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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).
The exterior Schwarzschild solution with r > r s is the one that is related to the gravitational fields of stars and planets. The interior Schwarzschild solution with 0 ≤ r < r s, which contains the singularity at r = 0, is completely separated from the outer patch by the singularity at r = r s. The Schwarzschild coordinates therefore give no ...
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 ...
In an isotropic chart (on a static spherically symmetric spacetime), the metric (aka line element) takes the form = + (+ (+ ())), < <, < <, < <, < < Depending on context, it may be appropriate to regard , as undetermined functions of the radial coordinate (for example, in deriving an exact static spherically symmetric solution of the Einstein field equation).
Schwarzschild's equation is the formula by which you may calculate the intensity of any flux of electromagnetic energy after passage through a non-scattering medium when all variables are fixed, provided we know the temperature, pressure, and composition of the medium.
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.
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.
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 ...
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