Schwarzschild radius
The Schwarzschild radius is a parameter in the Schwarzschild solution to Einstein's field equations that corresponds to the radius of a sphere in flat space that has the same surface area as that of the event horizon of a Schwarzschild black hole of a given mass. It is a characteristic quantity that may be associated with any quantity of mass. The Schwarzschild radius was named after the German astronomer Karl Schwarzschild, who calculated this solution for the theory of general relativity in 1916.
The Schwarzschild radius is given as
where G is the Newtonian constant of gravitation, M is the mass of the object, and c is the speed of light.
History
In 1916, Karl Schwarzschild obtained an exact solution to the Einstein field equations for the gravitational field outside a non-rotating, spherically symmetric body with mass . The solution contained terms of the form and, which have singularities 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 artifact of the particular system of coordinates that was used; while the one at 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 approximately and the Moon's is approximately.| Object | Mass | Schwarzschild radius | Actual radius | Schwarzschild density or |
| Milky Way | ||||
| SMBH in Phoenix A | ||||
| Ton 618 | ||||
| SMBH in NGC 4889 | ||||
| SMBH in Messier 87 | ||||
| SMBH in Andromeda Galaxy | ||||
| Sagittarius A* | ||||
| SMBH in NGC 4395 | ||||
| Potential intermediate black hole in HCN-0.009-0.044 | ||||
| Resulting intermediate black hole from GW190521 merger | ||||
| Sun | ||||
| Jupiter | ||||
| Saturn | ||||
| Neptune | ||||
| Uranus | ||||
| Earth | ||||
| Venus | ||||
| Mars | ||||
| Mercury | ||||
| Moon | ||||
| Human | ~ | |||
| Planck mass |
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 of the Milky Way is approximately 12 million kilometres. Its mass is about.
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.Micro black hole
A small mass has an extremely small Schwarzschild radius. A black hole of mass similar to that of Mount Everest,, would have a Schwarzschild radius much smaller than a nanometre. The Schwarzschild radius would be 2 × × / 2 . 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 have been formed in an early stage of the evolution of the universe, just after the Big Bang, when densities of matter 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 as follows:where:
- is the elapsed time for an observer at radial coordinate r within the gravitational field;
- is the elapsed time for an observer distant from the massive object ;
- is the radial coordinate of the observer ;
- is the Schwarzschild radius.
Compton wavelength intersection
Calculating the maximum volume and radius possible given a density before a black hole forms
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. Taking the input density as,For example, the density of water is. This means the largest amount of water you can have without forming a black hole would have a radius of .