The Schwarzschild radius or the gravitational radius is a physical parameter in the Schwarzschild solution to Einstein's field equations that corresponds to the radius defining the event horizon of a Schwarzschild black hole. It is a characteristic radius associated with any 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. [note 1] [1] [2]
In 1916, Karl Schwarzschild obtained the exact solution [3] [4] to Einstein's field equations for the gravitational field outside a non-rotating, spherically symmetric body with mass (see Schwarzschild metric). The solution contained terms of the form and , which becomes 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 artifact of the particular system of coordinates that was used; while the one at is a spacetime singularity and cannot be removed. [5] 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 [6] and Pierre-Simon Laplace. [7]
The Schwarzschild radius of an object is proportional to its mass. Accordingly, the Sun has a Schwarzschild radius of approximately 3.0 km (1.9 mi), [8] whereas Earth's is approximately 9 mm (0.35 in) [8] and the Moon's is approximately 0.1 mm (0.0039 in).
Object | Mass | Schwarzschild radius | Actual radius | Schwarzschild density or |
---|---|---|---|---|
Milky Way | 1.6×1042 kg | 2.4×1015 m (0.25 ly ) | 5×1020 m (52900 ly ) | 0.000029 kg/m3 |
SMBH in Phoenix A (one of the largest known black holes) | 2×1041 kg | 3×1014 m (~2000 AU) | 0.0018 kg/m3 | |
Ton 618 | 1.3×1041 kg | 1.9×1014 m (~1300 AU) | 0.0045 kg/m3 | |
SMBH in NGC 4889 | 4.2×1040 kg | 6.2×1013 m (~410 AU) | 0.042 kg/m3 | |
SMBH in Messier 87 [9] | 1.3×1040 kg | 1.9×1013 m (~130 AU) | 0.44 kg/m3 | |
SMBH in Andromeda Galaxy [10] | 3.4×1038 kg | 5.0×1011 m (3.3 AU) | 640 kg/m3 | |
Sagittarius A* (SMBH in Milky Way) [11] | 8.26×1036 kg | 1.23×1010 m (0.08 AU) | 1.068×106 kg/m3 | |
SMBH in NGC 4395 [12] | 7.1568×1035 kg | 1.062×109 m (1.53 R⊙) | 1.4230×108 kg/m3 | |
Potential intermediate black hole in HCN-0.009-0.044 [13] [14] | 6.3616×1034 kg | 9.44×108 m (14.8 R🜨) | 1.8011×1010 kg/m3 | |
Resulting intermediate black hole from GW190521 merger [15] | 2.823×1032 kg | 4.189×105 m (0.066 R🜨) | 9.125×1014 kg/m3 | |
Sun | 1.99×1030 kg | 2.95×103 m | 7.0×108 m | 1.84×1019 kg/m3 |
Jupiter | 1.90×1027 kg | 2.82 m | 7.0×107 m | 2.02×1025 kg/m3 |
Saturn | 5.683×1026 kg | 8.42×10−1 m | 6.03×107 m | 2.27×1026 kg/m3 |
Neptune | 1.024×1026 kg | 1.52×10−1 m | 2.47×107 m | 6.97×1027 kg/m3 |
Uranus | 8.681×1025 kg | 1.29×10−1 m | 2.56×107 m | 9.68×1027 kg/m3 |
Earth | 5.97×1024 kg | 8.87×10−3 m | 6.37×106 m | 2.04×1030 kg/m3 |
Venus | 4.867×1024 kg | 7.21×10−3 m | 6.05×106 m | 3.10×1030 kg/m3 |
Mars | 6.39×1023 kg | 9.47×10−4 m | 3.39×106 m | 1.80×1032 kg/m3 |
Mercury | 3.285×1023 kg | 4.87×10−4 m | 2.44×106 m | 6.79×1032 kg/m3 |
Moon | 7.35×1022 kg | 1.09×10−4 m | 1.74×106 m | 1.35×1034 kg/m3 |
Human | 70 kg | 1.04×10−25 m | ~5×10−1 m | 1.49×1076 kg/m3 |
Planck mass | 2.18×10−8 kg | 3.23×10−35 m (2 lP) | 1.54×1095 kg/m3 |
Class | Approx. mass | Approx. radius |
---|---|---|
Supermassive black hole | 105–1010 MSun | 0.001–400 AU |
Intermediate-mass black hole | 103MSun | 103 km ≈ REarth |
Stellar black hole | 10 MSun | 30 km |
Micro black hole | up to M Moon | up to 0.1 mm |
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 (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".
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.
A supermassive black hole (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 (2.1 × 1010) M☉ have been detected, such as NGC 4889.) [16] 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. [17] 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 (1.36 × 108 M☉), 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.[ citation needed ]
The Schwarzschild radius of the supermassive black hole at the Galactic Center of the Milky Way is approximately 12 million kilometres. [11] Its mass is about 4.1 million M☉.
Stellar black holes have much greater average densities than supermassive black holes. If one accumulates matter at nuclear density (the density of the nucleus of an atom, about 1018 kg/m3; neutron stars also reach this density), such an accumulation would fall within its own Schwarzschild radius at about 3 M☉ and thus would be a stellar black hole.
A small mass has an extremely small Schwarzschild radius. A black hole of mass similar to that of Mount Everest [18] [note 2] would have a Schwarzschild radius much smaller than a nanometre. [note 3] 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 of matter were extremely high. Therefore, these hypothetical miniature black holes are called primordial black holes.
Gravitational time dilation near a large, slowly rotating, nearly spherical body, such as the Earth or Sun can be reasonably approximated as follows: [19]
where:
The Schwarzschild radius () and the Compton wavelength () 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 ().
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 ρ,
For example, the density of water is 1000 kg/m3. This means the largest amount of water you can have without forming a black hole would have a radius of 400 920 754 km (about 2.67 AU).
Classification of black holes by type:
A classification of black holes by mass:
A black hole is a region of spacetime where gravity is so strong that nothing, including light and other electromagnetic waves, is capable of possessing enough energy to escape it. Einstein's theory of general relativity predicts that a sufficiently compact mass can deform spacetime to form a black hole. The boundary of no escape is called the event horizon. A black hole has a great effect on the fate and circumstances of an object crossing it, but it has no locally detectable features according to general relativity. In many ways, a black hole acts like an ideal black body, as it reflects no light. Moreover, quantum field theory in curved spacetime predicts that event horizons emit Hawking radiation, with the same spectrum as a black body of a temperature inversely proportional to its mass. This temperature is of the order of billionths of a kelvin for stellar black holes, making it essentially impossible to observe directly.
A neutron star is the collapsed core of a massive supergiant star, which had a total mass of between 10 and 25 solar masses (M☉), possibly more if the star was especially metal-rich. Except for black holes, neutron stars are the smallest and densest known class of stellar objects. Neutron stars have a radius on the order of 10 kilometers (6 mi) and a mass of about 1.4 M☉. They result from the supernova explosion of a massive star, combined with gravitational collapse, that compresses the core past white dwarf star density to that of atomic nuclei.
A gravitational singularity, spacetime singularity or simply singularity is a condition in which gravity is predicted to be so intense that spacetime itself would break down catastrophically. As such, a singularity is by definition no longer part of the regular spacetime and cannot be determined by "where" or "when". Gravitational singularities exist at a junction between general relativity and quantum mechanics; therefore, the properties of the singularity cannot be described without an established theory of quantum gravity. Trying to find a complete and precise definition of singularities in the theory of general relativity, the current best theory of gravity, remains a difficult problem. A singularity in general relativity can be defined by the scalar invariant curvature becoming infinite or, better, by a geodesic being incomplete.
Hawking radiation is the theoretical thermal black-body radiation released outside a black hole's event horizon. This is counterintuitive because once ordinary electromagnetic radiation is inside the event horizon, it cannot escape. It is named after the physicist Stephen Hawking, who developed a theoretical argument for its existence in 1974. Hawking radiation is predicted to be extremely faint and is many orders of magnitude below the current best telescopes' detecting ability.
In astronomy, the term compact object refers collectively to white dwarfs, neutron stars, and black holes. It could also include exotic stars if such hypothetical, dense bodies are confirmed to exist. All compact objects have a high mass relative to their radius, giving them a very high density, compared to ordinary atomic matter.
Messier 87 is a supergiant elliptical galaxy in the constellation Virgo that contains several trillion stars. One of the largest and most massive galaxies in the local universe, it has a large population of globular clusters—about 15,000 compared with the 150–200 orbiting the Milky Way—and a jet of energetic plasma that originates at the core and extends at least 1,500 parsecs, traveling at a relativistic speed. It is one of the brightest radio sources in the sky and a popular target for both amateur and professional astronomers.
A supermassive black hole 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 (M☉). Black holes are a class of astronomical objects that have undergone gravitational collapse, leaving behind spheroidal regions of space from which nothing can escape, not even light. Observational evidence indicates that almost every large galaxy has a supermassive black hole at its center. For example, the Milky Way galaxy has a supermassive black hole at its center, corresponding to the radio source Sagittarius A*. Accretion of interstellar gas onto supermassive black holes is the process responsible for powering active galactic nuclei (AGNs) and quasars.
In Einstein's theory of general relativity, the Schwarzschild metric is an exact solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assumption that the electric charge of the mass, angular momentum of the mass, and universal cosmological constant are all zero. The solution is a useful approximation for describing slowly rotating astronomical objects such as many stars and planets, including Earth and the Sun. It was found by Karl Schwarzschild in 1916.
The Kerr metric or Kerr geometry describes the geometry of empty spacetime around a rotating uncharged axially symmetric black hole with a quasispherical event horizon. The Kerr metric is an exact solution of the Einstein field equations of general relativity; these equations are highly non-linear, which makes exact solutions very difficult to find.
A stellar 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. They are the remnants of supernova explosions, which may be observed as a type of gamma ray burst. These black holes are also referred to as collapsars.
Sagittarius A*, abbreviated Sgr A*, is the supermassive black hole at the Galactic Center of the Milky Way. Viewed from Earth, it is located near the border of the constellations Sagittarius and Scorpius, about 5.6° south of the ecliptic, visually close to the Butterfly Cluster (M6) and Lambda Scorpii.
Fuzzballs are hypothetical objects in superstring theory, intended to provide a fully quantum description of the black holes predicted by general relativity.
The surface gravity, g, of an astronomical object is the gravitational acceleration experienced at its surface at the equator, including the effects of rotation. The surface gravity may be thought of as the acceleration due to gravity experienced by a hypothetical test particle which is very close to the object's surface and which, in order not to disturb the system, has negligible mass. For objects where the surface is deep in the atmosphere and the radius not known, the surface gravity is given at the 1 bar pressure level in the atmosphere.
In physics, there is a speculative hypothesis that, if there were a black hole with the same mass, charge and angular momentum as an electron, it would share other properties of the electron. Most notably, Brandon Carter showed in 1968 that the magnetic moment of such an object would match that of an electron. This is interesting because calculations ignoring special relativity and treating the electron as a small rotating sphere of charge give a magnetic moment roughly half the experimental value.
The following outline is provided as an overview of and topical guide to black holes:
The sphere of influence is a region around a supermassive black hole in which the gravitational potential of the black hole dominates the gravitational potential of the host galaxy. The radius of the sphere of influence is called the "(gravitational) influence radius".
In astrophysics, an extreme mass ratio inspiral (EMRI) is the orbit of a relatively light object around a much heavier object, that gradually spirals in due to the emission of gravitational waves. Such systems are likely to be found in the centers of galaxies, where stellar mass compact objects, such as stellar black holes and neutron stars, may be found orbiting a supermassive black hole. In the case of a black hole in orbit around another black hole this is an extreme mass ratio binary black hole. The term EMRI is sometimes used as a shorthand to denote the emitted gravitational waveform as well as the orbit itself.
Bahcall–Wolf cusp refers to a particular distribution of stars around a massive black hole at the center of a galaxy or globular cluster. If the nucleus containing the black hole is sufficiently old, exchange of orbital energy between stars drives their distribution toward a characteristic form, such that the density of stars, ρ, varies with distance from the black hole, r, as
A tidal disruption event (TDE) is an astronomical phenomenon that occurs when a star approaches sufficiently close to a supermassive black hole (SMBH) to be pulled apart by the black hole's tidal force, experiencing spaghettification. A portion of the star's mass can be captured into an accretion disk around the black hole, resulting in a temporary flare of electromagnetic radiation as matter in the disk is consumed by the black hole. According to early papers, tidal disruption events should be an inevitable consequence of massive black holes' activity hidden in galaxy nuclei, whereas later theorists concluded that the resulting explosion or flare of radiation from the accretion of the stellar debris could be a unique signpost for the presence of a dormant black hole in the center of a normal galaxy. Sometimes a star can survive the encounter with an SMBH, and a remnant is formed. These events are termed partial TDEs.
Abell 1201 BCG (short for Abell 1201 Brightest Cluster Galaxy) is a type-cD massive elliptical galaxy residing as the brightest cluster galaxy (BCG) of the Abell 1201 galaxy cluster. At a redshift of 0.169, this system is around 2.7 billion light-years from Earth, and offset about 11 kiloparsecs from the X-ray peak of the intracluster gas. With an ellipticity of 0.32±0.02, the stellar distribution is far from spherical. In solar units, the total stellar luminosity is 4×1011 L☉ in SDSS r-band, and 1.6×1012 L☉ in 2MASS K-band. Half the stars orbit within an effective radius of 15 kpc, and their central velocity dispersion is about 285 km s−1 within 5 kpc rising to 360 km s−1 at 20 kpc distance.
If Mount Everest is assumed* to be a cone of height 8850 m and radius 5000 m, then its volume can be calculated using the following equation:
volume = πr2h/3 [...] Mount Everest is composed of granite, which has a density of 2750 kg⋅m−3.