The Schwarzschild radius or the gravitational radius is a physical parameter in the
Schwarzschild solution
In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild solution) is an
exact solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assump ...
to
Einstein's field equation
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it.
The equations were published by Einstein in 1915 in the form ...
s that corresponds to the
radius
In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
defining the
event horizon
In astrophysics, an event horizon is a boundary beyond which events cannot affect an observer. Wolfgang Rindler coined the term in the 1950s.
In 1784, John Michell proposed that gravity can be strong enough in the vicinity of massive compact obj ...
of a Schwarzschild
black hole
A black hole is a region of spacetime where gravitation, gravity is so strong that nothing, including light or other Electromagnetic radiation, electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts t ...
. It is a characteristic radius associated with any quantity of mass. The Schwarzschild radius was named after the German astronomer
Karl Schwarzschild
Karl Schwarzschild (; 9 October 1873 – 11 May 1916) was a German physicist and astronomer.
Schwarzschild provided the first exact solution to the Einstein field equations of general relativity, for the limited case of a single spherical non-r ...
, who calculated this exact solution for the theory of
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
in 1916.
The Schwarzschild radius is given as
where ''G'' is the
gravitational constant
The gravitational constant (also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant), denoted by the capital letter , is an empirical physical constant involved in ...
, ''M'' is the object mass, and ''c'' is the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
.
History
In 1916,
Karl Schwarzschild
Karl Schwarzschild (; 9 October 1873 – 11 May 1916) was a German physicist and astronomer.
Schwarzschild provided the first exact solution to the Einstein field equations of general relativity, for the limited case of a single spherical non-r ...
obtained the exact solution to Einstein's field equations for the gravitational field outside a non-rotating, spherically symmetric body with mass (see
Schwarzschild metric
In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild solution) is an
exact solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assumpti ...
). The solution contained terms of the form and , which become
singular
Singular may refer to:
* Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms
* Singular homology
* SINGULAR, an open source Computer Algebra System (CAS)
* Singular or sounder, a group of boar, ...
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
A gravitational singularity, spacetime singularity or simply singularity is a condition in which gravity is so intense that spacetime itself breaks down catastrophically. As such, a singularity is by definition no longer part of the regular sp ...
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
In celestial mechanics, escape velocity or escape speed is the minimum speed needed for a free, non- propelled object to escape from the gravitational influence of a primary body, thus reaching an infinite distance from it. It is typically ...
was equal to the speed of light. It had been identified in the 18th century by
John Michell
John Michell (; 25 December 1724 – 21 April 1793) was an English people, English natural philosophy, natural philosopher and clergyman who provided pioneering insights into a wide range of scientific fields including astronomy, geology, opti ...
and
Pierre-Simon Laplace
Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...
.
Parameters
The Schwarzschild radius of an object is proportional to its mass. Accordingly, the
Sun
The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
has a Schwarzschild radius of approximately , whereas
Earth
Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
's is only about and the
Moon
The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width of ...
's is about . The
observable universe
The observable universe is a ball-shaped region of the universe comprising all matter that can be observed from Earth or its space-based telescopes and exploratory probes at the present time, because the electromagnetic radiation from these obj ...
'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
A black hole is a region of spacetime where gravitation, gravity is so strong that nothing, including light or other Electromagnetic radiation, electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts t ...
. The surface at the Schwarzschild radius acts as an
event horizon
In astrophysics, an event horizon is a boundary beyond which events cannot affect an observer. Wolfgang Rindler coined the term in the 1950s.
In 1784, John Michell proposed that gravity can be strong enough in the vicinity of massive compact obj ...
in a non-rotating body (a
rotating black hole
A rotating black hole is a black hole that possesses angular momentum. In particular, it rotates about one of its axes of symmetry.
All celestial objects – planets, stars (Sun), galaxies, black holes – spin.
Types of black holes
Ther ...
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
A supermassive black hole (SMBH or sometimes SBH) is the largest type of black hole, with its mass being on the order of hundreds of thousands, or millions to billions of times the mass of the Sun (). Black holes are a class of astronomical obj ...
(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
NGC 4889 (also known as Caldwell 35) is an E4 supergiant elliptical galaxy. It was discovered in 1785 by the British astronomer William Herschel, Frederick William Herschel I, who catalogued it as a bright, nebulous patch. The brightest galaxy wi ...
.) 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/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 (_, 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
A supermassive black hole (SMBH or sometimes SBH) is the largest type of black hole, with its mass being on the order of hundreds of thousands, or millions to billions of times the mass of the Sun (). Black holes are a class of astronomical obj ...
at the
Galactic Center
The Galactic Center or Galactic Centre is the rotational center, the barycenter, of the Milky Way galaxy. Its central massive object is a supermassive black hole of about 4 million solar masses, which is called Sagittarius A*, a compact rad ...
of the
Milky Way
The Milky Way is the galaxy that includes our Solar System, with the name describing the galaxy's appearance from Earth: a hazy band of light seen in the night sky formed from stars that cannot be individually distinguished by the naked eye ...
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
Nuclear density is the density of the nucleus of an atom. For heavy nuclei, it is close to the ''nuclear saturation density'' n_0=0.15\pm0.01 nucleons/ fm3, which minimizes the energy density of an infinite nuclear matter. The ''nuclear saturation ...
(the density of the nucleus of an atom, about 1018kg/m3;
neutron star
A neutron star is the collapsed core of a massive supergiant star, which had a total mass of between 10 and 25 solar masses, possibly more if the star was especially metal-rich. Except for black holes and some hypothetical objects (e.g. white ...
s also reach this density), such an accumulation would fall within its own Schwarzschild radius at about and thus would be a
stellar black hole
A stellar black hole (or stellar-mass black hole) is a black hole formed by the gravitational collapse of a star. They have masses ranging from about 5 to several tens of solar masses. The process is observed as a hypernova explosion or as a gam ...
.
Micro black hole
A small mass has an extremely small Schwarzschild radius. A mass similar to
Mount Everest
Mount Everest (; Tibetan: ''Chomolungma'' ; ) is Earth's highest mountain above sea level, located in the Mahalangur Himal sub-range of the Himalayas. The China–Nepal border runs across its summit point. Its elevation (snow heig ...
has a Schwarzschild radius much smaller than a
nanometre
330px, Different lengths as in respect to the molecular scale.
The nanometre (international spelling as used by the International Bureau of Weights and Measures; SI symbol: nm) or nanometer (American and British English spelling differences#-re ...
. 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
The Big Bang event is a physical theory that describes how the universe expanded from an initial state of high density and temperature. Various cosmological models of the Big Bang explain the evolution of the observable universe from the ...
, when densities were extremely high. Therefore, these hypothetical miniature black holes are called
primordial black hole
Primordial black holes (also abbreviated as PBH) are hypothetical black holes that formed soon after the Big Bang. Due to the extreme environment of the newly born universe, extremely dense pockets of sub-atomic matter had been tightly packed to ...
s.
Other uses
In gravitational time dilation
Gravitational time dilation
Gravitational time dilation is a form of time dilation, an actual difference of elapsed time between two events as measured by observers situated at varying distances from a gravitating mass. The lower the gravitational potential (the closer ...
near a large, slowly rotating, nearly spherical body, such as the Earth or Sun can be reasonably approximated as follows:Extract of page 208 /ref>
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 (and therefore outside of the gravitational field);
* is the radial coordinate of the observer (which is analogous to the classical distance from the center of the object);
* is the Schwarzschild radius.
Compton wavelength intersection
The Schwarzschild radius () and the
Compton wavelength
The Compton wavelength is a quantum mechanical property of a particle. The Compton wavelength of a particle is equal to the wavelength of a photon whose energy is the same as the rest energy of that particle (see mass–energy equivalence). It was ...
() corresponding to a given mass are similar when the mass is around one Planck mass (), when both are of the same order as the Planck length ().
See also
*
Black hole
A black hole is a region of spacetime where gravitation, gravity is so strong that nothing, including light or other Electromagnetic radiation, electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts t ...
, a general survey
*
Chandrasekhar limit
The Chandrasekhar limit () is the maximum mass of a stable white dwarf star. The currently accepted value of the Chandrasekhar limit is about ().
White dwarfs resist gravitational collapse primarily through electron degeneracy pressure, compared ...
, a second requirement for black hole formation
*
John Michell
John Michell (; 25 December 1724 – 21 April 1793) was an English people, English natural philosophy, natural philosopher and clergyman who provided pioneering insights into a wide range of scientific fields including astronomy, geology, opti ...
Primordial black hole
Primordial black holes (also abbreviated as PBH) are hypothetical black holes that formed soon after the Big Bang. Due to the extreme environment of the newly born universe, extremely dense pockets of sub-atomic matter had been tightly packed to ...
, a hypothetical leftover of the Big Bang
*
Stellar black hole
A stellar black hole (or stellar-mass black hole) is a black hole formed by the gravitational collapse of a star. They have masses ranging from about 5 to several tens of solar masses. The process is observed as a hypernova explosion or as a gam ...
, which could either be a static black hole or a rotating black hole
*
Supermassive black hole
A supermassive black hole (SMBH or sometimes SBH) is the largest type of black hole, with its mass being on the order of hundreds of thousands, or millions to billions of times the mass of the Sun (). Black holes are a class of astronomical obj ...
, which could also either be a static black hole or a rotating black hole
*
Visible universe
The observable universe is a ball-shaped region of the universe comprising all matter that can be observed from Earth or its space-based telescopes and exploratory probes at the present time, because the electromagnetic radiation from these obj ...
Virtual black hole
Virtual may refer to:
* Virtual (horse), a thoroughbred racehorse
* Virtual channel, a channel designation which differs from that of the actual radio channel (or range of frequencies) on which the signal travels
* Virtual function, a programming ...