Schwarzschild Metric - History

History

The Schwarzschild solution is named in honor of Karl Schwarzschild, who found the exact solution in 1916, a little more than a month after the publication of Einstein's theory of general relativity. It was the first exact solution of the Einstein field equations other than the trivial flat space solution. Schwarzschild had little time to think about his solution. He died shortly after his paper was published, as a result of a disease he contracted while serving in the German army during World War I.

Johannes Droste in 1916 independently produced the same solution as Schwarzschild, using a simpler more direct derivation.

In the early years of general relativity there was a lot of confusion about the nature of the singularities found in the Schwarzschild and other solutions of the Einstein field equations. In response to the singular nature of his solution, Schwarzschild (mistakenly) assumed that the outer most singularity (i.e. the one at r = r_s) must coincide with coordinate singularity of spherical coordinate system. To this end he maintained that the radial coordinate that appeared in his solution was not physical, and instead an alternative radial coordinate ρ, related to the one in his solution by r = (ρ3+r_s3)1/3, which puts the event horizon at the origin of the coordinate system.

A more complete analysis of the singularity structure was given by David Hilbert in the following year, identifying the singularities both at r = 0 and r = r_s. Although there was general consent that the singularity at r = 0 was a 'genuine' physical singularity, the nature of the singularity at r = r_s remained unclear.

In 1924 Arthur Eddington produced the first coordinate transformation (Eddington–Finkelstein coordinates) that showed that the singularity at r = r_s was a coordinate artifact, although he seems to have been unaware of the significance of this discovery. Later, in 1932, Georges Lemaître gave a different coordinate transformation (Lemaître coordinates) to the same effect and was the first to recognize that this implied that the singularity at r = r_s was not physical. In 1939 Howard Robertson showed that a free falling observer descending in the Schwarzschild metric would cross the r = r_s singularity in a finite amount of proper time even though this would take an infinite amount of time in terms of coordinate time t.

In 1950, John Synge produced a paper that showed the maximal analytic extension of the Schwarzschild metric, again showing that the singularity at r = r_s was a coordinate artifact. This result was later rediscovered by Martin Kruskal, who improved on Synge's result by providing a single set of coordinates that covered (almost) the entire spacetime. However due to the obscurity of the journals in which the papers of Lemaître and Synge were published their conclusions went unnoticed, with many of the major players in the field including Einstein believing that singularity at the Schwarzschild radius was physical.

Progress was only made in the 1960s when the more exact tools of differential geometry entered the field of general relativity allowing more exact definitions of what it means for a Lorentzian manifold to be singular. This led to definitive identification of the r = r_s singularity in the Schwarzschild metric as an event horizon (a hypersurface in spacetime that can only be crossed in one direction).