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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 ...
Gullstrand–Painlevé coordinates are a particular set of coordinates for the Schwarzschild metric – a solution to the Einstein field equations which describes a black hole. The ingoing coordinates are such that the time coordinate follows the proper time of a free-falling observer who starts from far away at zero velocity, and the spatial ...
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 ...
Unlike the nonlinear Poisson's equation Eq(7.b), Eq(8.b) is the linear Laplace equation; that is to say, superposition of given vacuum solutions to Eq(8.b) is still a solution. This fact has a widely application, such as to analytically distort a Schwarzschild black hole .
In the Schwarzschild solution, it is assumed that the larger mass M is stationary and it alone determines the gravitational field (i.e., the geometry of space-time) and, hence, the lesser mass m follows a geodesic path through that fixed space-time. This is a reasonable approximation for photons and the orbit of Mercury, which is roughly 6 ...
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.