History

The Kerr Solution: A Milestone in Black Hole Research

The Thrilling Revelation of Rotating Black Holes

“In my extensive scientific career spanning over forty-five years, the most profound moment has been realizing that an exact solution of Einstein’s equations, found by New Zealand mathematician Roy Kerr, accurately represents countless massive black holes throughout the universe.” — Subrahmanyan Chandrasekhar

Albert Einstein (1879–1955) is well-known for his contributions to our understanding of the cosmos. However, the period from 1960 to 1975, termed The Golden Age of Relativity, remains relatively less explored.

Shortly after Einstein unveiled his theory of relativity, he received a letter from a soldier named Karl Schwarzschild (1873–1916), stationed on the Russian front, who formulated a spacetime model consistent with relativity. However, his findings were limited to idealized scenarios, relying on simplified assumptions such as vacuum, spherical symmetry, lack of electric charge, and non-rotation. He applied field equations to illustrate how the central mass of a star would warp space, similar to how a cannonball distorts a rubber sheet.

Yet, nature does not operate under such constraints, as every celestial body rotates around an axis, prompting the exploration of spacetime with rotating gravitational masses.

Vitaly Ginzburg remarked: “One must grasp rotational effects in Einstein’s equations to conclusively assess the validity of this beautiful, albeit perplexing, theory.” — At the Gravitation and General Relativity meeting in Warsaw.

Roy Patrick Kerr was born on May 16, 1934, in the small New Zealand town of Kurow. He began his education at St. Andrew’s College in Christchurch, where his mathematical talents emerged. By 1951, he advanced directly to the third year of his mathematics major and graduated with first-class honors from the University of Canterbury. In 1955, he traveled to the University of Cambridge, funded by the Sir Arthur Sims Empire Scholarship, to pursue his doctorate.

While at Cambridge, Kerr took courses in quantum field theory and particle physics under the tutelage of Nobel Laureate Paul Dirac (1902–1984) and future Nobel Laureate Abdus Salam (1926–1996). Surprisingly, these renowned scholars had little influence on his doctoral work and subsequent research trajectory.

Moffat’s Influence

John Moffat, a professor emeritus at the University of Toronto, was at Trinity College, Cambridge, where he introduced Kerr to general relativity, igniting his interest in the dynamics of arbitrary spinning particles in spacetime. Kerr delved deeply into the literature on the theory, recognizing opportunities to contribute significantly due to the limitations of earlier research.

Moffat captivated Kerr with his unconventional background; he left school at 16 to pursue a career as an artist before shifting his focus to the cosmos in Copenhagen, where he quickly grasped concepts in general relativity and unified field theory. He even corresponded with Einstein, sharing his work on one of Einstein's significant theories.

He recounted: “In 1953, Einstein replied from Princeton, but the letter was in German. I rushed to my barber in Copenhagen to translate it. Over that summer and fall, we exchanged several letters, which caught the attention of physicist Niels Bohr and others, opening doors of opportunity for me.” — (Perimeter Institute for Theoretical Physics, 2005)

Kerr produced a comprehensive thesis on particle dynamics in general relativity, leading to several influential research papers.

Petrov Classification

He frequented Hermann Bondi’s (1919–2005) research group at King’s College London to establish connections within the relativity community. Felix Pirani (1928–2015), a former collaborator of Bondi, delivered a pivotal lecture on the algebraic symmetries of Einstein's field equations, drawing from Alexei Zinovievich Petrov’s work. This lecture proved instrumental for Kerr's groundbreaking research.

Researchers in relativity must contend with the daunting complexity of Einstein’s equations when fully expanded. Petrov demonstrated that simplifications are possible under certain conditions, reducing the number of terms to facilitate potential solutions.

At that time, Peter Bergmann (1915–2002), a German-American physicist, met Kerr during a visit to Cambridge and offered him a post-doctoral position at Syracuse University. Initially joining the group, he later accepted a senior position at Wright-Patterson Air Force Base in Dayton, Ohio.

During his two years at the Air Force Base, he began addressing the problem systematically, focusing on Petrov’s classification. Unable to locate Petrov's original papers, he collaborated with Joshua N. Goldberg (1925–2020) to reproduce the equations and results. Moreover, he felt invigorated by the community of relativists at the 1962 Gravitation and General Relativity meeting in Warsaw, gaining insights into ongoing research.

Alfred Schild (1921–1977), an influential Austrian-American physicist, founded the Center for Relativity at the University of Texas at Austin in 1962. This center became a flourishing hub for research, drawing bright young scientists.

Kerr and his family relocated to Austin at Schild's invitation, eager to solve Einstein's equations strategically. However, upon witnessing established figures like Ezra Ted Newman also striving toward this goal, he feared his efforts might be futile.

Nonetheless, Kerr remained convinced that spacetime should be structured to encompass shear-free geodesics, where light rays do not distort images as they travel.

A few months later, Newman and his team published a paper asserting that such solutions do not exist, which left Kerr disheartened by the implication that no shear-free spacetimes could be linked to observable gravitational sources. Despite his dismay, he suspected there might be flaws in Newman’s reasoning.

Kerr scrutinized the problem until he encountered a perplexing equation that did not equate to zero when its terms were summed. This revelation struck him as absurd, prompting him to rush to his colleagues, exclaiming, “They’re wrong! They’re wrong, and I can prove it.”

The Eureka Moment

The long-sought goal was to identify the “correct” coordinates for Einstein’s equations to accurately depict spacetime. However, even after fulfilling the shear-free condition and selecting appropriate coordinates, Kerr faced a challenging set of equations. Eventually, he added rotational symmetry and time independence to formulate equations that became manageable.

He sought to determine how this spacetime appeared to a distant observer. Mathematically, that solution was asymptotically flat, meaning the effects diminished as one moved farther from the central source.

The following day, he approached Schild to discuss the spacetime surrounding a rotating object. Alfred joined him as he calculated the angular momentum of this derived spacetime. After thirty minutes, Kerr declared, “Alfred, it’s spinning; its angular momentum is Ma.”

(Zero angular momentum indicates non-rotation, while any derived nonzero angular momentum suggests a spinning gravitational source.)

Schild was thrilled; the implications were clear.

Unlike Schwarzschild’s point singularity, Kerr’s spacetime featured a ring singularity, where the size of the ring correlates with the angular momentum of the spinning gravitational source. Although direct measurement of the angular momentum of a ring singularity is not feasible, it is understood that rotating masses cause spacetime to drag around them, known as the Lense-Thirring effect. This frame-dragging quantifies the rotation speed of the central mass.

In July 1963, Kerr submitted his concise paper for publication. It was accepted and published swiftly, albeit with minor typographical errors in the final version.

He was then promptly offered a tenured associate professorship at the University of Texas at Austin.

Kerr reflects: “Everyone attempting to solve the problem was approaching it from the front, but I sought a different perspective. Numerous new mathematical methods were emerging in relativity, and Josh Goldberg and I had experienced success with them. I aimed to understand the entire structure—the Bianchi identities, the Einstein equations, and these Tetrads—to see how they interrelated. It appeared promising, suggesting many solutions would emerge. However, I hit a wall. Teddy Newman and Roger Penrose were exploring similar methods, but Teddy had established an unpublished theorem that effectively proved my solution couldn't exist! Fortunately, my neighbor, also engaged in relativity, obtained a preprint, and after a quick scan, I identified the crucial part that indicated my solution was feasible! Following that, I worked fervently and found the solution within weeks.”

The relativity community reacted positively and promptly to one of the most significant achievements of the twentieth century. Eventually, Kerr’s solution was recognized as a comprehensive description of rotating black holes.

It is now regarded as “The most significant exact solution to any equation in physics.” Since its discovery, the solution has greatly advanced our understanding of gravitational theory and astrophysics.

The renowned Nobel Laureate Subrahmanyan Chandrasekhar (1919–1995) commented on this breakthrough in his book: “In my entire scientific life, extending over forty-five years, the most shattering experience has been the realization that an exact solution of Einstein’s equations of general relativity, discovered by New Zealand mathematician Roy Kerr, provides the absolutely exact representation of untold numbers of massive black holes that populate the universe. This awe in the face of beauty, this incredible fact that a discovery motivated by a quest for beauty in mathematics should find its exact counterpart in Nature, compels me to assert that beauty resonates with the human mind at its deepest and most profound levels.” — Truth and Beauty: Aesthetics and Motivations in Science.

Hawking discusses Kerr’s discovery in his book A Brief History of Time: “In 1963, Roy Kerr, a New Zealander, found a set of solutions to the equations of general relativity that described rotating black holes. These “Kerr” black holes rotate at a constant rate, with their size and shape determined solely by their mass and rotation speed. If rotation is absent, the black hole is perfectly spherical, aligning with the Schwarzschild solution. With non-zero rotation, the black hole bulges at its equator, similar to how Earth or the Sun bulge due to their own rotation, and the faster it spins, the more pronounced the bulge becomes.

Thus, it was hypothesized that any rotating body collapsing to form a black hole would ultimately stabilize into a stationary state described by the Kerr solution. In 1970, my colleague and fellow research student Brandon Carter at Cambridge took initial steps toward proving this hypothesis, demonstrating that any stationary rotating black hole possessing an axis of symmetry—like a spinning top—would have its size and shape dictated solely by its mass and rotation rate. Then, in 1971, I proved that any stationary rotating black hole must indeed have such an axis of symmetry. Ultimately, in 1973, David Robinson at King’s College, London, utilized Carter’s and my findings to confirm that the conjecture was correct: such a black hole must indeed be described by the Kerr solution.” — Stephen Hawking

In 1963, Schild recruited another notable relativist, Roger Penrose, to the center of relativity. Penrose quickly established a productive working relationship with Kerr. Meanwhile, Maarten Schmidt's discovery and distance measurement of quasars (highly luminous active galactic nuclei) fueled the desire for collaboration between relativists and astrophysicists.

A noteworthy counterintuitive event occurred during the first Texas Symposium on Relativistic Astrophysics, where three hundred attendees sought interdisciplinary insights into quasars.

Upon arriving, Kerr learned that Penrose—a more charismatic speaker—was set to present. Naturally, this decision frustrated him, leading to a complaint to the organizers. Despite Penrose's broader recognition and potential to engage the astrophysics audience, the symposium featured around fifty relativists and the majority being astrophysicists and astronomers, who had traveled to Texas with only a limited understanding of the significance of Kerr’s solution.

As Kerr began his ten-minute presentation, many attendees began to leave, while others engaged in side conversations. Only a small group of relativists listened intently, visibly displeased with the audience's lack of interest.

One relativist, Achilles Papapetrou (1907–1997), rose to express his frustration at the audience, recounting his thirty years of effort to discover such a solution and the repeated failures he faced. He also emphasized how this solution could bridge the relevance of general relativity to the physical realm.

Awards and Honors

Among the numerous accolades Kerr has received for his exceptional contributions, the Hector Medal awarded by the Royal Society of New Zealand in 1982 stands out. He was further honored with the Hughes Medal from the Royal Society of London in 1984.

The citation reads: “The Hughes Medal is awarded to Professor R P Kerr in recognition of his distinguished work on relativity, particularly for his discovery of the so-called Kerr black hole. In the early 1960s, Professor Kerr discovered a specific solution to Einstein’s field equations describing a structure now termed a Kerr black hole. This solution, notably complex and lacking the symmetry of prior solutions, became evident that any stationary black hole could be characterized by Kerr’s solution. His work is, therefore, crucial to general relativistic astrophysics, with all subsequent detailed research on black holes fundamentally relying on it. Professor Kerr has made additional significant contributions to general relativity theory, but the discovery of the Kerr black hole is so remarkable that it parallels the discovery of a new elementary particle in physics.”

He received the Rutherford Medal from the Royal Society of New Zealand in 1991 “for his outstanding discoveries in the extraterrestrial realm of black holes.” In 2004, a symposium focused on general relativity and quantum gravity was organized at the University of Canterbury in Christchurch to honor Kerr's 70th birthday, during which it was announced that he would receive the Marcel Grossman Award “for his fundamental contribution to Einstein’s theory of general relativity,” which he accepted in Berlin two years later.

Additionally, he was awarded the Albert Einstein Medal in 2013 “for his 1963 discovery of a solution to Einstein’s gravitational field equations.” In 2016, he received the Crafoord Prize from The Royal Swedish Academy of Sciences “for fundamental work on rotating black holes and their astrophysical implications.”

References

• The book: Cracking the Einstein Code: Relativity and the Birth of Black Hole Physics, published in 2009.
• Roy Kerr’s landmark paper Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics, published in Physical Review Letters.

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