Schwarzschild metric

In Einstein's theory of general relativity, the Schwarzschild solution (or the Schwarzschild vacuum) describes the gravitational field outside a spherical, non-rotating mass such as a (non-rotating) star, planet, or black hole. It is also a good approximation to the gravitational field of a slowly rotating body like the Earth or Sun. According to Birkhoff's theorem, the Schwarzchild solution is the most general static, spherically symmetric, vacuum solution of Einstein's field equations. A Schwarzschild black hole or static black hole is a black hole that has no charge or angular momentum. A Schwarzschild black hole has a Schwarzschild metric, and cannot be distinguished from any other Schwarzschild black hole except by its mass.

Embedding Schwarzschild space in Euclidean space

In general relativity mass changes the geometry of space. Space with mass is "curved", whereas empty space is flat (Euclidean). In some cases we can visualize the deviation from Euclidean geometry by mapping a 'curved' subspace of the 4-dimensional spacetime onto a Euclidean space with one dimension more.

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Suppose we choose the equatorial plane of a star, at a constant Schwarzschild time t=t_0 and heta=pi/2 and map this into three dimensions with the Euclidean metric (in cylindrical coordinates):

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:ds^2 = dz^2 + dr^2 + r^2dphi^2.,

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We will get a curved surface z= z(r) by writing the Euclidean metric in the form

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:ds^2 = left(1 + left(rac{dz}{dr} ight)^2 ight)dr^2 + r^2dphi^2

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where we have made the identification

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:dz = rac{dz}{dr}dr.

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We can then relate this to the Schwarzschild metric for the equatorial plane at a fixed time:

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:ds^2 = left(1-rac{2GM}{c^2 r} ight)^{-1} dr^2 + r^2dphi^2

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Which gives the following expression for z(r):

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:z(r) = int rac{dr}{sqrt{rac{c^2 r}{2GM}-1}} = 4GMsqrt{rac{c^2 r}{2GM}- 1} + mbox{ a constant}.

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