In three-dimensional space, the surface of a black hole must be a sphere. But a new result shows that in higher dimensions, an infinite number of configurations are possible.

- Recent mathematical proofs indicate that black holes can take on an infinite number of shapes in dimensions five and above, challenging the traditional view that they must be spherical.
- The study extends Stephen Hawking’s 1972 theorem, revealing new potential black hole shapes such as “black rings” and “black lenses,” particularly in higher dimensions.
- Researchers have developed new methods to prove the existence of complex black hole shapes by embedding them in higher-dimensional space-times and then reducing them to five dimensions.
- These findings are currently theoretical but suggest that detecting non-spherical black holes in particle colliders could provide evidence of higher dimensions in our universe.
- Ongoing research aims to determine if these complex black hole shapes can exist without matter fields and to explore the implications of higher-dimensional realms in general relativity.

The cosmos appears to favour round shapes. Planets and stars are typically spherical because gravity draws clouds of gas and dust towards the centre of mass. This principle also applies to black holes, or more precisely, to their event horizons, which are theorised to be spherical in a universe with three dimensions of space and one of time. However, if our universe possesses higher dimensions, as some theories suggest, could black holes adopt other shapes?

Mathematics provides a positive answer to this question. Over the past twenty years, researchers have occasionally identified exceptions to the rule that black holes must be spherical. A recent paper has significantly expanded on this, presenting a comprehensive mathematical proof that an infinite number of shapes are possible in dimensions five and above. This paper illustrates that Albert Einstein’s general relativity equations can produce a wide variety of unusual, higher-dimensional black holes.

This new work is purely theoretical and does not confirm the existence of such black holes in nature. However, if we were to detect such oddly shaped black holes, perhaps as tiny products of collisions in a particle collider, it would indicate that our universe has higher dimensions, according to Marcus Khuri, a geometer at Stony Brook University and co-author of the new study with Jordan Rainone, a recent Stony Brook maths Ph.D. “It’s now a matter of waiting to see if our experiments can detect any.”

**Black Hole Doughnut**

As with many black hole stories, this one begins with Stephen Hawking. Specifically, with his 1972 proof that the surface of a black hole, at a fixed moment in time, must be a two-dimensional sphere. While a black hole is a three-dimensional object, its surface has just two spatial dimensions.

Interest in extending Hawking’s theorem didn’t arise until the 1980s and ’90s when string theory, which suggests the existence of 10 or 11 dimensions, gained popularity. Physicists and mathematicians then started considering the implications of these extra dimensions for black hole topology.

Black holes are among the most perplexing predictions of Einstein’s equations — 10 interconnected nonlinear differential equations that are highly challenging to solve. Generally, they can only be solved under highly symmetrical, simplified conditions.

In 2002, three decades after Hawking’s result, physicists Roberto Emparan and Harvey Reall found a highly symmetrical black hole solution to the Einstein equations in five dimensions (four of space plus one of time). They called this a “black ring” — a three-dimensional surface resembling a doughnut.

It’s challenging to visualise a three-dimensional surface in a five-dimensional space, so imagine an ordinary circle. For each point on that circle, substitute a two-dimensional sphere. The combination of a circle and spheres results in a three-dimensional object that might resemble a solid, lumpy doughnut.

Such doughnut-like black holes could theoretically form if they spin at the right speed. “If they spin too fast, they would break apart, and if they don’t spin fast enough, they would revert to a ball,” Rainone explained. “Emparan and Reall found a sweet spot: their ring was spinning just fast enough to stay as a doughnut.”

This result inspired Rainone, a topologist, who remarked, “Our universe would be a boring place if every planet, star and black hole resembled a ball.”

**A New Focus**

In 2006, the concept of non-ball black holes gained traction. That year, Greg Galloway from the University of Miami and Richard Schoen from Stanford University extended Hawking’s theorem to describe all possible black hole shapes in dimensions beyond four. These included the familiar sphere, the previously demonstrated ring, and a class of objects called lens spaces.

Lens spaces are a particular type of mathematical construction significant in both geometry and topology. “Among all possible shapes the universe could present in three dimensions,” Khuri said, “the sphere is the simplest, and lens spaces are the next simplest case.”

Khuri describes lens spaces as “folded-up spheres.” Imagine starting with a simpler shape — a circle. Divide this circle into upper and lower halves. Then move every point in the bottom half to the point in the top half that’s directly opposite. This process leaves just the upper semicircle and two antipodal points, which must be joined to form a smaller circle with half the original circumference.

Next, consider two dimensions, where it becomes more complex. Start with a two-dimensional sphere and move every point on the bottom half up so it touches the antipodal point on the top half. You’re left with the top hemisphere, but the equatorial points must also be identified with each other, resulting in a highly contorted surface.

When discussing lens spaces, mathematicians usually refer to the three-dimensional variety. Start with a solid globe that includes surface and interior points. Run longitudinal lines from the north to the south pole, splitting the globe into two hemispheres. Points on one hemisphere are then identified with their antipodal points on the other.

You can have more longitudinal lines and various ways of connecting the sectors they define. In a lens space denoted L(p, q), p indicates the number of sectors, and q shows how these sectors are identified. An L(2, 1) lens space has two sectors (hemispheres) with one way to identify points, antipodally.

With more sectors, there are more ways to connect them. In an L(4, 3) lens space, four sectors exist, and each upper sector matches its lower counterpart three sectors over: upper sector 1 goes to lower sector 4, upper sector 2 to lower sector 1, and so forth. “Think of this process as twisting the top to find the correct place on the bottom to glue,” Khuri explained. “The amount of twisting is determined by q.” As more twisting becomes necessary, the resulting shapes become increasingly complex.

“People often ask me: How do I visualise these things?” said Hari Kunduri, a mathematical physicist at McMaster University. “The answer is, I don’t. We treat these objects mathematically, demonstrating the power of abstraction, which allows us to work without drawing pictures.”

**All the Black Holes**

In 2014, Kunduri and James Lucietti from the University of Edinburgh proved the existence of a black hole of the L(2, 1) type in five dimensions.

The Kunduri-Lucietti solution, which they refer to as a “black lens,” has several significant features. It describes an “asymptotically flat” space-time, where the curvature of space-time, high near a black hole, approaches zero at infinity. This characteristic helps ensure the results are physically relevant. “It’s not hard to make a black lens,” Kunduri noted. “The hard part is ensuring space-time is flat at infinity.”

Just as rotation keeps Emparan and Reall’s black ring intact, the Kunduri-Lucietti black lens must also spin. However, Kunduri and Lucietti used a “matter” field — a type of electric charge — to maintain their lens’s shape.

In their December 2022 paper, Khuri and Rainone generalised the Kunduri-Lucietti result as far as possible. They first proved the existence of black holes with lens topology L(p, q) in five dimensions, for any values of p and q greater than or equal to 1, as long as p is greater than q, and p and q have no common prime factors.

Then they extended their findings. They demonstrated that they could produce a black hole in any lens space shape, with any values of p and q (meeting the same conditions), in any higher dimension, leading to an infinite number of possible black holes in infinite dimensions. However, Khuri noted that “when you go to dimensions above five, the lens space is just one piece of the total topology.” The black hole is even more complex than the already challenging lens space it contains.

The Khuri-Rainone black holes can rotate but don’t have to. Their solution also pertains to an asymptotically flat space-time. Khuri and Rainone required a different type of matter field, consisting of particles associated with higher dimensions, to maintain the shape of their black holes and prevent defects or irregularities. The black lenses they constructed, like the black ring, have two independent rotational symmetries (in five dimensions) to simplify solving the Einstein equations. “It is a simplifying assumption, but a reasonable one,” Rainone said. “Without it, we don’t have a paper.”

“It’s excellent and original work,” Kunduri remarked. “They demonstrated that all the possibilities presented by Galloway and Schoen can be realised once the rotational symmetries are considered.”

Galloway was particularly impressed by the strategy Khuri and Rainone devised. To prove the existence of a five-dimensional black lens with given p and q, they first embedded the black hole in a higher-dimensional space-time, where proving its existence was easier due to more room to manoeuvre. Then, they reduced the space-time to five dimensions while maintaining the desired topology. “It’s a brilliant idea,” Galloway said.

The procedure Khuri and Rainone introduced is remarkable for its generality, applying to all possibilities at once, Kunduri said.

As for future research, Khuri is exploring whether lens black hole solutions can exist and remain stable in a vacuum without matter fields. A 2021 paper by Lucietti and Fred Tomlinson concluded that some matter field is necessary, but their argument was based on computational evidence rather than a mathematical proof, leaving the question open, according to Khuri.

Meanwhile, a larger mystery remains. “Are we really living in a higher-dimensional realm?” Khuri asked. Physicists have predicted that tiny black holes might someday be produced at the Large Hadron Collider or another higher-energy particle accelerator. If a collider-produced black hole could be detected during its brief lifetime and observed to have a non-spherical topology, it would provide evidence that our universe has more than three dimensions of space and one of time.

Such a discovery could also address another academic issue. “General relativity has traditionally been a four-dimensional theory,” Khuri said. By exploring black holes in five and higher dimensions, “we are betting that general relativity is valid in higher dimensions. If any exotic black holes are detected, it would confirm our bet.”