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This diagram gives the route to find the Schwarzschild solution by using the weak field approximation. The equality on the second row gives g 44 = −c 2 + 2GM/r, assuming the desired solution degenerates to Minkowski metric when the motion happens far away from the blackhole (r approaches to positive infinity).
Johannes Droste in 1916 [11] independently produced the same solution as Schwarzschild, using a simpler, more direct derivation. [12] 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 Schwarzschild's ...
The solution was proposed independently by Paul Painlevé in 1921 [1] and Allvar Gullstrand [2] in 1922. It was not explicitly shown that these solutions were simply coordinate transformations of the usual Schwarzschild solution until 1933 in Lemaître's paper, [3] although Einstein immediately believed that to be true.
The Schwarzschild radius was named after the German astronomer Karl Schwarzschild, who calculated this exact solution for the theory of general relativity in 1916. The Schwarzschild radius is given as r s = 2 G M c 2 , {\displaystyle r_{\text{s}}={\frac {2GM}{c^{2}}},} where G is the gravitational constant , M is the object mass, and c is the ...
See Deriving the Schwarzschild solution for a more detailed derivation of this expression. Depending on context, it may be appropriate to regard a and b as undetermined functions of the radial coordinate (for example, in deriving an exact static spherically symmetric solution of the Einstein field equation). Alternatively, we can plug in ...
In Einstein's theory of general relativity, the interior Schwarzschild metric (also interior Schwarzschild solution or Schwarzschild fluid solution) is an exact solution for the gravitational field in the interior of a non-rotating spherical body which consists of an incompressible fluid (implying that density is constant throughout the body) and has zero pressure at the surface.
The Schwarzschild metric is named in honour of its discoverer Karl Schwarzschild, who found the solution in 1915, only about 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 solution in Schwarzschild coordinates, with two space dimensions suppressed, leaving just the time t and the distance from the center r. In red the incoming null geodesics. In blue outcoming null geodesics. In green the null light cones on which borders light moves, while massive objects move inside the cones.