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{{Properties_of_mass}}
The '''Schwarzschild radius''' (sometimes historically referred to as the '''gravitational radius''') is a physical parameter that shows up in the [[Schwarzschild solution]] to [[Einstein's field equation]]s, corresponding to the [[radius]] defining the [[event horizon]] of a Schwarzschild [[black hole]]. It is a characteristic radius associated with every quantity of mass. The ''Schwarzschild radius'' was  named after the [[Germany|German]] astronomer [[Karl Schwarzschild]], who calculated this exact solution for the theory of [[general relativity]] in 1916.

The Schwarzschild radius is given as

:<math> r_s = \frac{2 G M}{c^2} </math>

where G is the [[gravitational constant]], M is the object mass, and c is the [[speed of light]].<ref>{{cite book |last=Kutner |first=Marc |title=Astronomy: A Physical Perspective |publisher=[[Cambridge University Press]] |date=2003 |page=148 |url=https://books.google.co.uk/books?id=2QVmiMW0O0MC&pg=PA148#v=onepage&q&ffalse|isbn=9780521529273 }}</ref>

==History==
In 1916, [[Karl Schwarzschild]] obtained the exact solution<ref>K. Schwarzschild, "Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie", ''Sitzungsberichte der Deutschen Akademie der Wissenschaften zu Berlin, Klasse fur Mathematik, Physik, und Technik'' (1916) pp 189.</ref><ref>K. Schwarzschild, "Über das Gravitationsfeld einer Kugel aus inkompressibler Flussigkeit nach der Einsteinschen Theorie", ''Sitzungsberichte der Deutschen Akademie der Wissenschaften zu Berlin, Klasse fur Mathematik, Physik, und Technik'' (1916) pp 424.</ref> to Einstein's field equations for the gravitational field outside a non-rotating, spherically symmetric body with mass <math>M</math> (see [[Schwarzschild metric]]). The solution contained terms of the form <math>1-{r_s}/r </math> and <math>\frac{1}{1-{r_s}/r} </math>, which become [[Mathematical singularity|singular]] at <math> r=0</math> and <math> r=r_s</math> respectively. The <math>r_s</math> has come to be known as the ''Schwarzschild radius''. The physical significance of these [[Mathematical singularity|singularities]] was debated for decades. It was found that the one at <math> r=r_s</math> is a coordinate singularity, meaning that it can be removed by a change of coordinates, while the one at <math> r=0</math> is physical, and cannot be removed.<ref>{{cite book |last1=Wald |first1=Robert |title=General Relativity |url=https://archive.org/details/generalrelativit0000wald |url-access=registration |date=1984 |publisher=The University of Chicago Press |isbn=978-0-226-87033-5 |pages=[https://archive.org/details/generalrelativit0000wald/page/152 152–153]}}</ref> 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]]<ref name="Schaffer">{{cite journal |last1=Schaffer |first1=Simon |title=John Mitchell and Black Holes |journal=Journal for the History of Astronomy |date=1979 |volume=10 |pages=42–43 |url=http://adsbit.harvard.edu//full/1979JHA....10...42S/0000042.000.html |accessdate=4 June 2018|bibcode=1979JHA....10...42S |doi=10.1177/002182867901000104 }}</ref> and by 19th century astronomers such as [[Pierre-Simon Laplace]].<ref>Colin Montgomery, Wayne Orchiston and Ian Whittingham, [http://www.narit.or.th/en/files/2009JAHHvol12/2009JAHH...12...90M.pdf "Michell, Laplace and the origin of the Black Hole Concept"], ''Journal of Astronomical History and Heritage'', '''12'''(2), 90–96 (2009).</ref>

==Parameters==
The Schwarzschild radius of an object is proportional to the mass. Accordingly, the [[Sun]] has a Schwarzschild radius of approximately {{convert|3.0|km|mi|abbr=on}}, whereas [[Earth]]'s is only about {{convert|9|mm|in|abbr=on}} and the [[Moon]]'s is about {{convert|0.1|mm|in|abbr=on}}. The [[observable universe]]'s mass has a Schwarzschild radius of approximately 13.7 billion light-years.<ref name="Valev">{{cite arXiv |last=Valev |first=Dimitar |eprint=1008.0933 |class=physics.gen-ph |title=Consequences from conservation of the total density of the universe during the expansion |date=October 2008}}</ref><ref>{{cite book |first=Michel Marie |last=Deza |first2=Elena |last2=Deza |title=Encyclopedia of Distances |edition=2nd |url=https://books.google.com/books?id=QxX2CX5OVMsC&pg=PA452 |doi=10.1007/978-3-642-30958-8 |location=Heidelberg |publisher=Springer Science & Business Media |date=Oct 28, 2012 |page=452 |isbn=978-3-642-30958-8 |accessdate=8 December 2014 }}</ref>
{| style="border:1px solid gray; border-collapse:collapse" cellpadding="5" cellspacing="0"
|- 
! align="left" style="border: 1px solid gray"   | Object
! align="center" style="border: 1px solid gray" | Mass: <math>M</math>   
! align="center" style="border: 1px solid gray" | Schwarzschild radius: <math>\frac{2GM}{c^2}</math>
! align="center" style="border: 1px solid gray" | Schwarzschild density: <math>\frac{3c^6}{32\pi G^3M^2}</math>or <math>\frac{3c^2}{8\pi Gr^2}</math>
|-
| [[Observable universe]]<ref name="Valev" />
| align="right" | 8.8{{e|52}} kg    
| align="right" | 1.3{{e|26}} m (13.7 billion [[light-year|ly]]) 
| align="right" | 9.5{{e|-27}} kg/m<sup>3</sup>
|-
| [[Milky Way]]  
| align="right" | 1.6{{e|42}} kg    
| align="right" | 2.4{{e|15}} m (~0.25 [[light-year|ly]]) 
| align="right" | 0.000029&nbsp;kg/m<sup>3</sup>
|-
| [[Supermassive black hole|SMBH]] in [[NGC 4889]]
| align="right" | 4.2{{e|40}} kg
| align="right" | 6.2{{e|13}} m
| align="right" | 0.042&nbsp;kg/m<sup>3</sup>
|-
| [[Supermassive black hole|SMBH]] in [[Messier 87]]<ref name="Event Horizon Telescope Collaboration et al. 2019">{{cite journal
| last1 = Event Horizon Telescope Collaboration
| first1 = 
| date = 2019
| title = First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole
| journal = [[Astrophysical Journal Letters]]
| volume = 875
| issue = 1
| pages = 1–17
| bibcode = 2019ApJ...875L...1E
| doi = 10.3847/2041-8213
}}
6.5(7) × 10<sup>9</sup> {{Solar mass}} = 1.29(14){{e|40}} kg.
</ref>
| align="right" | 1.3{{e|40}} kg
| align="right" | 1.9{{e|13}} m
| align="right" | 0.44&nbsp;kg/m<sup>3</sup>
|-
| SMBH in [[Andromeda Galaxy]]<ref name="Bender et al 2005">{{cite journal
| last1 = Bender
| first1 = Ralf
| last2 = Kormendy
| first2 = John
| last3 = Bower
| first3 = Gary
 | display-authors=3
| last4 = Green
| first4 = Richard
| last5 = Thomas
| first5 = Jens
| last6 = Danks
| first6 = Anthony C.
| last7 = Gull
| first7 = Theodore
| last8 = Hutchings
| first8 = J. B.
| last9 = Joseph
| first9 = C. L.
| date = 2005
| title = HST STIS Spectroscopy of the Triple Nucleus of M31: Two Nested Disks in Keplerian Rotation around a Supermassive Black Hole
| journal = [[Astrophysical Journal]]
| volume = 631
| issue = 1
| pages = 280–300
| bibcode = 2005ApJ...631..280B
| arxiv = astro-ph/0509839
| doi = 10.1086/432434
| last10 = Kaiser
| first10 = M. E.
| last11 = Lauer
| first11 = Tod R.
| last12 = Nelson
| first12 = Charles H.
| last13 = Richstone
| first13 = Douglas
| last14 = Weistrop
| first14 = Donna
| last15 = Woodgate
| first15 = Bruce
}}
1.7(6) × 10<sup>8</sup> {{Solar mass}} = 0.34(12){{e|39}} kg.
</ref>
| align="right" | 3.4{{e|38}} kg
| align="right" | 5.0{{e|11}} m
| align="right" | 640&nbsp;kg/m<sup>3</sup>
|-
| [[Sagittarius A*|Sagittarius A* (SMBH)]]
| align="right" | 8.2{{e|36}} kg
| align="right" | 1.2{{e|10}} m
| align="right" | 1.1{{e|6}} kg/m<sup>3</sup>
|-
| [[Sun]]
| align="right" | 1.99{{e|30}} kg
| align="right" | 2.95{{e|3}} m
| align="right" | 1.84{{e|19}} kg/m<sup>3</sup>
|-
| [[Jupiter]]
| align="right" | 1.90{{e|27}} kg
| align="right" | 2.82 meters
| align="right" | 2.02{{e|25}} kg/m<sup>3</sup>
|-
| [[Earth]]
| align="right" | 5.97{{e|24}} kg
| align="right" | 8.87{{e|-3}} m
| align="right" | 2.04{{e|30}} kg/m<sup>3</sup>
|-
| [[Moon]]
| align="right" | 7.35{{e|22}} kg
| align="right" | 1.09{{e|-4}} m
| align="right" | 1.35{{e|34}} kg/m<sup>3</sup>
|-
| [[Human]]
| align="right" | 70 kilograms
| align="right" | 1.04{{e|-25}} m
| align="right" | 1.49{{e|76}} kg/m<sup>3</sup>
|-
| [[Big Mac]]
| align="right" | 0.215 kilograms
| align="right" | 3.19{{e|-28}} m
| align="right" | 1.58{{e|81}} kg/m<sup>3</sup>
|-
| [[Planck mass]]
| align="right" | 2.18{{e|-8}} kg
| align="right" | 3.23{{e|-35}} m
| align="right" | 1.54{{e|95}} kg/m<sup>3</sup>
|-
|}

==Derivation==
{{main|Deriving the Schwarzschild solution}}

==Black hole classification by Schwarzschild radius==
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. As the 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]] (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 {{Solar mass|(2.1 × 10<sup>10</sup>)}} have been detected, such as [[NGC 4889]].)<ref>{{cite journal|title=Two ten-billion-solar-mass black holes at the centres of giant elliptical galaxies|last=McConnell|first=Nicholas J.|date=2011-12-08|journal=Nature|volume=480|issue=7376|doi=10.1038/nature10636|pmid = 22158244|pages=215–218|arxiv=1112.1078|bibcode=2011Natur.480..215M|df=}}</ref> Unlike [[Stellar black hole|stellar mass black holes]], supermassive black holes have comparatively low average densities.  (Note that a 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.<ref>{{cite book|author=Robert H. Sanders|title=Revealing the Heart of the Galaxy: The Milky Way and its Black Hole|url=https://books.google.com/books?id=C1dzAwAAQBAJ|year=2013|publisher=Cambridge University Press|isbn=978-1-107-51274-0|page=[https://books.google.com.ph/books?id=C1dzAwAAQBAJ&pg=PA36 36]}}</ref>  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 [[kilogram per cubic metre|kg/m<sup>3</sup>]], 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 {{Solar mass|(1.36 × 10<sup>8</sup>)}}, 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|date=June 2014}}

The Schwarzschild radius of the [[Sagittarius A*|supermassive black hole]] at the [[Galactic Center]] is approximately 12 million kilometres.<ref name="Ghez08">{{cite journal
 | author = Ghez, A. M.
 | display-authors=et al.
 | title = Measuring Distance and Properties of the Milky Way's Central Supermassive Black Hole with Stellar Orbits
 | journal = Astrophysical Journal
 | date = December 2008
 | volume = 689 | issue = 2 | pages = 1044–1062
 | arxiv=0808.2870
 | bibcode = 2008ApJ...689.1044G
 | doi = 10.1086/592738
}}</ref>

===Stellar black hole===
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 10<sup>18</sup> [[kilogram per cubic metre|kg/m<sup>3</sup>]]; [[neutron star]]s also reach this density), such an accumulation would fall within its own Schwarzschild radius at about {{Solar mass|3}} and thus would be a [[stellar black hole]].

===Primordial black hole===
A small mass has an extremely small Schwarzschild radius. A mass similar to [[Mount Everest]]<ref name="SST">{{cite web
 |url         = http://chemist.sg/mole/Mount%20Everest%20M&Ms.pdf
 |title       = How does the mass of one mole of M&M's compare to the mass of Mount Everest?
 |date        = March 2003
 |website     = 
 |publisher   = School of Science and Technology, Singapore
 |access-date  = 8 December 2014
 |quote       = 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:<br /> volume <nowiki>=</nowiki> {{pi}}r<sup>2</sup>h/3 [...] Mount Everest is composed of granite, which has a density of 2750 kg m<sup>−3</sup>.
 |url-status     = dead
 |archive-url  = https://web.archive.org/web/20141210070657/http://chemist.sg/mole/Mount%20Everest%20M%26Ms.pdf
 |archive-date = 10 December 2014
 |df          = 
}}</ref>{{refn|group=note|Using these values,<ref name="SST"/> one can calculate a mass estimate of 6.3715e14 kg.}} has a Schwarzschild radius much smaller than a [[nanometre]].{{refn|group=note|One can calculate the Schwarzschild radius: 2 × 6.6738e-11 m<sup>3</sup> kg<sup>−1</sup> s<sup>−2</sup> × 6.3715e14 kg / (299 792 458 m s<sup>−1</sup>)<sup>2</sup> {{=}} 9.46e-13 m, or 9.46e-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 the [[Big Bang]], when densities were extremely high. Therefore, these hypothetical miniature black holes are called [[primordial black hole]]s.

== Other uses ==

=== In gravitational time dilation ===

[[Gravitational time dilation]] near a large, slowly rotating, nearly spherical body, such as the Earth or Sun can be reasonably approximated using the Schwarzschild radius as follows:

:<math> \frac{t_r}{t} = \sqrt{1 - \frac{r_\mathrm{s}}{r}} </math>

where:
: {{var|t<sub>r</sub>}} is the elapsed time for an observer at radial coordinate ''r'' within the gravitational field;
: {{var|t}} is the elapsed time for an observer distant from the massive object (and therefore outside of the gravitational field);
: {{var|r}} is the radial coordinate of the observer (which is analogous to the classical distance from the center of the object);
: {{math|''r''<sub>s</sub>}} 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.

=== 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:

:<math>mg = \frac{G M m}{r^2}\quad\Rightarrow\quad g r^2 = G M</math>
and
:<math>r_\mathrm{s} = \frac{2 G M}{c^2}\quad\Rightarrow\quad r_\mathrm{s} c^2 = 2 G M</math>

Therefore, on dividing above by below:
: <math> \frac{g}{r_\mathrm{s}} \left( \frac{r}{c} \right)^2 = \frac{1}{2} </math>
where:

: {{var|g}} is the gravitational acceleration at radial coordinate ''r'';
: {{math|''r''<sub>s</sub>}} is the Schwarzschild radius of the gravitating central body;
: {{var|r}} is the radial coordinate;
: {{var|c}} is the [[speed of light]] in vacuum.

On the surface of the Earth:

: <math> \frac{ 9.80665 \ \mathrm{m} / \mathrm{s}^2 }{ 8.870056 \ \mathrm{mm} } \left( \frac{6375416 \ \mathrm{m} }{299792458 \ \mathrm{m} / \mathrm{s} } \right)^2 = \left( 1105.59 \ \mathrm{s}^{-2} \right)  \left( 0.0212661 \ \mathrm{s} \right)^2 = \frac{1}{2}.</math>

=== In Keplerian orbits ===
For all [[circular orbit]]s around a given central body:

:<math>\frac{m v^2}{r} = \text{Centripetal force} = \text{Gravitational force} = \frac{G M m}{r^2}</math>
Therefore,
:<math>r v^2 = G M, </math>
but
:<math>r_\mathrm{s} c^2 = 2 G M</math> (derived above)
Therefore,
: <math> \frac{r}{r_\mathrm{s}} \left( \frac{v}{c} \right)^2 = \frac{1}{2} </math>
where:
: {{var|r}} is the orbit [[radius]];
: {{math|''r''<sub>s</sub>}} is the Schwarzschild radius of the gravitating central body;
: {{var|v}} is the [[orbital speed]];
: {{var|c}} is the [[speed of light]] in vacuum.

This equality can be generalized to [[elliptic orbit]]s as follows:

: <math> \frac{a}{r_\mathrm{s}} \left( \frac{2 \pi a}{c T} \right)^2 = \frac{1}{2} </math>
where:
: {{var|a}} is the [[semi-major axis]];
: {{var|T}} is the [[orbital period]].

For the [[Earth]], as a planet orbiting the [[Sun]]:

: <math>\frac{1 \,\mathrm{AU}}{2953.25\,\mathrm m} \left( \frac{2 \pi \,\mathrm{AU}}{\mathrm{light \,year}} \right)^2 = \left(50 655 379.7 \right) \left(9.8714403 \times 10^{-9} \right)= \frac{1}{2}.</math>

=== 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:

: <math> \frac{r}{r_s} \left( \frac{v}{c} \sqrt{1 - \frac{r_s}{r}} \right)^2 = \frac{1}{2} </math>

: <math> \frac{r}{r_s} \left( \frac{v}{c} \right)^2 \left(1 - \frac{r_s}{r} \right) = \frac{1}{2} </math>

: <math> \left( \frac{v}{c} \right)^2 \left( \frac{r}{r_s} - 1 \right) = \frac{1}{2}.</math>

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]].

=== Schwarzschild radius for Planck mass ===
For the [[Planck mass]] <math>m_{\rm P} = \sqrt{\hbar c/G}</math>, the Schwarzschild radius <math>r_{\rm S} = 2 l_{\rm P}</math> and the [[Compton wavelength]] <math>\lambda_{\rm C} = 2 \pi l_{\rm P}</math> are of the same order as the [[Planck length]] <math>l_{\rm P} = \sqrt{\hbar G/c^3}</math>.

==See also==
* [[Black hole]], a general survey
* [[Chandrasekhar limit]], a second requirement for black hole formation
* [[John Michell]]
Classification of black holes by type:
* [[Schwarzschild black hole|Static or Schwarzschild black hole]]
* [[Rotating black hole|Rotating or Kerr black hole]]
* [[Charged black hole|Charged black hole or Newman black hole and Kerr-Newman black hole]]
A classification of black holes by mass:
* [[Micro black hole]] and extra-dimensional black hole
* [[Primordial black hole]], a hypothetical leftover of the Big Bang
* [[Stellar black hole]], which could either be a static black hole or a rotating black hole
* [[Supermassive black hole]], which could also either be a static black hole or a rotating black hole
* [[Visible universe]], if its density is the [[Friedmann equations#Density parameter|critical density]], as a [[black hole cosmology|hypothetical black hole]]

==Notes==
{{Reflist|group=note}}

==References==
{{Reflist|30em}}

{{Black holes}}

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