Math Proof Draws New Boundaries Around Black Hole Formation

Quata Magazine

Steve Nadis

Contributing writer

August 16, 2023

The modern notion of a black hole has been with us since February 1916, three months after Albert Einstein unveiled his theory of gravity. That’s when the physicist Karl Schwarzschild, in the midst of fighting in the German army during World War I, published a paper with astonishing implications: If enough mass is confined within a perfectly spherical region (bounded by the “Schwarzschild radius”), nothing can escape such an object’s intense gravitational pull, not even light itself. At the center of this sphere lies a singularity where density approaches infinity and known physics goes off the rails.

In the 100-plus years since, physicists and mathematicians have explored the properties of these enigmatic objects from the perspective of both theory and experiment. So it may be surprising to hear that “if you took a region of space with a bunch of matter spread out in it and asked a physicist if that region would collapse to form a black hole, we don’t yet have the tools to answer that question,” said Marcus Khuri, a mathematician at Stony Brook University.

Don’t despair. Khuri and three colleagues — Sven Hirsch at the Institute for Advanced Study, Demetre Kazaras at Michigan State University, and Yiyue Zhang at the University of California, Irvine — have released a new paper that brings us closer to determining the presence of black holes based solely on the concentration of matter. In addition, their paper proves mathematically that higher-dimensional black holes — those of four, five, six or seven spatial dimensions — can exist, which is not something that could confidently have been said before.

To put the recent paper in context, it might be worth backing up to 1964, the year Roger Penrose began introducing the singularity theorems that earned him a share of the 2020 Nobel Prize in Physics. Penrose proved that if space-time has something called a closed trapped surface — a surface whose curvature is so extreme that outward-going light gets wrapped around and turned inward — then it must also contain a singularity.

It was a monumental result, in part because Penrose brought powerful new tools from geometry and topology to the study of black holes and other phenomena in Einstein’s theory. But Penrose’s work did not spell out what it takes to create a closed trapped surface in the first place.

In 1972, the physicist Kip Thorne took a step in that direction by formulating the hoop conjecture. Thorne recognized that figuring out whether a nonspherical object — one lacking the symmetry assumed in Schwarzschild’s pioneering efforts — would collapse into a black hole would be “much harder to compute [and] indeed far beyond my talents.” (Thorne would go on to win the 2017 Nobel Prize in Physics.) Yet he felt his conjecture might make the problem more manageable. The basic idea is to first determine the mass of a given object and from that compute the critical radius of a hoop that the object must fit within — no matter how the hoop is oriented — to make the formation of a black hole inevitable. It would be like showing that a hula hoop that fits around your waist could also — if rotated 360 degrees — fit around your entire elongated body, including your feet and head. If the object fits, it will collapse to a black hole.

“The hoop conjecture is not well defined,” Kazaras commented. “Thorne intentionally used vague wording in the hopes that others would provide a more precise statement.”

In 1983, the mathematicians Richard Schoen and Shing-Tung Yau* obliged, proving an important version of the hoop conjecture, subsequently referred to as the black hole existence theorem. Schoen and Yau showed — in a clear-cut mathematical argument — just how much matter must be crammed into a given volume to induce the space-time curvature necessary to create a closed trapped surface.

Kazaras praised the Schoen-Yau work for its originality and generality; their technique could reveal whether any configuration of matter, regardless of symmetry considerations, was destined to become a black hole. But their approach had a major drawback. The way they measured the size of a given region of space — by determining the radius of the fattest torus, or doughnut, that could fit inside — was, to many observers, “cumbersome and nonintuitive,” Kazaras said, and hence impractical.

The recent paper offers an alternative. One of Schoen and Yau’s major innovations was to recognize that an equation devised by the physicist Pong Soo Jang, which originally had nothing to do with black holes, can “blow up” — go to infinity — at certain points in space. Amazingly, where it blows up coincides with the location of a closed trapped surface. So if you want to find such a surface, first figure out where the Jang equation goes to infinity. “In high school, we often try to solve an equation when the solution is equal to zero,” explained the mathematician Mu-Tao Wang of Columbia University. “In this case, we’re trying to solve the [Jang] equation such that the solution is infinite.”

Hirsch, Kazaras, Khuri and Zhang also rely on the Jang equation. But in addition to a torus, they use a cube — one that can be seriously deformed. This approach “is akin to Thorne’s idea, using square hoops instead of traditional circular hoops,” Khuri said. It draws upon the “cube inequality” developed by the mathematician Mikhail Gromov. This relationship connects the size of a cube to the curvature of space in and around it.

The new paper shows that if you can find a cube somewhere in space such that the matter concentration is large compared to the size of the cube, then a trapped surface will form. “This measurement is much easier to check” than one involving a torus, said Pengzi Miao, a mathematician at the University of Miami, “because all you need to compute is the distance between the cube’s two nearest opposing faces.”

Mathematicians can also build doughnuts (tori) and cubes in higher dimensions. In order to extend their proof of black hole existence to these spaces, Hirsch and colleagues built upon geometric insights that have been developed in the four decades since Schoen and Yau’s 1983 paper. The team was unable to go beyond seven spatial dimensions because singularities start cropping up in their results. “Getting around those singularities is a common sticking point in geometry,” Khuri said.


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The logical next step, he said, is to prove black hole existence based on “quasi-local mass,” which includes the energy coming from both matter and gravitational radiation, rather than from matter alone. That’s no simple task, in part because there is no universally agreed-upon definition of quasi-local mass.

Meanwhile, another question looms: To create a black hole of three spatial dimensions, must an object be compressed in all three directions, as Thorne insisted, or could compression in two directions or even just one be enough? All evidence points to Thorne’s statement being true, Khuri said, though it is not yet proved. Indeed, it is just one of many open questions that persist about black holes after they first manifested more than a century ago in a German soldier’s notebook.

Correction: August 17,2023
Demetre Kazaras recently moved from Duke University to Michigan State University. His affiliation has been updated.




At a press conference in Hong Kong earlier this week, the Shaw Prize Foundation awarded the 2023 Shaw Prize in Mathematical Sciences in equal shares to Vladimir Drinfeld, Harry Pratt Judson Distinguished Service Professor of Mathematics at the University of Chicago, and Shint-Tung Yau, Director of Yau Mathematical Sciences Center at Tsinghua University and Professor Emeritus at Harvard University. The two laureates were chosen for their contributions related to mathematical physics, arithmetic and differential geometry, and to Kähler geometry.

Yau was an active member of Harvard’s Department of Mathematics for 35 years before becoming professor emeritus in 2022. He has been a leader in the fields of mathematics and physics since the early years of his academic career, with a reputation as a thinker of unrivaled technical power. A press release by the Mathematical Sciences Selection Committee highlighted the systematically partial differential equation methods in differential geometry that Yau developed. With these, he solved the Calabi conjecture**, the existence of Hermitian Yang-Mills connections alongside Karel Uhlenbeck, and the positive mass conjecture with Richard Schoen. Yau introduced geometric methods to important problems in general relativity, which led to Shoen-Yau’s black hole existence theorem and to an intrinsic definition of quasi-local mass in general relativity.

Yau’s work on the existence of Kähler-Einstein metric led to the solution to the Calabi conjecture and to the concept of Calabi-Yau manifolds, which are cornerstones in string theory and complex geometry. Similarly, the Strominger-Yau-Zaslow construction has had a major impact on mirror symmetry. Yau’s collaboration with Peter Li on heat kernel estimates and differential Harnack inequalities has changed the analysis of geometric equations on manifolds and has influenced the development of optimal transportation and Hamilton’s work on Ricci flow. Yau contributed to the fusion of geometry and analysis, now known as geometric analysis. His work has had a deep and lasting impact on both mathematics and theoretical physics.

The Shaw Prize is awarded annually in three disciplines: Astronomy, Life Sciences and Medicine, and Mathematical Sciences. Each prize carries a gold medal, a certificate, and a monetary award of U.S. $1.2 million. This will be the 20th year that the Prize has been awarded, and the presentation ceremony is scheduled for November 12, 2023 in Hong Kong.

Read the full profile of Harvard Professor Emeritus Shing-Tung Yau in our 2021-2022 Newsletter.


** Calabi-Yau - First proposed by Eugenio Calabi in 1954 and a proof was published in 1978 by S.T. Yau. One direct consequence of this theorem is the existence of Ricci flat Ka ̈hler manifolds, now also called as Calabi-Yau manifolds. These are Ka ̈hler manifolds with trivial canonical line bundle. These special surfaces find applications in String theory as well. In these notes we go through Yau’s proof in detail.

Calabi-Yau manifolds are compact, complex Kähler manifolds that have trivial first Chern classes (over ℝ). In most cases, we assume that they have finite fundamental groups. By the conjecture of Calabi (1957) proved by Yau (1977; 1979), there exists on every Calabi-Yau manifold a Kähler metric with vanishing Ricci curvature.

Currently, research on Calabi-Yau manifolds is a central focus in both mathematics and mathematical physics. It is partially propelled by the prominent role the Calabi-Yau threefolds play in superstring theories. While many beautiful properties of Calabi-Yau manifolds have been discovered, more questions have been raised and probed. The landscape of various constructions, theories, conjectures, and above all the fast pace of progress in this subject, have made the research of Calabi-Yau manifolds an extremely active research field both in mathematics and in mathematical physics.

By the late 1960s, many were doubtful of the Calabi conjecture. Some attempted to use a reduction theorem of Cheeger and Gromoll (1971) to construct counterexamples to the Conjecture. Using the reduction theorem and assuming the conjecture, Yau announced the following splitting theorem in his 1973 lecture at the Stanford geometry conference: Every compact Kähler manifold with vanishing Ricci curvature can be covered by a metric product of a torus and a simply connected manifold with a Ricci-flat Kähler metric. He then used this theorem to produce a "counterexample" to the conjecture. The "counterexample" was soon discovered to be flawed; Yau withdrew his Stanford lecture. (The flaw was due to the mistaken assumption that manifolds with numerically non-negative anti-canonical divisor admits a first Chern form which is pointwise non-negative.)

In 1976, Yau (Yau, 1977; Yau, 1979) proved the Calabi conjecture by solving the complex Monge-Ampère equation for a real valued function ϕ det(gij¯+∂2ϕ∂zi∂z¯j)=efdet(gij¯), where ef is any smooth function of average 1 and gij¯+∂i∂j¯ϕ is required to be positive definite. The solution ϕ of the above equation ensures that the new Kähler metric ω+−1‾‾‾√∂∂¯ϕ can attain any Ricci (curvature) form in the class referred to in the Calabi conjecture.


Calabi-Yau manifolds have a property which is very interesting to physics. Einstein's equations show that spacetime curves according to the distribution of energy and momentum. But what if space is all empty? By Yau's theorem, not only is flat space a solution but so are Calabi-Yau manifolds. Calabi-Yau spaces are possible candidates for the shape of extra spatial dimensions in String Theory. Here, one is particularly interested in (complex) three-dimensional Calabi-Yau manifolds such as the quintic, known as threefolds. Every example of a Calabi-Yau threefold then gives rise to a different universe with different sets of elementary particles and particle interactions. This makes the question of classification, that is, which Calabi-Yau threefolds there are and what their properties are, tremendously interesting. However, to date it is not even clear whether there are finitely many. So far, physicists and mathematicians have constructed roughly half a billion examples.

calabi-yau manifold.png

Calabi-Yau manifold - Andart II

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