TheSchwarzschild radius or thegravitational radius is a physical parameter in theSchwarzschild solution toEinstein's field equations that corresponds to theradius defining theevent horizon of a Schwarzschildblack hole. It is a characteristic radius associated with any quantity of mass. The Schwarzschild radius was named after the German astronomerKarl Schwarzschild, who calculated this exact solution for the theory ofgeneral relativity in 1916.
The Schwarzschild radius is given aswhereG is thegravitational constant,M is the object mass, andc is thespeed 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 (seeSchwarzschild metric). The solution contained terms of the form and, which becomesingular at and respectively. The has come to be known as theSchwarzschild radius. The physical significance of thesesingularities 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 aspacetime 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 theescape velocity was equal to the speed of light. It had been identified in the 18th century byJohn Michell[6] andPierre-Simon Laplace.[7]
The Schwarzschild radius of an object is proportional to its mass. Accordingly, theSun has a Schwarzschild radius of approximately 3.0 km (1.9 mi),[8] whereasEarth's is approximately 9 mm (0.35 in)[8] and theMoon'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 inPhoenix A (one of the largestknown 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 inNGC 4889 | 4.2×1040 kg | 6.2×1013 m (~410 AU) | 0.042 kg/m3 | |
| SMBH inMessier 87[9] | 1.3×1040 kg | 1.9×1013 m (~130 AU) | 0.44 kg/m3 | |
| SMBH inAndromeda 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 inNGC 4395[12] | 7.1568×1035 kg | 1.062×109 m (1.53 R⊙) | 1.4230×108 kg/m3 | |
| Potentialintermediate black hole inHCN-0.009-0.044[13][14] | 6.3616×1034 kg | 9.44×108 m (14.8 R🜨) | 1.8011×1010 kg/m3 | |
| Resultingintermediate black hole fromGW190521 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 |
The simplest way of deriving the Schwarzschild radius comes from the equality of the modulus of a spherical solid mass' rest energy with its gravitational energy:
So, the Schwarzschild radius reads as
| Class | Approx. mass | Approx. radius |
|---|---|---|
| Supermassive black hole | 105–1011MSun | 0.002–2000AU |
| Intermediate-mass black hole | 103MSun | 3 x 103 km ≈RMars |
| Stellar black hole | 10MSun | 30 km |
| Micro black hole | up toMMoon | up to 0.1 mm |
Any object whose radius is smaller than its Schwarzschild radius is called ablack hole.[16]: 410 The surface at the Schwarzschild radius acts as anevent horizon in a non-rotating body (arotating 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.
Asupermassive 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 asNGC 4889.)[17] Unlikestellar 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.[citation needed]
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.[18] 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, 997kg/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.[19]
The Schwarzschild radius of thesupermassive black hole at theGalactic Center of theMilky 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 atnuclear density (the density of the nucleus of an atom, about 1018kg/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 astellar black hole.[citation needed]
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A small mass has an extremely small Schwarzschild radius. A black hole of mass similar to that ofMount Everest,[20]6.3715×1014 kg, would have a Schwarzschild radius much smaller than ananometre.[citation needed] The Schwarzschild radius would be 2 ×6.6738×10−11 m3⋅kg−1⋅s−2 ×6.3715×1014 kg / (299792458 m⋅s−1)2 =9.46×10−13 m =9.46×10−4 nm. 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 theBig Bang, when densities of matter were extremely high. Therefore, these hypothetical miniature black holes are calledprimordial black holes.[citation needed]
When moving to thePlanck scale ≈ 10−35 m, it is convenient to write the gravitational radius in the form, (see alsovirtual black hole).[21]
Gravitational time dilation near a large, slowly rotating, nearly spherical body, such as the Earth or Sun can be reasonably approximated as follows:[22]where:
The Schwarzschild radius () and theCompton wavelength () corresponding to a given mass are similar when the mass is around onePlanck mass (), when both are of the same order as thePlanck length ().
Thus, or, which is another form of theHeisenberg uncertainty principle on thePlanck scale. (See alsoVirtual black hole).[21][23]
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 asρ,
For example, the density of water is1000 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 2.67AU).
Classification of black holes by type:
A classification of black holes by mass:
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 of2750 kg⋅m−3.
