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Mathematicians Find an Infinity of Possible Black Hole Shapes

The cosmos seems to have a preference for things that are round. Planets and stars tend to be spheres because gravity pulls clouds of gas and dust toward the center of mass. The same holds for black holes — or, to be more precise, the event horizons of black holes — which must, according to theory, be spherically shaped in a universe with three dimensions of space and one of time.

But do the same restrictions apply if our universe has higher dimensions, as is sometimes postulated — dimensions we cannot see but whose effects are still palpable? In those settings, are other black hole shapes possible?

The answer to the latter question, mathematics tells us, is yes. Over the past two decades, researchers have found occasional exceptions to the rule that confines black holes to a spherical shape.

Now a new paper goes much further, showing in a sweeping mathematical proof that an infinite number of shapes are possible in dimensions five and above. The paper demonstrates that Albert Einstein’s equations of general relativity can produce a great variety of exotic-looking, higher-dimensional black holes.

The new work is purely theoretical. It does not tell us whether such black holes exist in nature. But if we were to somehow detect such oddly shaped black holes — perhaps as the microscopic products of collisions at a particle collider — “that would automatically show that our universe is higher-dimensional,” said Marcus Khuri, a geometer at Stony Brook University and co-author of the new work along with Jordan Rainone, a recent Stony Brook math Ph.D. “So it’s now a matter of waiting to see if our experiments can detect any.”

Black Hole Doughnut
As with so many stories about black holes, 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.)

Little thought was given to extending Hawking’s theorem until the 1980s and ’90s, when enthusiasm grew for string theory — an idea that requires the existence of perhaps 10 or 11 dimensions. Physicists and mathematicians then started to give serious consideration to what these extra dimensions might imply for black hole topology.

Black holes are some of the most perplexing predictions of Einstein’s equations — 10 linked nonlinear differential equations that are incredibly challenging to deal with. In general, they can only be explicitly solved under highly symmetrical, and hence simplified, circumstances.

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

It’s difficult to picture a three-dimensional surface in a five-dimensional space, so let’s instead imagine an ordinary circle. For every point on that circle, we can substitute a two-dimensional sphere. The result of this combination of a circle and spheres is a three-dimensional object that might be thought of as a solid, lumpy doughnut.

In principle, such doughnutlike black holes could form if they were spinning at just the right speed. “If they spin too fast, they would break apart, and if they don’t spin fast enough, they would go back to being a ball,” Rainone said. “Emparan and Reall found a sweet spot: Their ring was spinning just fast enough to stay as a doughnut.”

Learning about that result gave hope to Rainone, a topologist, who said, “Our universe would be a boring place if every planet, star and black hole resembled a ball.”

A New Focus
In 2006, the non-ball black hole universe really began to flower. That year, Greg Galloway of the University of Miami and Richard Schoen of Stanford University generalized Hawking’s theorem to describe all possible shapes that black holes could potentially assume in dimensions beyond four. Included among the allowable shapes: the familiar sphere, the previously demonstrated ring, and a broad class of objects called lens spaces.

Lens spaces are a particular type of mathematical construction that has long been important in both geometry and topology. “Among all possible shapes the universe could throw at us in three dimensions,” Khuri said, “the sphere is the simplest, and lens spaces are the next-simplest case.”

Khuri thinks of lens spaces as “folded-up spheres. You are taking a sphere and folding it up in a very complicated way.” To understand how this works, start with a simpler shape — a circle. Divide this circle into upper and lower halves. Then move every point in the bottom half of the circle to the point in the top half that’s diametrically opposite to it. That leaves us with just the upper semicircle and two antipodal points — one at each end of the semicircle. These must be glued to each other, creating a smaller circle with half the circumference of the original.

Next, move to two dimensions, where things begin to get complicated. Start with a two-dimensional sphere — a hollow ball — and move every point on the bottom half up so that it’s touching the antipodal point on the top half. You’re left with just the top hemisphere. But the points along the equator also have to be “identified” (or attached) with one another, and because of all the crisscrossing required, the resulting surface will become extremely contorted.

When mathematicians talk about lens spaces, they are usually referring to the three-dimensional variety. Again, let’s start with the simplest example, a solid globe that includes the surface and interior points. Run longitudinal lines down the globe from the north to the south pole. In this case, you have only two lines, which split the globe into two hemispheres (East and West, you might say). You can then identify points on one hemisphere with the antipodal points on the other.

But you can also have many more longitudinal lines and many different ways of connecting the sectors that they define. Mathematicians keep track of these options in a lens space with the notation L(p, q), where p tells you the number of sectors the globe is divided into, while q tells you how those sectors are to be identified with one another. A lens space labeled L(2, 1) indicates two sectors (or hemispheres) with just one way to identify points, which is antipodally.

If the globe is split into more sectors, there are more ways to knit them together. For example, in an L(4, 3) lens space, there are four sectors, and every upper sector is matched to its lower counterpart three sectors over: upper sector 1 goes to lower sector 4, upper sector 2 goes to lower sector 1, and so forth. “One can think of this [process] as twisting the top to find the correct place on the bottom to glue,” Khuri said. “The amount of twisting is determined by q.” As more twisting becomes necessary, the resulting shapes can get increasingly elaborate.

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

All the Black Holes
In 2014, Kunduri and James Lucietti of 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 a couple of important features. Their solution describes an “asymptotically flat” space-time, meaning that the curvature of space-time, which would be high in the vicinity of a black hole, approaches zero as one moves toward infinity. This characteristic helps ensure that the results are physically relevant. “It’s not so hard to make a black lens,” Kunduri noted. “The hard part is doing that and making space-time flat at infinity.”

Just as rotation keeps Emparan and Reall’s black ring from collapsing on itself, the Kunduri-Lucietti black lens must spin as well. But Kunduri and Lucietti also used a “matter” field — in this case, a type of electric charge — to hold their lens together.

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

Then they went further. They found that they could produce a black hole in the shape of any lens space — any values of p and q (satisfying the same stipulations), in any higher dimension — yielding an infinite number of possible black holes in an infinite number of dimensions. There is one caveat, Khuri pointed out: “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 visually 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. However, Khuri and Rainone needed a somewhat different kind of matter field — one that consists of particles associated with higher dimensions — to preserve the shape of their black holes and prevent defects or irregularities that would compromise their result. The black lenses they constructed, like the black ring, have two independent rotational symmetries (in five dimensions) to make the Einstein equations easier to solve. “It is a simplifying assumption, but one that is not unreasonable,” Rainone said. “And without it, we don’t have a paper.”

“It’s really nice and original work,” Kunduri said. “They showed that all the possibilities presented by Galloway and Schoen can be explicitly realized,” once the aforementioned rotational symmetries are taken into account.

Galloway was particularly impressed by the strategy invented by Khuri and Rainone. To prove the existence of a five-dimensional black lens of a given p and q, they first embedded the black hole in a higher-dimensional space-time where its existence was easier to prove, in part because there is more room to move around in. Next, they contracted their space-time to five dimensions while keeping the desired topology intact. “It’s a beautiful idea,” Galloway said.

The great thing about the procedure that Khuri and Rainone introduced, Kunduri said, “is that it’s very general, applying to all possibilities at once.”

As for what’s next, Khuri has begun looking into whether lens black hole solutions can exist and remain stable in a vacuum without matter fields to support them. A 2021 paper by Lucietti and Fred Tomlinson concluded that it’s not possible — that some kind of matter field is needed. Their argument, however, was not based on a mathematical proof but on computational evidence, “so it is still an open question,” Khuri said.

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

Such a finding could clear up another, somewhat more academic issue. “General relativity,” Khuri said, “has traditionally been a four-dimensional theory.” In exploring ideas about black holes in dimensions five and above, “we are betting on the fact that general relativity is valid in higher dimensions. If any exotic [nonspherical] black holes are detected, that would tell us our bet was justified.””

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