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In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild solution) is an exact solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assumption that the electric charge of the mass, angular momentum of the mass, and universal cosmological constant are all zero.
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]
Karl Schwarzschild (German: [kaʁl ˈʃvaʁtsʃɪlt] ⓘ; 9 October 1873 – 11 May 1916) was a German physicist and astronomer.. Schwarzschild provided the first exact solution to the Einstein field equations of general relativity, for the limited case of a single spherical non-rotating mass, which he accomplished in 1915, the same year that Einstein first introduced general relativity.
The function () is clearly singular at r=2M as it must be to remove that singularity in the Schwarzschild metric. Other choices for a ( r ) {\displaystyle a(r)} lead to other coordinate charts for the Schwarzschild vacuum; a general treatment is given in Francis & Kosowsky.
This metric has a coordinate singularity at the Schwarzschild radius =. Georges Lemaître was the first to show that this is not a real physical singularity but simply a manifestation of the fact that the static Schwarzschild coordinates cannot be realized with material bodies inside the Schwarzschild radius.
The transformation between Schwarzschild coordinates and Kruskal–Szekeres coordinates defined for r > 2GM and < < can be extended, as an analytic function, at least to the first singularity which occurs at =. Thus the above metric is a solution of Einstein's equations throughout this region.
In deriving the Schwarzschild metric, it was assumed that the metric was vacuum, spherically symmetric and static. The static assumption is unneeded, as Birkhoff's theorem states that any spherically symmetric vacuum solution of Einstein's field equations is stationary; the Schwarzschild solution thus follows
The de Sitter–Schwarzschild spacetime is a combination of the two, and describes a black hole horizon spherically centered in an otherwise de Sitter universe. An observer who hasn't fallen into the black hole, and who can still see the black hole despite the inflation is sandwiched between the two horizons.