Schwarzschild Radius Formula For The Schwarzschild Radius

of an exact solution for the gravitational field outside a static, spherically symmetric star (see Schwarzschild metric, which is a solution of the

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

1 Formula for the Schwarzschild radius

The Schwarzschild radius is proportional to the mass, with a proportional constant involving the gravitational constantAccording to the law of universal gravitation, the attractive force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. The constant of proportionality is called , the gr and the speed of light. The formula for the Schwarzschild radius is

2 Average density within the Schwarschild radius

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

so the larger the mass the smaller the average density of matter is, if squeezed to within its Schwarzschild radius.

3 Classification by Schwarzschild radius

3.1 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 holeA supermassive black hole is a black hole with a mass in the range of millions or billions solar masses. The Milky Way is believed to have one supermassive black hole, at the Galactic Center. Most if not all galaxies are thought to host a supermassive bla 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.

3.2 Stellar black hole

If one accumulates matter at nuclear density (the density of the nucleus of an atom, about 10¹⁸ kg/m³Kilogram per cubic metre is the SI measure of density and is represented as kg/m³, where kg stands for kilogram and m³ stands for cubic metre. The density of water is about 1000 kg/m³, since a cubic metre of water weighs about a tonne. kg/m³" is sometimes; neutron starThis article is about the celestial body. Neutron Star was a 1966 Hugo award winning short story by Larry Niven A neutron star is a compact star in which the weight of the star is carried by the pressure of free neutrons. It is a so called degenerate stars 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.

3.3 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 nanometreA nanometre ( American spelling: nanometer is 10−9 metres—or one billionth of a metre. Its symbol is nm . It is an SI measure of length, commonly used in measuring the wavelengths of visible light, ultraviolet radiation and gamma rays; amongst other. 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 holes.

4 See also

black hole, a general survey

Classifification of black holes by type:

A classification of black holes by mass:

micro black hole and extra-dimensional black hole
primordial black hole, a hypothetical leftover of the Big Bang
stellar black hole, which could either be a static black hole or a rotating black hole
supermassive black hole, which could also either be a static black hole or a rotating black hole

Black holes

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