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In general relativity, the Oppenheimer–Snyder model is a solution to the Einstein field equations based on the Schwarzschild metric describing the collapse of an object of extreme mass into a black hole.^[1] It is named after physicists J. Robert Oppenheimer and Hartland Snyder, who published it in 1939.^[2]

During the collapse of a star to a black hole the geometry on the outside of the sphere is the Schwarzschild geometry. However the geometry inside is, curiously enough, the same Robertson-Walker geometry as in the rest of the observable universe.^[3]

Albert Einstein, who had developed his theory of general relativity in 1915, initially denied the possibility of black holes,^[4] even though they were a genuine implication of the Schwarzschild metric, obtained by Karl Schwarzschild in 1916, the first known non-trivial exact solution to Einstein's field equations.^[1] In 1939, Einstein published "On a Stationary System with Spherical Symmetry Consisting of Many Gravitating Masses" in the Annals of Mathematics, claiming to provide "a clear understanding as to why these 'Schwarzschild singularities' do not exist in physical reality."^[4][5]

Months after the issuing of Einstein's article,^[4] J. Robert Oppenheimer and his student Hartland Snyder studied this topic with their paper "On Continued Gravitational Contraction" making the opposite argument as Einstein's.^[6][5] They showed when a sufficiently massive star runs out of thermonuclear fuel, it will undergo continued gravitational contraction and become separated from the rest of the universe by a boundary called the event horizon, which not even light can escape. This paper predicted the existence of what are today known as black holes.^[1][7] The term "black hole" was coined decades later, in the fall of 1967, by John Archibald Wheeler at a conference held by the Goddard Institute for Space Studies in New York City;^[7] it appeared for the first time in print the following year.^[8] Oppenheimer and Snyder used Einstein's own theory of gravity to prove how black holes could develop for the first time in contemporary physics, but without referencing the aforementioned article by Einstein.^[4] Oppenheimer and Snyder did, however, refer to an earlier article by Oppenheimer and Volkoff on neutron stars, improving upon the work of Lev Davidovich Landau.^[7] Previously, and in the same year, Oppenheimer and three colleagues, Richard Tolman, Robert Serber, and George Volkoff, had investigated the stability of neutron stars, obtaining the Tolman-Oppenheimer-Volkoff limit.^[9][10][11] Oppenheimer would not revisit the topic in future publications.^[12]

The Oppenheimer–Snyder model of continued gravitational collapse is described by the line element^[13] $${\displaystyle ds^{2}=-d\tau ^{2}+A^{2}(\eta )\left({\frac {dR^{2}}{1-2M{\frac {R_{-}^{2}}{R_{b}^{2}}}{\frac {1}{R_{+}}}}}+R^{2}d\Omega ^{2}\right)}$$ The quantities appearing in this expression are as follows:

• The coordinates are ${\displaystyle (\tau ,R,\theta ,\phi )}$ where ${\displaystyle \theta ,\phi }$ are coordinates for the 2-sphere.
• ${\displaystyle R_{b}}$ is a positive quantity, the "boundary radius", representing the boundary of the matter region.
• ${\displaystyle M}$ is a positive quantity, the mass.
• ${\displaystyle R_{-}=\mathrm {min} (R,R_{b})}$ and ${\displaystyle R_{+}=\mathrm {max} (R,R_{b})}$.
• ${\displaystyle \eta }$ is defined implicitly by the equation

$${\displaystyle \tau (\eta ,R)={\frac {1}{2}}{\sqrt {\frac {R_{+}^{3}}{2M}}}(\eta +\sin \eta ).}$$

• ${\displaystyle A(\eta )={\frac {1+\cos \eta }{2}}}$.

This expression is valid both in the matter region ${\displaystyle R<R_{b}}$, and the vacuum region ${\displaystyle R>R_{b}}$, and continuously transitions between the two.

Kip Thorne recalled that physicists were initially skeptical of the model, viewing it as "truly strange" at the time.^[12] He explained further, "It was hard for people of that era to understand the paper because the things that were being smoked out of the mathematics were so different from any mental picture of how things should behave in the universe."^[14] Oppenheimer himself thought little of this discovery.^[2] However, some considered the model's discovery to be more significant than Oppenheimer did,^[2] and model would later be described as forward thinking.^[12] Freeman Dyson thought it was Oppenheimer's greatest contribution to science. Lev Davidovich Landau added the Oppenheimer-Snyder paper to his "golden list" of classic papers.^[2] John Archibald Wheeler was initially an opponent of the model until the late 1950s,^[1][12] when he was asked to teach a course on general relativity at Princeton University.^[8] Wheeler claimed at a conference in 1958 that the Oppenheimer-Snyder model had neglected the many features of a realistic star. However, he later changed his mind completely after being informed by Edward Teller that a computer simulation ran by Stirling Colgate and his team at the Lawrence Livermore National Laboratory had shown a sufficiently heavy star would undergo continued gravitational contraction in a manner similar to the idealized scenario described by Oppenheimer and Snyder.^[1] Wheeler subsequently played a key role in reviving interest in general relativity in the United States, and popularized the term "black hole" in the late 1960s.^[8] Various theoretical physicists pursued this topic^[5] and by the late 1960s and early 1970s, advances in observational astronomy, such as radio telescopes, changed the attitude of the scientific community.^[14] Pulsars had already been discovered and black holes were no longer considered mere textbook curiosities.^[15] Cygnus X-1, the first solid black-hole candidate, was discovered by the Uhuru X-ray space telescope in 1971.^[1] Jeremy Bernstein described it as "one of the great papers in twentieth-century physics."^[14]

After winning the Nobel Prize in Physics in 2020, Roger Penrose would credit the Oppenheimer–Snyder model as one of his inspirations for research.^[16][12]

The Hindu wrote in 2023:^[17]

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.

• In the 2023 film Oppenheimer, an interaction between Oppenheimer and his student Snyder occurs as their paper was published on the same day as the Invasion of Poland.^[17][18]

• Tolman–Oppenheimer–Volkoff equation
• Timeline of gravitational physics and relativity