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.

The Schwarzschild metric

In Schwarzschild coordinates, the Schwarzschild metric can be put into the form (see deriving the Schwarzschild solution)

Related Topics:
Schwarzschild coordinates - Deriving the Schwarzschild solution

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

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where G is the gravitational constant, M is interpreted as the mass of the gravitating object, and

Related Topics:
Gravitational constant - Mass

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:dOmega^2 = d heta^2+sin^2 heta dphi^2,

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is the standard metric on the 2-sphere (i.e. the standard element of solid angle). The constant

Related Topics:
2-sphere - Solid angle

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:r_s = rac{2GM}{c^2}

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is called the Schwarzschild radius and plays an important role in the Schwarzschild solution.

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The Schwarzschild metric is a solution to vacuum field equations, meaning that it is only valid outside the gravitating body. That is, for a spherical body of radius R the solution is valid for r > R. (Although, if R is less then the Schwarzschild radius r_s then the solution describes a black hole; see below.) In order to describe the gravitational field both inside and outside the gravitating body the Schwarzschild solution must be matched with some suitable interior solution at r = R.

Related Topics:
Vacuum field equations - Black hole

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Note that as M o 0 or r ightarrowinfty one recovers the Minkowski metric:

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:ds^{2} = -c^2dt^2 + dr^2 + r^2 dOmega^2.,

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Intuitively, this makes sense, as far away from any gravitating bodies we expect space to be nearly flat. Metrics with this property are called asymptotically flat.

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