.
Extremal black holes are predicted to have a surface gravity of zero, thus they do not emit standard Hawking radiation and would not evaporate in a vacuum. This specific characteristic of extremal black holes is known as “zero temperature.”
Researchers at the University of Cambridge ran numerical simulations exploring the possibility that extremal rotating black holes could be formed in finite time in vacuum gravity (i.e., a scenario in which spacetime evolves without the presence of matter, gas, or radiation).
Their paper, published in Physical Review Letters, suggests that, in vacuum gravity, an extremal black hole could form from a pre-existing Schwarzschild black hole (i.e., a non-rotating black hole with a mass but no electric charge), violating the 3rd law of black hole mechanics.
“One of the most profound insights in the history of black holes was Jacob Bekenstein’s proposal that they are thermodynamic objects, obeying what are called the four laws of black hole mechanics (as formulated by Bardeen, Carter, and Hawking in 1973)—in correspondence with the four laws of thermodynamics,” John R. Crump, first author of the paper, told Phys.org.
“Accordingly, they have entropy, given by the black hole’s surface area, and a temperature, given by the surface gravity. In line with thermodynamics, we expect that the entropy should only increase with time (the famous 2nd law) and we also expected that it should not be possible to reduce their temperature to absolute zero (the 3rd law).”
In 1986, physicist Werner Israel published a paper that offered theoretical “proof” for the 3rd law of black hole mechanics. The proof he provided, however, included an assumption that appeared physically reasonable, but that was recently found to be flawed.
“In 2022, Ryan Unger and Christoph Kehle sent shockwaves through the community when they realized that this assumption was flawed and published their thunderbolt of a paper that proved that the 3rd law is in fact false—under some conditions,” said Crump.
“Namely, the black holes they studied are electrically charged and reach zero temperature by absorbing idealized charged matter. This was huge news, and the natural question on everyone’s minds was ‘if the 3rd law is false for charged black holes, what about for black holes without any matter at all?'”
Simulating black holes in five dimensions
Building on the earlier theoretical study by Unger and Kehle, the researchers set out to further explore a scenario that might violate the 3rd law of black hole mechanics. Specifically, they explored the possibility that a black hole could reach zero temperature in finite time simply by absorbing gravitational waves.
“Readers may be wondering why our result is for black holes in five dimensions (5d) rather than four, as nobody I’ve met has ever accidentally turned around and ended up in an as-yet unnoticed extra spatial dimension,” explained Crump.
“The reason for this is a technical one—surprisingly, it’s considerably easier to set up the problem in 5d because there is a symmetry that can be exploited, and this symmetry isn’t there in 4d.
“On a deeper level, showing this result for vacuum gravity in 5d is still a coup that enriches our understanding of general relativity—after 111 years it feels as though Einstein’s theory of gravity usually only becomes more mysterious rather than less.”
The team relied on a tool called characteristic gluing, which was also employed by Unger and Kehle in their paper. As suggested by its name, this approach entails “gluing” together (i.e., connecting) different spacetime regions into a larger spacetime that satisfies Einstein’s equations.
“Characteristic gluing can be used to ‘glue’ together different patches of spacetime—if you can find something that correctly interpolates between those patches,” said Crump.
“In our case, we glued an initial region containing a Schwarzschild black hole to a final region with a zero-temperature black hole (called an extremal Myers-Perry black hole),” said Crump. “To be really thorough, we also did the gluing with an initial region that didn’t contain a black hole at all—the extremal black hole was formed through collapsing gravitational waves alone.”
Unger and Kehle tackled this problem using mathematical equations alone. In contrast, Crump and his colleagues also relied on numerical simulations and computational tools, particularly artificial neural networks (ANNs).
“We trained a neural network to find the interpolating region of spacetime,” explained Crump. “Instead of outputting next-word predictions from input prompts, like conventional large language models (LLMs) do, our neural networks take coordinates as input and output the spacetime geometry. To ‘train’ them, we used the same methods that are used to train ChatGPT and other conversational platforms based on LLMs.”
Reshaping fundamental physics theories
The results of the team’s numerical simulations suggest that in 5d the 3rd law of black hole mechanics is false. This 3rd law violation holds irrespective of the theoretical model of matter that one is relying on when performing analyses.
“Whether your matter model is something like the standard model, or dark matter, or charged fluids, or anything like that, the 3rd law is still false because gravity is all you need to make the black hole reach zero temperature,” explained Crump.
“Zero temperature black holes are strange, exotic objects with many quirky properties (which is part of the reason that the 3rd law was originally thought to be true), so the fact that they can be formed opens the door to all sorts of phenomena. Hopefully, this is a rabbit hole that will uncover even more secrets of general relativity.”
This recent study could soon inspire further theoretical research simulating the behavior of black holes and other cosmological objects in five-dimensional vacuum gravity. Meanwhile, Crump and his colleagues plan to run further numerical simulations to test the third law of black hole mechanics in four-dimensional vacuum gravity
“This is technically more challenging than in 5d, and—as with anything that you’re trying to disprove—it’s not clear yet whether this is even possible,” added Crump.
“It might be the case that the 3rd law miraculously holds true in 4d vacuum gravity, in which case it would be a fool’s errand trying to disprove it … but we have our suspicions, and hopefully we don’t end up fooled.”
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Publication details
John R. V. Crump et al, Violation of the Third Law of Black Hole Mechanics in Vacuum Gravity, Physical Review Letters (2026). DOI: 10.1103/gbg1-pjgq.
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