section_path: History/General relativity
section_name: General relativity
paragraph_text: General relativity
  
  In 1915, Albert Einstein developed his theory of general relativity, having earlier shown that gravity does influence light's motion. Only a few months later, Karl Schwarzschild found a solution to the Einstein field equations that describes the gravitational field of a point mass and a spherical mass. A few months after Schwarzschild, Johannes Droste, a student of Hendrik Lorentz, independently gave the same solution for the point mass and wrote more extensively about its properties. This solution had a peculiar behaviour at what is now called the Schwarzschild radius, where it became singular, meaning that some of the terms in the Einstein equations became infinite. The nature of this surface was not quite understood at the time.
  
  In 1924, Arthur Eddington showed that the singularity disappeared after a change of coordinates. In 1933, Georges Lemaître realised that this meant the singularity at the Schwarzschild radius was a non-physical coordinate singularity. Arthur Eddington commented on the possibility of a star with mass compressed to the Schwarzschild radius in a 1926 book, noting that Einstein's theory allows us to rule out overly large densities for visible stars like Betelgeuse because "a star of 250 million km radius could not possibly have so high a density as the Sun. Firstly, the force of gravitation would be so great that light would be unable to escape from it, the rays falling back to the star like a stone to the earth. Secondly, the red shift of the spectral lines would be so great that the spectrum would be shifted out of existence. Thirdly, the mass would produce so much curvature of the spacetime metric that space would close up around the star, leaving us outside (i.e., nowhere)."
  
  In 1931, Subrahmanyan Chandrasekhar calculated, using special relativity, that a non-rotating body of electron-degenerate matter above a certain limiting mass (now called the Chandrasekhar limit at 1.4 M) has no stable solutions. His arguments were opposed by many of his contemporaries like Eddington and Lev Landau, who argued that some yet unknown mechanism would stop the collapse. They were partly correct: a white dwarf slightly more massive than the Chandrasekhar limit will collapse into a neutron star, which is itself stable.
  
  In 1939, Robert Oppenheimer and others predicted that neutron stars above another limit, the Tolman–Oppenheimer–Volkoff limit, would collapse further for the reasons presented by Chandrasekhar, and concluded that no law of physics was likely to intervene and stop at least some stars from collapsing to black holes. Their original calculations, based on the Pauli exclusion principle, gave it as 0.7 M. Subsequent consideration of neutron-neutron repulsion mediated by the strong force raised the estimate to approximately 1.5 M to 3.0 M. Observations of the neutron star merger GW170817, which is thought to have generated a black hole shortly afterward, have refined the TOV limit estimate to ~ 2.17 M.
  
  Oppenheimer and his co-authors interpreted the singularity at the boundary of the Schwarzschild radius as indicating that this was the boundary of a bubble in which time stopped. This is a valid point of view for external observers, but not for infalling observers. The hypothetical collapsed stars were called "frozen stars", because an outside observer would see the surface of the star frozen in time at the instant where its collapse takes it to the Schwarzschild radius.
  
  Also in 1939, Einstein attempted to prove that black holes were impossible in his publication "On a Stationary System with Spherical Symmetry Consisting of Many Gravitating Masses", using his theory of general relativity to defend his argument. Months later, Oppenheimer and his student Hartland Snyder provided the Oppenheimer–Snyder model in their paper "On Continued Gravitational Contraction", which predicted the existence of black holes. In the paper, which made no reference to Einstein's recent publication, Oppenheimer and Snyder used Einstein's own theory of general relativity to show the conditions on how a black hole could develop, for the first time in contemporary physics.