# Penrose: Mathematician won the prize in physics

On October 6, 2020, the Royal Swedish Academy of Sciences announced the winners of the 2020 Nobel Prize in Physics. British mathematical physicist Roger Penrose won half of the prize for “discovering the formation of black holes is a solid prediction of general relativity”; German astrophysicist Reinhard Genzel and American astronomer Andrea Gates shared the other half of the bonus for “discovering a supermassive compact object in the center of our galaxy”.
The two award-winning studies are all about black holes. The former is purely theoretical research, and the latter is observational, which complements each other. In this article, we will give a brief introduction to Penrose.

Black Hole concept
as background, we first introduce the “black hole” concept. The origin of this concept is often traced back to the British geologist John Michel. In 1783, based on Newton’s law of universal gravitation, Michel obtained a result that can be deduced by middle school students today, that is, if a planet with the same density as the sun is hundreds of times larger in diameter than the sun, the gravitational force will be so powerful Nor can it escape from its surface (thus it will look “black”). In 1796, French mathematician Pierre-Simon Laplace also got the same result. These results are often regarded as the buds of the concept of black holes.
However, the black holes of Michelle and Laplace have little in common with the black holes in general relativity we call now, except for the “radius” parameter. For example, the “black” of the former is just that light cannot escape to a distance, but it can still be seen at close distances, while the latter is not; even the so-called “radius” of “exactly the same” parameter has mutual meanings. It is completely different. The former is the distance from the center of the black hole to the surface. The latter does not have such a meaning. It is just the abbreviation of the perimeter divided by 2π (this is not the same as the former in general relativity). One thing). As for various more subtle other characteristics, they are unique to black holes in general relativity (hereinafter referred to as “black holes”). Therefore, Penrose once said: “The concept of black holes can actually only be derived from the special properties of general relativity, and does not appear in Newton’s theory.”
Schwarzschild’s solution
So, how is the concept of black holes “derived from the special properties of general relativity”? This has to go back to January 1916. At that time, although it was less than two months after Einstein proposed the general theory of relativity, a German physicist named Karl Schwarzschild obtained a strict solution of the general theory of relativity—now called the Schwarzschild solution .
The Schwarzschild solution describes a spherically symmetrical space-time. It has two very striking features-both appear as “0” in the denominator, which makes the mathematical expression meaningless: one of them appears on the ball At the center of symmetry, the other appears on a spherical surface. The radius of this spherical surface is called the “Schwarzschild radius”. After a long period of research, physicists have gradually understood the true meaning of these two characteristics: the former is called a “singularity”, has “pathological” properties such as infinite curvature of time and space, and will Let the laws of physics fail; the latter is called the “event horizon” or “horizon” for short. Although it was once regarded as a singularity, it is actually just a defect in the specific coordinates used by Schwarzschild.
Singularity and horizon are the two main characteristics of black holes. Therefore, the advent of the Schwarzschild solution can be regarded as the earliest prediction of black holes by general relativity. But this kind of prediction only shows that the general theory of relativity can describe the main characteristics of black holes in principle and can allow things like singularities and horizons, but it cannot tell us whether any physical process actually produces such things. If not, then the so-called “in principle possible” is still nothing but a spectacle.
So, is there actually any physical process that can produce a black hole? In 1939, a study by American physicist Robert Oppenheimer and his student Hartland Sind took an important step towards answering this question.
Oppenheimer and Sind studied the collapse process of stars after they run out of nuclear fuel (so that there is no longer radiation pressure to counter gravity). They found that for observers in the distance, when the star collapses close to the horizon, The wavelength of the light emitted from the surface of the star will become longer and longer, and the collapse process will appear slower and slower until it “freezes.” Because of this strange effect, black holes have an early name called “frozen stars.” But this effect does not mean that black holes cannot be formed. It is just like a video tape that stops slowly after half of the box, making it impossible to see the ending, but it does not mean that the ending did not happen. What’s more, Oppenheimer and Sind also discovered that for observers who collapsed with the star, the collapse will cross the horizon without stopping, and produce singularities within a limited time, thus showing that black holes can be formed. of.
With these studies, can it be considered that the formation of black holes is a prediction of general relativity? Not yet. Because both the Schwarzschild solution and the research of Oppenheimer and Sind rely on a symmetry that cannot be strictly achieved in the real world-spherical symmetry.
To make matters worse, because general relativity is a very complex theory, symmetry was essential to almost all similar studies at that time. For example, the New Zealand mathematician Roy Kerr obtained a solution of general relativity describing a rotating black hole in 1963-called the Kerr solution. This solution is much more common than the Schwarzschild solution, but it also depends on a kind of symmetry. Sex-axial symmetry. Although most stars are close to axisymmetric or even spherical, they can never be strictly axisymmetric or spherical. Similarly, many cosmological solutions of general relativity also rely on symmetry-such as uniformity and isotropy. These symmetries cannot be strictly achieved in the real world.
Soviet school of
general, physicists are not strictly not achieve this on carping in symmetry, symmetry is because they study the real world, the most powerful tool that a friend is not an exaggeration. But the formation of black holes is an exception, because as mentioned earlier, one of the main characteristics of black holes is singularity, and singularity will invalidate the laws of physics. Since the laws of physics are what physicists have to eat, facing the serious consequences of the failure of the laws of physics, even this friend of symmetry can be abandoned.
Therefore, many physicists attribute the problem to symmetry and believe that singularities do not exist. All processes that seem to produce singularities and black holes are due to the introduction of symmetry. As long as symmetry is discarded, singularities and black holes are Can be “destroyed”-that is, will not be formed. Representatives who hold this view are Soviet physicists EM Livschitz and IM Karatnikov-let’s call it the “Soviet School”.
In the early 1960s, the “Soviet School” conducted an in-depth study of general relativity while discarding symmetry, trying to prove that singularities would not form. They even thought they had completed the proof for a while, and wrote it into the famous book “Theoretical Physics Course” co-authored by Lev Landau and Li Fushizi. Since the advent of the Schwarzschild solution, due to other considerations, more physicists have been skeptical of the existence of singularities and black holes, including Einstein himself.
Mathematics Penrose
It is against this background, Penrose in the fall of 1964 (age 33) substantially involved in the study of black holes.

The American physicist Kip Thorne once told some interesting stories about Penrose when he was young, and we share one as an introduction to him. Penrose’s parents’ professional fields are related to medicine (father is a professor of human genetics, mother is a doctor), and therefore, they hope that at least one of their four children can take medicine as a career. But when Penrose chose his major, one of his two brothers had chosen physics (later became a well-known statistical physicist) and the other had chosen chess (later won 10 British championships), and his sister was still too young , He wants to choose mathematics. Seeing that the hope of “at least one person can take medicine as a career” was about to fail, Penrose’s parents intervened in his professional choice. At the request of Penrose’s father, the university he applied for gave him a special mathematics test. There are a total of 12 questions in the test, and ordinary students can do one or two is considered good, while Penrose did all 12 questions correctly. In this way, he won a “license” to learn mathematics from his parents.
As a student of the Department of Mathematics, Penrose received a PhD in mathematics in 1957 with a research in the field of geometry. But long before he got his degree, he was also interested in astronomy and physics because of listening to the radio lectures of British astrophysicist Fred Hoyle and acquainting with British physicist Dennis Sharma. . Hoyle was the main supporter of one of the popular cosmological models at the time, the “steady state model”, and Shama also studied it. Influenced by them, Penrose also studied the steady state model. In addition, the acquaintance with Xia Ma paved the way for his later cooperation with Stephen Hawking-because Hawking’s doctoral supervisor was Xia Ma.
The steady state model is a model of the universe that quickly failed and was abandoned by most astronomers in the 1960s. Before that, although it was popular, it also faced some problems. Since the steady-state model also relies on symmetry, similar to the case of the singularity, some proponents of the steady-state model also attribute the problem to symmetry, but the direction of their efforts is exactly the opposite, hoping to abandon symmetry Sex to “save” the steady state model. Affected by this hope, Penrose also studied the steady-state model without symmetry, but found no substantial difference-that is to say, the problem of the steady-state model remains regardless of whether there is symmetry or not. exist. This experience has a lot of enlightenment for Penrose’s later black hole research, because it shows that abandoning symmetry may not have the effect that people hope. In this case, will the singularity still exist regardless of the existence of symmetry? This consideration made Penrose’s subsequent black hole research completely different from the “Soviet School”.
Involvement in black holes
In the fall of 1964, Penrose began to substantially intervene in black hole research. It was a strange celestial body that had just been discovered the previous year—that is, in 1963—that tempted him to intervene. This kind of celestial body, which was quickly called a “quasar”, was much brighter than galaxies, but its linearity Only one millionth of a galaxy (thus it looks similar to a star-from which the name “quasar” comes from), which necessarily contains a highly dense structure.
Preliminary analysis shows that the most likely “luminescence mechanism” of this “quasar” is that a giant black hole swallows matter including stars (the matter emits intense radiation before being swallowed). This new discovery, which supports the existence of black holes, and the enlightenment from the study of the steady state model mentioned above, enabled Penrose to intervene in black hole research from a goal contrary to the “Soviet School”-namely Explore the universality of the formation of singularities without relying on symmetry (rather than trying to prove that singularities will not form).
Penrose’s research not only aims at the opposite of the “Soviet School”, but also uses completely different methods. The research of the “Soviet School” is biased towards examples, and is dedicated to solving the general theory of relativity without symmetry in order to find examples where singularities will not form; while Penrose explores the universality of the formation of singularities, not Specific examples are not devoted to solving general relativity. Since general relativity is not solved, then factors such as the shape and size of the planet are not important.
Since general relativity is a highly geometrical theory, the formation of singularities is a highly geometrical problem in the nature of time and space. Those who are familiar with mathematics know that in geometric problems, if the shape, size and other factors are not important, then the rest is the so-called topological properties. Therefore, Penrose’s research has adopted a large number of topological methods-he himself called “light topology”, which happens to be his strength as a mathematician-and a mathematician with a doctorate in the field of geometry research.
Inspiration comes
targets Although determined, means corporations are the strengths of the study of the singularity is still very abstruse, take some inspiration. Penrose later described the origin of one of his important inspirations in this research. It was in the late autumn of 1964. One day shortly after he began to explore the singularity problem, Penrose and the mathematical physicist Ivor Robinson were walking on the street while discussing the problem (the problem is not related to singularities and black holes. relationship). When crossing an intersection, they met a red light and stopped—and also paused the discussion. In that short interval, an important inspiration for proving the singularity theorem appeared. Later that day, Penrose recalled his thoughts carefully in the office and finally “discovered” the inspiration clearly.
With the help of that inspiration, after months of hard work, Penrose proved an important result—it is the earliest one of a large class of theorems now called “Singularity Theorems”, based on “Gravitational Collapse and Space-time Singularity” The topic was published in 1965. To put it simply, Penrose’s singularity theorem contains several components-the basic structure of all other singularity theorems afterwards: first it assumes that matter has certain properties, secondly imposes certain requirements on time and space itself, and finally It is assumed that the distribution of matter satisfies certain conditions; under these three types of premises, Penrose proved that the formation of singularities is universal and inevitable—especially, it does not depend on symmetry.
Penrose was not the first to use topological means to study the structure of time and space. More than ten years before him, two Soviet mathematicians started research in this area and created a powerful and beautiful method. It is a pity that one of these two mathematicians was promoted (as an administrative leader, spending more and more time on administrative affairs), and the other was imprisoned (being labeled as “anti-Soviet”). Members of the “Group”) eventually stopped research in this direction, so it did not have an impact. Penrose’s research is different, not only got beautiful results, but also quickly attracted attention.
Great debate
in that year Penrose singularity theorems prove – that 1965 “Third International Conference of General Relativity and Gravitational” held in London, England. This conference gathered the world’s top experts on the theory of general relativity. Even Livschiz and Karatnikov of the “Soviet School” came to London across the “Cold War” gap and reported on their fundamentally negative strangeness. Point research. This meeting therefore became the first “collision” between Penrose’s singularity theorem and the negative result of the “Soviet School”.
Although the “collision” did not immediately tell the winner, Penrose’s research attracted the interest of several young physicists with profound knowledge in geometry and topology, including Sharma’s graduate student Hawking, who was also studying at the time. The singularities are nothing but cosmological singularities rather than black hole singularities; and the American theoretical physicist Robert Jeroch, who was the same age as Hawking. In the following years, Penrose, Hawking, Jeroche and others have proved more singularity theorems under a variety of different premises, making the existence of singularity theorems, singularities and black holes more and more Recognition.

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This recognition finally shook the “Soviet School”.
In September 1969, American physicist Thorne visited the Soviet Union. Taking this opportunity, Livschitz handed Thorne a manuscript to be secretly brought to the United States for publication (because—according to Thorne’s account—all academic manuscripts in the Soviet Union at that time were automatically regarded as secret documents, unless they were lengthy. The decryption review cannot be communicated with international counterparts). In that manuscript, the “Soviet School” admitted that their denial of the singularity was wrong and stated that it would revise the “Theoretical Physics Course”.
The confession of the
“Soviet School” in cooperation with Hawking cleared the main doubts about the existence of singularities and black holes. But Penrose’s singularity theorem itself still has some shortcomings.
This shortcoming is reflected in the premise. As mentioned earlier, the premises of the singularity theorem are divided into three categories. Among them, the nature of matter is that the energy density cannot be negative. This is not disputed in the classical physics to which general relativity belongs; the conditions for the distribution of matter are: Such as the collapse of massive stars can be achieved, so there is no problem; but the requirements imposed on time and space itself appear too strong. In fact, this requirement—specifically, the existence of a so-called Cauchy hypersurface in time and space—is so strong that it is not only extremely unlikely to be confirmed by observations, but also theoretically controversial. Even Penrose himself The singularity theorem contained a counterexample in another paper published almost simultaneously.
This shortcoming is known to Penrose himself, as well as Hawking and Jeroche, who followed his footsteps in studying the singularity theorem. For example, Hawking has stated in his autobiography that what Penrose and his own early singularity theorem may prove is the absence of Cauchy hypersurfaces rather than the existence of singularities and black holes; Penrose himself also admitted in subsequent research, There is no reason to assume the existence of Cauchy hypersurfaces in general relativity. The reason why the singularity theorem has become a large class of theorems is largely to eliminate the inadequacies in the premise. Among the singularity theorems after 1965, Penrose, Hawking, Jeroche and others tried to adapt mainly to the premise of the theorem.
In the end, Penrose and Hawking collaborated and published a paper entitled “Gravitational Collapse and Singularities in Cosmology” in 1970, and proposed a new “version” now called “Hawking-Penrose Singularity Theorem” “. This “version” expresses the singularity theorem with a more empirical basis, and thus more physically realistic premises, and covers both black hole singularities and cosmological singularities.
Anyone who has studied logic knows that in order for a logical reasoning to ensure the correct conclusion, not only the reasoning must be strict, but the premise must also be established. Quite similarly, the solidity of a theorem describing the physical world depends not only on the rigor of reasoning, but also on the reality of the premises, both of which are indispensable. In this sense, the “Hawking-Penrose Singularity Theorem” has more realistic premises and more solid conclusions. Among all the singularity theorems, if you ask which one is best known as “the discovery of the formation of black holes is a solid prediction of general relativity,” the answer is none other than the Hawking-Penrose singularity theorem.
Mathematical theorem
is worth noting that this study black holes Penrose compared with other studies of the Nobel Prize in Physics, as well as most other physics research, there is a very unique place, that is, it studies other than those Closer to the theorem of pure mathematics—it’s just based on the general theory of relativity.
This is because Penrose’s black hole study only made a “solid prediction” for general relativity. Whether that prediction is overturned or confirmed, it is the general theory of relativity that affects the research, not Penrose’s research, which is correct. It only depends on the correctness of its mathematical reasoning. The 2020 Nobel Prize in Physics has been jokingly called by some people as winning the physics prize for astronomical research, but in fact, this half of Penrose’s research can be said to have won the physics prize for mathematical research. It is not uncommon for astronomical research to receive physics awards, and it is almost a precedent for mathematical research to receive physics awards.