In physics and astronomy, especially in the theory of gravitation, general relativity, the Schwarzschild radius or gravitational radius is a characteristic radius associated with every mass. It was found in 1916 by Karl Schwarzschild and results from his discovery of an exact solution for the gravitational field outside a static, spherically symmetric star (see Schwarzschild metric, which is a solution of the Einstein field equations). The Schwarzschild radius is proportional to the mass. The Sun has a Schwarzschild radius of about 3 km, the Earth about 3 cm.
An object smaller than the Schwarzschild radius, is called a black hole. The surface at the Schwarzschild radius acts as an event horizon. Neither light nor particles can escape through this surface from the region inside, hence the name black hole.
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2 Average density within the Schwarschild radius |
The equation for the Schwarzschild radius is
where
It is interesting to see what the average density of an amount of matter of mass is, if squeezed inside a volume with radius equal to the Schwarzschild radius.
The volume for a sphere of radius scales as the third power of the radius, . The Schwarzschild radius is proportional to the mass , so the volume inside the Schwarzschild radius scales as . The average density scales as
A few simple results are as follows.
If one accumulates matter of normal density (say 1000 kg/m^{3}, such as water, which also happens to be about the same as the average density of the Sun) up to about 300,000 times the mass of the Sun, such an accumulation will fall inside its own Schwarzschild radius and thus it would be a supermassive black hole of 300,000 solar masses (Supermassive black holes are seen up to a few billion solar masses).
If one accumulates matter at a density inside the nucleus of an atom (say about 10^{18} kg/m^{3}) (like inside neutron stars), such an accumulation would fall within its own Schwarzschild radius at about 3 solar masses and thus would be a stellar black hole.
Conversely, small mass have an extremely small Schwarzschild radius. A mass as big as the Mount Everest has a Schwarzschild radius smaller than a nanometer. Its average density at that size would so high that no known mechanisme could form such a primordial black hole. They might possibly be formed in an early stage of the evolution of the universe, just after the Big Bang, when densities are extremely high.