Schwarzschild radius
Encyclopedia
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 radius
associated with every quantity of mass
.
In 1915, Karl Schwarzschild
obtained an exact solution to Einstein's field equations for the gravitational field outside a non-rotating, spherically symmetric body (see Schwarzschild metric
). 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 hole
did not occur until the second half of the 20th century.
The Schwarzschild radius of an object is proportional to the mass. Accordingly, the Sun
has a Schwarzschild radius of approximately 3 km (1.9 mi) while the Earth
's is only about 9.0 mm, the size of a peanut
. The observable universe
's mass has Schwarzschild radius of approximately 10 billion light years.
An 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. (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". The Schwarzschild radius of the (currently hypothesized) supermassive black hole
at our Galactic Center
would be approximately 13.3 million kilometres.
and the speed of light:
where:
The proportionality constant, 2G/c^{2}, is approximately , or .
An object of any density can be large enough to fall within its own Schwarzschild radius,
where:
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 stellar-sized 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
dispersion
of a galaxy
bulge
is called the M-sigma relation
.
(the density of the nucleus of an atom, about 10^{18} kg/m^{3}; neutron star
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 hole
.
. 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 hole
s.
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.
where:
On the surface of the Earth:
s around a given central body:
where:
This equality can be generalized to elliptic orbits as follows:
where: is the semi-major axis
; is the orbital period
.
For the Earth orbiting the Sun:
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.
Classification of black holes by type:
A classification of black holes by mass:
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...
associated with every quantity of mass
Mass
Mass 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 Schwarzschild
Karl Schwarzschild
Karl 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 non-rotating mass, which he accomplished...
obtained an exact solution to Einstein's field equations for the gravitational field outside a non-rotating, spherically symmetric body (see Schwarzschild metric
Schwarzschild metric
In Einstein's theory of general relativity, the Schwarzschild solution describes the gravitational field outside a spherical, uncharged, non-rotating 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 hole
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...
did not occur until the second half of the 20th century.
The Schwarzschild radius of an object is proportional to the mass. Accordingly, the Sun
Sun
The 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 Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest 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 peanut
Peanut
The 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 universe
Observable universe
In 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}) | |
---|---|---|
Universe Universe The 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 ly Light-year A light-year, also light year or lightyear is a unit of length, equal to just under 10 trillion kilometres... ) |
8 |
Milky Way Milky Way The Milky Way is the galaxy that contains the Solar System. This name derives from its appearance as a dim un-resolved "milky" glowing band arching across the night sky... |
2.08 (~0.2 ly Light-year A light-year, also light year or lightyear is a unit of length, equal to just under 10 trillion kilometres... ) |
3.72 |
Sun Sun The 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 |
Earth Earth Earth is the third planet from the Sun, and the densest and fifth-largest 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 hole
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...
. The surface at the Schwarzschild radius acts as an event horizon
Event horizon
In 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 non-rotating body. (A rotating black hole
Rotating 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...
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 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...
at our Galactic Center
Galactic Center
The 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 constantGravitational 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...
and the speed of light:
where:
- is the Schwarzschild radius;
- is the gravitational constantGravitational 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...
; - is the mass of the gravitating object;
- is the speed of lightSpeed of lightThe 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, 2G/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 holeSupermassive 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...
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 stellar-sized 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
dispersion
Velocity dispersion
In 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 galaxy
Galaxy
A 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...
bulge
Bulge (astronomy)
In 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 M-sigma relation
M-sigma relation
The M-sigma 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
Nuclear 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 star
Neutron star
A 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 hole
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...
.
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 nanometreNanometre
A 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 Bang
Big Bang
The 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 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....
s.
The Schwarzschild radius in gravitational time dilation
Gravitational time dilationGravitational time dilation
Gravitational 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 lightSpeed of lightThe 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 orbitCircular orbit
A 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 radiusRadiusIn 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 speedOrbital speedThe 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 lightSpeed of lightThe 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 semi-major axis
Semi-major axis
The 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 period
Orbital period
The 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 holeBlack 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...
, a general survey - Chandrasekhar limitChandrasekhar limitWhen 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 MichellJohn MichellJohn 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 holeRotating 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...
- charged black hole or Newman black hole and Kerr-Newman black holeCharged black holeA 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 holeMicro black holeMicro 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 extra-dimensional black hole - primordial black holePrimordial 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....
, a hypothetical leftover of the Big Bang - stellar black holeStellar 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...
, which could either be a static black hole or a rotating black hole - supermassive black holeSupermassive 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...
, which could also either be a static black hole or a rotating black hole - visible universe, if its density is the critical density