Schwarzschild radius
The Schwarzschild radius is a physical parameter that shows up in the Schwarzschild solution to Einstein's field equations, corresponding to the radius defining the event horizon of a Schwarzschild black hole. It is a characteristic radius associated with every 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
where G is the gravitational constant, M is the object mass, and c is the speed of light.
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 . The solution contained terms of the form and, which become singular at and respectively. The 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 is a coordinate singularity, meaning that it is an artefact of the particular system of coordinates that were used, while the one at is physical, 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 by 19th century astronomers such as Pierre-Simon Laplace.
Parameters
The Schwarzschild radius of an object is proportional to the 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.Derivation
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. 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 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. Unlike stellar mass black holes, supermassive black holes have comparatively low average densities. 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, 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 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, such an accumulation would fall within its own Schwarzschild radius at about and thus would be a stellar black hole.Primordial 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
near a large, slowly rotating, nearly spherical body, such as the Earth or Sun can be reasonably approximated using the Schwarzschild radius as follows:where:
The results of the Pound–Rebka experiment in 1959 were found to be consistent with predictions made by general relativity. By measuring Earth's gravitational time dilation, this experiment indirectly measured Earth's Schwarzschild radius.
In Newtonian gravitational fields
The Newtonian gravitational field near a large, slowly rotating, nearly spherical body can be reasonably approximated using the Schwarzschild radius as follows:and
Therefore, on dividing above by below:
where:
On the surface of the Earth:
In Keplerian orbits
For all circular orbits around a given central body:Therefore,
but
Therefore,
where:
This equality can be generalized to elliptic orbits as follows:
where:
For the Earth, as a planet orbiting the Sun:
Relativistic circular orbits and the photon sphere
The Keplerian equation for circular orbits can be generalized to the relativistic equation for circular orbits by accounting for time dilation in the velocity term:This final equation indicates that an object orbiting at the speed of light would have an orbital radius of 1.5 times the Schwarzschild radius. This is a special orbit known as the photon sphere.