The
Schwarzschild radius (sometimes historically referred to as the
gravitational radius) is the distance from the center of an object such that, if all the mass of the object were compressed within that sphere, the escape speed from the surface would equal the speed of light. Once a stellar remnant collapses within this radius, light cannot escape and the object is no longer visible. It is a characteristic
radiusIn classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...
associated with every quantity of
massMass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
.
In 1915,
Karl SchwarzschildKarl Schwarzschild was a German physicist. He is also the father of astrophysicist Martin Schwarzschild.He is best known for providing the first exact solution to the Einstein field equations of general relativity, for the limited case of a single spherical nonrotating mass, which he accomplished...
obtained an exact solution to Einstein's field equations for the gravitational field outside a nonrotating, spherically symmetric body (see
Schwarzschild metricIn Einstein's theory of general relativity, the Schwarzschild solution describes the gravitational field outside a spherical, uncharged, nonrotating mass such as a star, planet, or black hole. It is also a good approximation to the gravitational field of a slowly rotating body like the Earth or...
). Using the definition
, the solution contained a term of the form
; where the value of
making this term singular has come to be known as the
Schwarzschild radius. The physical significance of this
singularity, and whether this singularity could ever occur in nature, was debated for many decades; a general acceptance of the possibility of a
black holeA black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...
did not occur until the second half of the 20th century.
The Schwarzschild radius of an object is proportional to the mass. Accordingly, the
SunThe Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...
has a Schwarzschild radius of approximately 3 km (1.9 mi) while the
EarthEarth is the third planet from the Sun, and the densest and fifthlargest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...
's is only about 9.0 mm, the size of a
peanutThe peanut, or groundnut , is a species in the legume or "bean" family , so it is not a nut. The peanut was probably first cultivated in the valleys of Peru. It is an annual herbaceous plant growing tall...
. The
observable universeIn Big Bang cosmology, the observable universe consists of the galaxies and other matter that we can in principle observe from Earth in the present day, because light from those objects has had time to reach us since the beginning of the cosmological expansion...
's mass has Schwarzschild radius of approximately 10 billion light years.

(m) 
(g/cm^{3}) 
UniverseThe Universe is commonly defined as the totality of everything that exists, including all matter and energy, the planets, stars, galaxies, and the contents of intergalactic space. Definitions and usage vary and similar terms include the cosmos, the world and nature... 
4.46 (~10B lyA lightyear, also light year or lightyear is a unit of length, equal to just under 10 trillion kilometres... ) 
8 
Milky WayThe Milky Way is the galaxy that contains the Solar System. This name derives from its appearance as a dim unresolved "milky" glowing band arching across the night sky... 
2.08 (~0.2 lyA lightyear, also light year or lightyear is a unit of length, equal to just under 10 trillion kilometres... ) 
3.72 
SunThe Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields... 
2.95 
1.84 
EarthEarth is the third planet from the Sun, and the densest and fifthlargest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets... 
8.87 
2.04 
An object whose radius is smaller than its Schwarzschild radius is called a
black holeA black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...
. The surface at the Schwarzschild radius acts as an
event horizonIn general relativity, an event horizon is a boundary in spacetime beyond which events cannot affect an outside observer. In layman's terms it is defined as "the point of no return" i.e. the point at which the gravitational pull becomes so great as to make escape impossible. The most common case...
in a nonrotating body. (A
rotating black holeA rotating black hole is a black hole that possesses spin angular momentum.Types of black holes:There are four known, exact, black hole solutions to Einstein's equations, which describe gravity in General Relativity. Two of these rotate...
operates slightly differently.) Neither light nor particles can escape through this surface from the region inside, hence the name "black hole". The Schwarzschild radius of the (currently hypothesized)
supermassive black holeA supermassive black hole is the largest type of black hole in a galaxy, in the order of hundreds of thousands to billions of solar masses. Most, and possibly all galaxies, including the Milky Way, are believed to contain supermassive black holes at their centers.Supermassive black holes have...
at our
Galactic CenterThe Galactic Center is the rotational center of the Milky Way galaxy. It is located at a distance of 8.33±0.35 kpc from the Earth in the direction of the constellations Sagittarius, Ophiuchus, and Scorpius where the Milky Way appears brightest...
would be approximately 13.3 million kilometres.
Formula for the Schwarzschild radius
The Schwarzschild radius is proportional to the mass with a proportionality constant involving the
gravitational constantThe gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...
and the speed of light:

where:
 is the Schwarzschild radius;
 is the gravitational constant
The gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...
;
 is the mass of the gravitating object;
 is the speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
in vacuum.
The proportionality constant, 2
G/
c^{2}, is approximately , or .
An object of any density can be large enough to fall within its own Schwarzschild radius,

where:
 is the volume of the object;
 is its density.
Supermassive black hole
Assuming constant density, the Schwarzschild radius of a body is proportional to its mass, but the radius is proportional to the cube root of the volume and hence the mass. Therefore, as one accumulates matter at normal density (1 g/cm
^{3}, for example, the density of water), its Schwarzschild radius increases more quickly than its radius. At around 150,000,000 times the mass of the Sun, such an accumulation will fall inside its own Schwarzschild radius and thus it would be a
supermassive black holeA supermassive black hole is the largest type of black hole in a galaxy, in the order of hundreds of thousands to billions of solar masses. Most, and possibly all galaxies, including the Milky Way, are believed to contain supermassive black holes at their centers.Supermassive black holes have...
of 150,000,000 solar masses. (Supermassive black holes up to 18 billion solar masses have been observed.) The supermassive black hole in the center of our galaxy (4.5 ± 0.4 million solar masses) constitutes observationally the most convincing evidence for the existence of black holes in general.
It is thought that large black holes like these don't form directly in one collapse of a cluster of stars.
Instead they may start as a stellarsized black hole and grow larger by the accretion of matter and other black holes. An empirical correlation between the size of supermassive black holes and the stellar
velocity
dispersionIn astronomy, the velocity dispersion σ, is the range of velocities about the mean velocity for a group of objects, such as a cluster of stars about a galaxy...
of a
galaxyA galaxy is a massive, gravitationally bound system that consists of stars and stellar remnants, an interstellar medium of gas and dust, and an important but poorly understood component tentatively dubbed dark matter. The word galaxy is derived from the Greek galaxias , literally "milky", a...
bulgeIn astronomy, a bulge is a tightly packed group of stars within a larger formation. The term almost exclusively refers to the central group of stars found in most spiral galaxies...
is called the
Msigma relationThe Msigma relation is an empirical correlation between the stellar velocity dispersion \sigma of a galaxy bulge and the mass M of the supermassive black hole atthe galaxy's center.The relation can be expressed mathematically as...
.
Stellar black hole
If one accumulates matter at
nuclear densityNuclear density is the density of the nucleus of an atom, averaging about 4×1017 kg/m³. The descriptive term nuclear density is also applied to situations where similarly high densities occur, such as within neutron stars....
(the density of the nucleus of an atom, about 10
^{18} kg/m^{3};
neutron starA neutron star is a type of stellar remnant that can result from the gravitational collapse of a massive star during a Type II, Type Ib or Type Ic supernova event. Such stars are composed almost entirely of neutrons, which are subatomic particles without electrical charge and with a slightly larger...
s also reach this density), such an accumulation would fall within its own Schwarzschild radius at about 3 solar masses and thus would be a
stellar black holeA stellar black hole is a black hole formed by the gravitational collapse of a massive star. They have masses ranging from about 3 to several tens of solar masses...
.
Primordial black hole
Conversely, a small mass has an extremely small Schwarzschild radius. A mass similar to Mount Everest has a Schwarzschild radius smaller than a
nanometreA nanometre is a unit of length in the metric system, equal to one billionth of a metre. The name combines the SI prefix nano with the parent unit name metre .The nanometre is often used to express dimensions on the atomic scale: the diameter...
. 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 BangThe Big Bang theory is the prevailing cosmological model that explains the early development of the Universe. According to the Big Bang theory, the Universe was once in an extremely hot and dense state which expanded rapidly. This rapid expansion caused the young Universe to cool and resulted in...
, when densities were extremely high. Therefore these hypothetical miniature black holes are called
primordial black holeA primordial black hole is a hypothetical type of black hole that is formed not by the gravitational collapse of a large star but by the extreme density of matter present during the universe's early expansion....
s.
The Schwarzschild radius in gravitational time dilation
Gravitational time dilationGravitational time dilation is the effect of time passing at different rates in regions of different gravitational potential; the lower the gravitational potential, the more slowly time passes...
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:
 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.
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.
The 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:

where:
 is the gravitational acceleration at radial coordinate "r";
 is the Schwarzschild radius of the gravitating central body;
 is the radial coordinate;
 is the speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
in vacuum.
On the surface of the Earth:

The Schwarzschild radius in Keplerian orbits
For all
circular orbitA circular orbit is the orbit at a fixed distance around any point by an object rotating around a fixed axis.Below we consider a circular orbit in astrodynamics or celestial mechanics under standard assumptions...
s around a given central body:

where:
 is the orbit radius
In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...
;
 is the Schwarzschild radius of the gravitating central body;
 is the orbital speed
The orbital speed of a body, generally a planet, a natural satellite, an artificial satellite, or a multiple star, is the speed at which it orbits around the barycenter of a system, usually around a more massive body...
;
 is the speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
in vacuum.
This equality can be generalized to elliptic orbits as follows:

where:
is the
semimajor axisThe major axis of an ellipse is its longest diameter, a line that runs through the centre and both foci, its ends being at the widest points of the shape...
;
is the
orbital periodThe orbital period is the time taken for a given object to make one complete orbit about another object.When mentioned without further qualification in astronomy this refers to the sidereal period of an astronomical object, which is calculated with respect to the stars.There are several kinds of...
.
For the Earth 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.
See also
 black hole
A black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...
, a general survey
 Chandrasekhar limit
When a star starts running out of fuel, it usually cools off and collapses into one of three compact forms, depending on its total mass:* a White Dwarf, a big lump of Carbon and Oxygen atoms, almost like one huge molecule...
, a second requirement for black hole formation
 John Michell
John Michell was an English natural philosopher and geologist whose work spanned a wide range of subjects from astronomy to geology, optics, and gravitation. He was both a theorist and an experimenter....
Classification of black holes by type:
 Schwarzschild or static black hole
 rotating or Kerr black hole
A rotating black hole is a black hole that possesses spin angular momentum.Types of black holes:There are four known, exact, black hole solutions to Einstein's equations, which describe gravity in General Relativity. Two of these rotate...
 charged black hole or Newman black hole and KerrNewman black hole
A charged black hole is a black hole that possesses electric charge. Since the electromagnetic repulsion in compressing an electrically charged mass is dramatically greater than the gravitational attraction , it is not expected that black holes with a significant electric charge will be formed in...
A classification of black holes by mass:
 micro black hole
Micro black holes are tiny black holes, also called quantum mechanical black holes or mini black holes, for which quantum mechanical effects play an important role....
and extradimensional black hole
 primordial black hole
A primordial black hole is a hypothetical type of black hole that is formed not by the gravitational collapse of a large star but by the extreme density of matter present during the universe's early expansion....
, a hypothetical leftover of the Big Bang
 stellar black hole
A stellar black hole is a black hole formed by the gravitational collapse of a massive star. They have masses ranging from about 3 to several tens of solar masses...
, which could either be a static black hole or a rotating black hole
 supermassive black hole
A supermassive black hole is the largest type of black hole in a galaxy, in the order of hundreds of thousands to billions of solar masses. Most, and possibly all galaxies, including the Milky Way, are believed to contain supermassive black holes at their centers.Supermassive black holes have...
, which could also either be a static black hole or a rotating black hole
 visible universe, if its density is the critical density