Schwarzschild radius
The Schwarzschild radius or the gravitational radius 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.
The Schwarzschild radius was named after the German astronomer Karl Schwarzschild, who calculated this exact solution for the theory of general relativity in 1916.
where G is the gravitational constant, M is the object mass, and c is the speed of light.
[note 1][1][2] In 1916, Karl Schwarzschild obtained the exact solution[3][4] to Einstein's field equations for the gravitational field outside a non-rotating, spherically symmetric body with mass
[5] 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[6] and Pierre-Simon Laplace.
[7] The Schwarzschild radius of an object is proportional to its mass.
[16]: 410 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.
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 (2.1 × 1010) M☉ have been detected, such as NGC 4889.
)[17] Unlike stellar mass black holes, supermassive black holes have comparatively low average densities.
[18] 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/m3, 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 (1.36 × 108 M☉), 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.
[19] The Schwarzschild radius of the supermassive black hole at the Galactic Center of the Milky Way is approximately 12 million kilometres.
A black hole of mass similar to that of Mount Everest,[20] 6.3715×1014 kg, would have a Schwarzschild radius much smaller than a nanometre.
[citation needed] The Schwarzschild radius would be 2 × 6.6738×10−11 m3⋅kg−1⋅s−2 × 6.3715×1014 kg / (299792458 m⋅s−1)2 = 9.46×10−13 m = 9.46×10−4 nm.
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 of matter were extremely high.
[21] Gravitational time dilation near a large, slowly rotating, nearly spherical body, such as the Earth or Sun can be reasonably approximated as follows:[22]
, which is another form of the Heisenberg uncertainty principle on the Planck scale.
[21][23] The Schwarzschild radius equation can be manipulated to yield an expression that gives the largest possible radius from an input density that doesn't form a black hole.
This means the largest amount of water you can have without forming a black hole would have a radius of 400 920 754 km (about 2.67 AU).
