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The Schwarzschild radius is proportional to the mass.
The Sun has a Schwarzschild radius of approximately 3 km, the Earth 's being approximately 9 mm.

An object smaller than its Schwarzschild radius is called a
Black Hole . The surface at the Schwarzschild radius acts as an
Event Horizon in a static body. (A Rotating Black Hole operates slightly differently.) Neither light nor particles can escape through this surface from the region inside, hence the name "black hole". The Schwarzschild radius of the Supermassive Black Hole at our Galactic Center is approximately 7.8 million km. The Schwarzschild radius of a sphere with a uniform density equal to the Critical Density is equal to the radius of the Visible Universe .

Connie Willis's Hard Science Fiction short story "The Schwarzschild Radius" offers both an accessible and accurate explanation of the phenomenon which makes it surprisingly applicable to not-so-scientific pursuits.

It is coincidental that ''Schwarzschild'' also means "black shield", which is befitting for the idea of a black hole.


FORMULA FOR THE SCHWARZSCHILD RADIUS


The Schwarzschild radius is proportional to the mass, with a proportionality constant involving the Gravitational Constant and the speed of light.
The formula for the Schwarzschild radius can be found by setting the Escape Velocity to the speed of light, and is
: r_s = rac{2Gm}{c^2}

where

: r_s is the Schwarzschild radius

: G is the Gravitational Constant , that is 6.67 × 10-11 N m2 / kg2;

: ''m'' is the mass of the gravitating object; and

: ''c''² is the Speed Of Light squared, that is (299,792,458 m/s)² = 8.98755 × 1016 m²/s².

The proportionality constant, 2G/c^2, can be approximated as 1.48 × 10-27 m / kg.

This means that the equation can be approximately written as

: r_s = m imes 1.48 imes 10^{-27}

with r_s in meters and m in kilograms.

Note that although the result is correct, General Relativity must be used to truly derive the Schwarzschild radius. It is only a coincidence that Newtonian Physics produces the same result.


CLASSIFICATION BY SCHWARZSCHILD RADIUS



Supermassive black hole


If one accumulates matter of normal density (say 1000 kg/m³, 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 up to a few billion solar masses are thought to exist). The supermassive black hole in the center of our galaxy (2.5 million solar masses) constitutes observationally the most convincing evidence for the existence of black holes in general.
It is thought that large black holes like these don't form directly in one collapse of a cluster of stars.
Instead they may start as a stellar-sized black hole and grow larger by the accretion of matter and other black holes. The larger the mass of a galaxy, the larger is the mass of the supermassive black hole in its center.


Stellar black hole


If one accumulates matter at Nuclear Density (the density of the nucleus of an atom, about 1018 Kg/m³ ; Neutron Star s also reach this density), such an accumulation would fall within its own Schwarzschild radius at about 3 solar masses and thus would be a Stellar Black Hole .


Primordial black hole


Conversely, a small mass has an extremely small Schwarzschild radius. A mass as big as Mount Everest has a Schwarzschild radius smaller than a Nanometre . Its average density at that size would be so high that no known mechanism could form such extremely compact objects. Such black holes might possibly be formed in an early stage of the evolution of the universe, just after the Big Bang , when densities were extremely high. Therefore these hypothetical baby black holes are called Primordial Black Hole s.


SEE ALSO

  • Black Hole , a general survey

    • Classification of black holes by type: