Tag: Schwarzschild limit

There is a scene in Christopher Nolan’s new film, Oppenheimer, where the eponymous physicist is thronged by his adoring pupils after his paper is published. They have gathered to celebrate the ‘black hole paper’ that J. Robert Oppenheimer wrote with his student Hartland Snyder. “The world will remember the day,” one of his students says.  

The world of physics does indeed remember the paper. While Oppenheimer is remembered in history as the “father of the atomic bomb”, his greatest contribution as a physicist was on the physics of black holes. The work of Oppenheimer and Hartland Snyder helped transform black holes from figments of mathematics to real, physical possibilities – something to be found in the cosmos out there.

Exceptionally versatile

At the time of this work, Oppenheimer was a professor of physics at the University of California, Berkeley. The Manhattan Project and the atomic bomb were still some years in his future. He was unknown to the public but the community of physicists knew him as the man who had established the finest school of theoretical physics outside of Europe.

Before Oppenheimer, it was customary for young American theoretical physicists to go to Europe, which had become the mecca of physics. Oppenheimer had made the pilgrimage himself in his youth and studied with some of the pioneers of quantum theory, such as Max Born and Wolfgang Pauli. After Oppenheimer joined Berkeley, many of the best young American physicists flocked there instead, drawn to his genius.

Oppenheimer and his students worked on a wide range of topics – from cosmic rays to nuclear physics, from quantum electrodynamics to astrophysics. Each student worked on a different topic, and the exceptionally versatile Oppenheimer oversaw it all.

For most physicists, who prefer to dig deep into one or two topics at a time, this would be a nightmare scenario. But Oppenheimer thrived on it.

When he later became the scientific director of the Manhattan Project at Los Alamos, his versatility helped him oversee diverse aspects of building the world’s first nuclear weapons.

From darkness to light

Among his students, Snyder was regarded as the most proficient at hard mathematical problems. He would go on to make important contributions to accelerator physics and noncommutative field theory. Oppenheimer gave him the problem of black hole formation to solve.

In their collaboration, Oppenheimer provided the vision and Snyder fleshed it out. Together, they brought black holes to life.

The possibility of black holes had been discovered shortly after Albert Einstein developed his theory of general relativity, in 1915. According to this theory, matter warps the fabric of spacetime around it. To determine the exact amount of warp, physicists have to solve a set of equations known as Einstein’s equations. The first person to find such a solution to these equations was the German physicist Karl Schwarzschild: he computed the warping outside a perfectly spherical mass.

Schwarzschild’s solution contained a surprise. He found that if you compute the warping near spheres of the same mass but of smaller and smaller radii, the warp keeps increasing. Below a certain critical radius, the neighbouring spacetime would curve into a pocket from which not even light can escape. That is, if a certain amount of mass was packed into a small enough radius, a black hole would exist around it.

Most physicists dismissed the possibility of black holes as mathematical fiction. They pointed to the fact that there was no known way by which matter could be squeezed so tight that a black hole would form.

A remarkably accurate picture

The next step came from the astrophysicist Subrahmanyan Chandrasekhar. His work showed that black holes could be formed when certain stars run out of fuel and collapse under their own weight. But a description of a star imploding and forming a black hole was still missing.

This is where Oppenheimer and Snyder came in. Oppenheimer had already made a foray into the topic of stellar collapse, which convinced him of the inevitability of black holes. Together with his student George Volkoff, he was able to significantly extend Chandrasekhar’s result.  Now, with Snyder, Oppenheimer set out to provide a mathematical description of the birth of a black hole.

The collapse of a star is an enormously complicated process that would have been impossible to fully understand mathematically. But Oppenheimer had a talent for zeroing in on the essential features of a problem. He told Snyder to solve the problem for a perfectly spherical star with no internal forces. Unrealistic though their model was, Oppenheimer and Snyder’s final result was remarkably accurate.

But even with the simplifications, the problem was not easy. Oppenheimer and Snyder had to work out how the contraction of the star would affect the spacetime inside it. (Unbeknownst to them, the problem had been solved a year earlier by an Indian physicist named Bishveshwar Datt).

For all their simplifications, their final result provided a remarkably accurate picture of the birth of a black hole. It showed that a black hole would inevitably form once the star collapsed into its critical radius. It also showed that the star would continue to implode, eventually reaching infinite density, creating a singularity.

Oppenheimer and Snyder’s work also produced a striking demonstration of the relativity of time for different observers. For an observer on an imploding star that was as heavy as our Sun, it would take mere hours to shrink to the size of the critical radius. But for an observer outside, it would take an eternity. They would see the collapse get slower and slower as the star shrank to become smaller and smaller, never quite crossing the critical radius.

The Oppenheimer-Snyder paper should have closed the debate on black holes. Unfortunately, most physicists were not ready to accept the weirdness of black holes yet and argued that the idealisations that Oppenheimer and Snyder had made were too unrealistic. Oppenheimer himself lost all interest in the subject: he would change the topic whenever someone tried to discuss it with him.

It was only after Roger Penrose proved the inevitability of black hole formation that the importance of the Oppenheimer-Snyder paper became recognised. But by then, both the authors were deceased.

Into the black hole

One has to wonder how Oppenheimer’s life and career would have panned out under different circumstances, if he could have continued at Berkeley as the brilliant teacher and physicist that he was. But that was not to be.

Oppenheimer’s and Snyder’s paper was published by the journal Physical Review on September 1, 1939. Two other notable events took place that day. First, Niels Bohr and John Wheeler published a paper that explained nuclear fission and demonstrated the utility of the isotope uranium-235 to produce nuclear chain reactions. Second, Adolf Hitler’s army invaded Poland, starting World War II.

The course of history from that point on was perhaps as inevitable as the collapse of a star into a black hole. Oppenheimer was caught in its relentless pull, never to escape.

Nirmalya Kajuri is an assistant professor of physics in IIT Mandi.