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.

Singularities and black holes

The Schwarzschild solution appears to have singularities at r = 0 and r=r_s; some of the metric components blow up at these radii. Since the Schwarzschild metric is only expected to be valid for radii larger than the radius R of the gravitating body, there is no problem as long as R > r_s. For ordinary stars and planets this is always the case. For example, the radius of the Sun is approximately 700,000 km, while its Schwarzschild radius is only 3 km.

Related Topics:
Singularities - Sun

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One might naturally wonder what happens when the radius R becomes less than or equal to the Schwarzschild radius r_s. It turns out that the Schwarzchild solution still makes sense in this case, although it has some rather odd properties. The apparent singularity at r = r_s is an illusion; it is an example of what is called a coordinate singularity. As the name implies, the singularity arises from a bad choice of coordinates. By choosing another set of suitable coordinates one can show that the metric is well-defined at the Schwarzschild radius. See, for example, Eddington-Finkelstein coordinates or Kruskal coordinates.

Related Topics:
Coordinate singularity - Eddington-Finkelstein coordinates - Kruskal coordinates

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This case r = 0 is different, however. If one asks that the solution be valid for all r one runs into a true physical singularity, or gravitational singularity, at the origin. To see that this is a true singularity one must look at quantities that are independent of the choice of coordinates. One such important quantity is the Kretschmann invariant, which is given by

Related Topics:
Gravitational singularity - Kretschmann invariant

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:R^{abcd}R_{abcd}= rac{12 r_s^2}{r^6}

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At r=0 the curvature blows-up (becomes infinite) indicating the presence of a singularity. At this point the metric, and space-time itself, is no longer well-defined. For a long time it was thought that such a solution was non-physical. However, a greater understanding of general relativity led to the realization that such singularities were a generic feature of the theory and not just an exotic special case. Such solutions are now believed to exist and are termed black holes.

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The Schwarzschild solution, taken to be valid for all r > 0, is called a Schwarzschild black hole. It is a perfectly valid solution of the Einstein field equations, although it has some rather bizarre properties. For r < r_s the Schwarzschild radial coordinate r becomes timelike and the time coordinate t becomes spacelike. A curve at constant r is no longer a possible worldline of a particle or observer, not even if a force is exerted to try to keep it there; this occurs not just because the gravitational field is strong but because spacetime has been curved so much that the direction of cause and effect (the particle's future light cone) points into the singularity. The surface r = r_s demarcates what is called the event horizon of the black hole. It represents the point past which light can no longer escape the gravitational field. Any physical object whose radius R becomes less than or equal to the Schwarzschild radius will undergo gravitational collapse and become a black hole.

Related Topics:
Schwarzschild black hole - Worldline - Light cone - Event horizon - Gravitational collapse

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