Schwarzschild radius
The Schwarzschild radius (sometimes erroneously referred to as the gravitational radius, which does not have the factor of 2) is a physical parameter in the Schwarzschild solution to Einstein's field equations that corresponds to the radius defining the event horizon of a Schwarzschild black hole. It is a characteristic radius associated with any quantity of mass. The Schwarzschild radius was named after the German astronomer Karl Schwarzschild, who calculated this exact solution for the theory of general relativity in 1916. The Schwarzschild radius is given as formula_1 where "G" is the gravitational constant, "M" is the object  mass, and "c" is the speed of light. In natural units, the gravitational constant and the speed of light are both taken to be unity, so the Schwarzschild radius is formula_2. History. In 1916, Karl Schwarzschild obtained the exact solution to Einstein's field equations for the gravitational field outside a non-rotating, spherically symmetric body with mass formula_3 (see Schwarzschild metric). The solution contained terms of the form formula_4 and formula_5, which become singular at formula_6 and formula_7 respectively. The formula_8 has come to be known as the "Schwarzschild radius". The physical significance of these singularities was debated for  decades. It was found that the one at formula_9 is a coordinate singularity, meaning that it is an artifact of the particular system of coordinates that were used; while the one at formula_10 is a spacetime singularity and cannot be removed. The Schwarzschild radius is nonetheless a physically relevant quantity, as noted above and below. This expression had previously been calculated, using Newtonian mechanics, as the radius of a spherically symmetric body at which the escape velocity was equal to the speed of light. It had been identified in the 18th century by John Michell and Pierre-Simon Laplace . Parameters. The Schwarzschild radius of an object is proportional to its mass. Accordingly, the Sun has a Schwarzschild radius of approximately , whereas Earth's is only about and the Moon's is about . The observable universe's mass has a Schwarzschild radius of approximately 13.7 billion light-years. Black hole classification by Schwarzschild radius. Any object whose radius is smaller than its Schwarzschild radius is called a black hole. The surface at the Schwarzschild radius acts as an event horizon in a non-rotating 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". Black holes can be classified based on their Schwarzschild radius, or equivalently, by their density, where density is defined as mass of a black hole divided by the volume of its Schwarzschild sphere. As the Schwarzschild radius is linearly related to mass, while the enclosed volume corresponds to the third power of the radius, small black holes are therefore much more dense than large ones. The volume enclosed in the event horizon of the most massive black holes has an average density lower than main sequence stars. Supermassive black hole . A supermassive black hole (SMBH) is the largest type of black hole, though there are few official criteria on how such an object is considered so, on the order of hundreds of thousands to billions of solar masses. (Supermassive black holes up to 21 billion have been detected, such as NGC 4889.) Unlike stellar mass black holes, supermassive black holes have comparatively low average densities. (Note that a (non-rotating) black hole is a spherical region in space that surrounds the singularity at its center; it is not the singularity itself.) With that in mind, the average density of a  supermassive black hole can be less than the density of water. The Schwarzschild radius of a body is proportional to its mass and therefore to its volume, assuming that the body has a constant mass-density. In contrast, the physical radius of the body is proportional to the cube root of its volume. Therefore, as the body accumulates matter at a given fixed density (in this example, 997 kg/m, the density of water), its Schwarzschild radius will increase more quickly than its physical radius. When a body of this density has grown to around 136 million solar masses , its  physical radius would be overtaken by its Schwarzschild radius, and thus it would form a supermassive black hole. It is thought that supermassive black holes like these do not form immediately from the singular collapse of a cluster of stars. Instead they may begin life as smaller, stellar-sized black holes and grow larger by the accretion of matter, or even of other black holes. The Schwarzschild radius of the supermassive black hole at the Galactic Center of the Milky Way is approximately 12 million kilometres. Stellar black hole. Stellar black holes have much greater average densities than supermassive black  holes. If one accumulates matter at nuclear density (the density of the nucleus of an atom, about 10 kg/m; neutron stars also reach this density), such an accumulation would fall within its own Schwarzschild radius at about and thus would be a stellar black hole. Micro black hole. A small mass has an extremely small Schwarzschild radius. A mass similar to Mount Everest has a Schwarzschild radius much 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 miniature black holes are called primordial black holes. Other uses. In gravitational time dilation. Gravitational time dilation near a large, slowly rotating, nearly spherical body, such as the Earth or Sun can be reasonably as follows: formula_11 where: Schwarzschild radius for Planck mass. For the Planck mass formula_12, the Schwarzschild radius formula_13 and the Compton wavelength formula_14 are of the same order as the Planck length formula_15. Classification of black holes by type: A classification of black  holes by mass: