Oppenheimer–Snyder model

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. It is named after physicists J. Robert Oppenheimer and Hartland Snyder, who published it in 1939.

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.

History

Albert Einstein, who had developed his theory of general relativity in 1915, initially denied the possibility of black holes, 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. In 1939, Einstein published "On a Stationary System with Spherical Symmetry Consisting of Many Gravitating Masses" in the Annals of Mathematics, using his theory to argue that black holes were not possible.

Months after the issuing of Einstein's article, J. Robert Oppenheimer and his student Hartland Snyder studied this topic with their paper "On Continued Gravitational Contraction". 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. 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; it appeared for the first time in print the following year. 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. 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. 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. Oppenheimer would not revisit the topic in future publications.

Model

The Oppenheimer–Snyder model of continued gravitational collapse is described by the line element

$ds^{2}=-d{\tau ^{2}}+A^{2}({\eta })\left({\frac {dR^{2}}{1-2M{\frac {R_{-}^{2}}{R_{b}^{2}}}{\frac {1}{R_{+}}}}}+R^{2}d\Omega ^{2}\right)$.

The quantities appearing in this expression are as follows:

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

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

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

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

Reception and legacy

Physicists were initially skeptical of the model, with Kip Thorne saying that the community saw the model as "truly strange" at the time. Oppenheimer himself thought little of this discovery; however, some considered the model's discovery to be more significant than Oppenheimer did. 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. John Archibald Wheeler was initially an opponent model until the 1950s, when he was asked to teach a course on general relativity at Princeton University. Wheeler played a key role in reviving interest in general relativity in the United States, and popularized the term "black hole" in the late 1960s.

The model would later be described as forward thinking. After winning the Nobel Prize in Physics in 2020, Roger Penrose would credit the Oppenheimer–Snyder model as one of his inspirations for research.

The Hindu wrote in 2023:

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 popular culture

• 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.

See also

• Tolman–Oppenheimer–Volkoff equation