View source for Schwarzschild radius

{{Short description|Radius of the event horizon of a Schwarzschild black hole}}
{{use dmy dates |date=September 2023}}

[[File:Triangle of everything simplified 2 triangle of everything - Planck Units.png | thumb|upright=1.5 | In a mass-radius plot, the Schwarzschild radius represents the upper limit of any object that can exist (the [[Compton Wavelength]] and [[Hubble radius]] being other 2 limits forming a triangle). It's intersection with the Compton Wavelength defines all [[Planck Units]]. ]]

The '''Schwarzschild radius''' or the '''gravitational radius''' is a physical parameter in the [[Schwarzschild solution]] to [[Einstein's field equation]]s 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
<math display="block"> r_\text{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]].{{refn|group=note|In [[geometrized unit system]]s, ''G'' and ''c'' are both taken to be unity, which reduces this equation to <math> r_\text{s} = 2M</math>.}}<ref>{{Cite book |last=Kutner |first=Marc Leslie |url=https://archive.org/details/astronomyphysica00kutn/ |title=Astronomy: a physical perspective |date=2003 |publisher=[[Cambridge University Press]] |isbn=978-0-521-82196-4 |edition=2nd |location=Cambridge, U.K.; New York |pages=148}}</ref><ref>{{Cite book |last=Guidry |first=M. W. |title=Modern general relativity: black holes, gravitational waves, and cosmology |date=2019 |publisher=Cambridge University Press |isbn=978-1-107-19789-3 |location=Cambridge; New York, NY |pages=92}}</ref>

==History==
In 1916, [[Karl Schwarzschild]] obtained the exact solution<ref>{{cite journal | url=https://ui.adsabs.harvard.edu/abs/1916SPAW.......189S/abstract |bibcode=1916SPAW.......189S |title=Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie |last1=Schwarzschild |first1=Karl |journal=Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften |date=1916 |page=189 }}</ref><ref>{{cite journal |url=https://ui.adsabs.harvard.edu/abs/1916skpa.conf..424S/abstract |page=424 |bibcode=1916skpa.conf..424S |title=Über das Gravitationsfeld einer Kugel aus inkompressibler Flüssigkeit nach der Einsteinschen Theorie |last1=Schwarzschild |first1=Karl |journal=Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin |date=1916 }}</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_\text{s}}/r </math> and <math>\frac{1}{1-{r_\text{s}}/r} </math>, which become [[Mathematical singularity|singular]] at <math> r = 0</math> and <math> r=r_\text{s}</math> respectively. The <math>r_\text{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_\text{s}</math> is a coordinate singularity, meaning that it is an artifact of the particular system of coordinates that was used; while the one at <math> r=0</math> is a [[spacetime singularity]] 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 Michell 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 |access-date=4 June 2018|bibcode=1979JHA....10...42S |doi=10.1177/002182867901000104 |s2cid=123958527 }}</ref> and [[Pierre-Simon Laplace]].<ref>{{cite journal |bibcode=2009JAHH...12...90M |url=http://www.narit.or.th/en/files/2009JAHHvol12/2009JAHH...12...90M.pdf |archive-url=https://web.archive.org/web/20140502005017/http://www.narit.or.th/en/files/2009JAHHvol12/2009JAHH...12...90M.pdf |archive-date=2 May 2014 |title=Michell, Laplace and the origin of the black hole concept |last1=Montgomery |first1=Colin |last2=Orchiston |first2=Wayne |last3=Whittingham |first3=Ian |journal=Journal of Astronomical History and Heritage |date=2009 |volume=12 |issue=2 |page=90 |doi=10.3724/SP.J.1440-2807.2009.02.01 |s2cid=55890996 }}</ref>

==Parameters==
The Schwarzschild radius of an object is proportional to its mass. Accordingly, the [[Sun]] has a Schwarzschild radius of approximately {{convert|3.0|km|mi|abbr=on}},<ref name="Anderson">{{cite encyclopedia |title=V.C The Schwarzschild Field, Event Horizons, and Black Holes |encyclopedia=Encyclopedia of Physical Science and Technology (Third Edition) |editor1-last=Meyer |editor1-first=Robert A. |date=2001 |last=Anderson |first=James L. |publisher=[[Academic Press]] |location=Cambridge, Massachusetts |isbn=978-0-12-227410-7 |url=https://www.sciencedirect.com/topics/physics-and-astronomy/schwarzschild-radius |access-date=23 October 2023}}</ref> whereas [[Earth]]'s is approximately {{convert|9|mm|in|abbr=on}}<ref name="Anderson" /> and the [[Moon]]'s is approximately {{convert|0.1|mm|in|abbr=on}}.

{| class="wikitable mw-collapsible mw-collapsed" style="text-align:right;"
|+Object's Schwarzschild radius
|- 
! style="text-align:left;" | Object
! Mass <math display="inline">M</math>
! Schwarzschild radius <math display="inline">\frac{2GM}{c^2}</math>
! Actual radius <math display="inline">r</math>
! Schwarzschild density <math display="inline">\frac{3c^6}{32\pi G^3M^2}</math> or <math display="inline">\frac{3c^2}{8\pi Gr^2}</math>
|-
| style="text-align:left;" | [[Milky Way]]
| {{val|1.6|e=42|u=kg}}
| {{val|2.4|e=15|u=m}} ({{val|0.25|u=[[light-year|ly]]}})
| {{val|5|e=20|u=m}} ({{val|52900|u=[[light-year|ly]]}})
| {{val|0.000029|u=kg/m3}}
|-
| style="text-align:left;" | [[Supermassive black hole|SMBH]] in [[Phoenix Cluster#Supermassive black hole|Phoenix A]] (one of the largest [[List of most massive black holes|known black holes]])
| {{val|2|e=41|u=kg}}
| {{val|3|e=14|u=m}} (~2000&nbsp;[[astronomical unit|AU]])
|
| {{val|0.0018|u=kg/m3}}
|-
| style="text-align:left;" | [[Ton 618]]
| {{val|1.3|e=41|u=kg}}
| {{val|1.9|e=14|u=m}} (~1300&nbsp;[[astronomical unit|AU]])
|
| {{val|0.0045|u=kg/m3}}
|-
| style="text-align:left;" | [[Supermassive black hole|SMBH]] in [[NGC 4889]]
| {{val|4.2|e=40|u=kg}}
| {{val|6.2|e=13|u=m}} (~410&nbsp;[[astronomical unit|AU]])
|
| {{val|0.042|u=kg/m3}}
|-
| style="text-align:left;" | [[Supermassive black hole|SMBH]] in [[Messier 87]]<ref name="Event Horizon Telescope Collaboration et al. 2019">{{cite journal
 | last1 = Event Horizon Telescope Collaboration
 | 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 = L1
 | bibcode = 2019ApJ...875L...1E
 | doi = 10.3847/2041-8213/AB0EC7
 | arxiv = 1906.11238
 | doi-access = free
}}
{{val|6.5|(7)|e=9|u={{Solar mass}}}} = {{val|1.29|(14)|e=40|u=kg}}.
</ref>
| {{val|1.3|e=40|u=kg}}
| {{val|1.9|e=13|u=m}} (~130&nbsp;[[astronomical unit|AU]])
|
| {{val|0.44|u=kg/m3}}
|-
| style="text-align:left;" | [[Supermassive black hole|SMBH]] in [[Andromeda Galaxy]]<ref name="Bender et al 2005">{{cite journal
 | display-authors = 3
 | last1 = Bender | first1 = Ralf
 | last2 = Kormendy | first2 = John
 | last3 = Bower | first3 = Gary
 | 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.
 | 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
 | 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
 | s2cid = 53415285
}}
{{val|1.7|(6)|e=8|u={{Solar mass}}}} = {{val|0.34|(12)|e=39|u=kg}}.
</ref>
| {{val|3.4|e=38|u=kg}}
| {{val|5.0|e=11|u=m}} (3.3&nbsp;[[astronomical unit|AU]])
|
| {{val|640|u=kg/m3}}
|-
| style="text-align:left;" | [[Sagittarius A*|Sagittarius A* (SMBH in Milky Way)]]<ref name="Ghez08" />
| {{val|8.26|e=36|u=kg}}
| {{val|1.23|e=10|u=m}} (0.08&nbsp;[[astronomical unit|AU]])
|
| {{val|1.068|e=6|u=kg/m3}}
|-
| style="text-align:left;" |[[Supermassive black hole|SMBH]] in [[NGC 4395]]<ref>{{Cite journal |last1=Peterson |first1=Bradley M. |last2=Bentz |first2=Misty C. |last3=Desroches |first3=Louis-Benoit |last4=Filippenko |first4=Alexei V. |last5=Ho |first5=Luis C. |last6=Kaspi |first6=Shai |last7=Laor |first7=Ari |last8=Maoz |first8=Dan |last9=Moran |first9=Edward C. |last10=Pogge |first10=Richard W. |last11=Quillen |first11=Alice C. |date=2005-10-20 |title=Multiwavelength Monitoring of the Dwarf Seyfert 1 Galaxy NGC 4395. I. A Reverberation-Based Measurement of the Black Hole Mass |journal=The Astrophysical Journal |volume=632 |issue=2 |pages=799–808 |doi=10.1086/444494 |arxiv=astro-ph/0506665 |bibcode=2005ApJ...632..799P |hdl=1811/48314 |s2cid=13886279 |issn=0004-637X}}</ref>
| {{val|7.1568|e=35|u=kg}} 
| {{val|1.062|e=9|u=m}} (1.53&nbsp;[[Solar radius|''R''<sub>⊙</sub>]])
|
| {{val|1.4230|e=8|u=kg/m3}}
|-
| style="text-align:left;" |Potential [[Intermediate-mass black hole|intermediate black hole]] in [[HCN-0.009-0.044]]<ref>{{Cite web |last= |first= |date=1 March 2019 |title=Hiding black hole found |url=https://phys.org/news/2019-03-black-hole.html |access-date=2022-06-15 |website=[[phys.org]] |language=en}}</ref><ref>{{cite journal |doi=10.3847/2041-8213/aafb07 |doi-access=free |title=Indication of Another Intermediate-mass Black Hole in the Galactic Center |date=2019 |last1=Takekawa |first1=Shunya |last2=Oka |first2=Tomoharu |last3=Iwata |first3=Yuhei |last4=Tsujimoto |first4=Shiho |last5=Nomura |first5=Mariko |journal=The Astrophysical Journal |volume=871 |issue=1 |pages=L1 |arxiv=1812.10733 |bibcode=2019ApJ...871L...1T }}</ref>
| {{val|6.3616|e=34|u=kg}}
| {{val|9.44|e=8|u=m}} (14.8&nbsp;[[Earth radius|''R''<sub>🜨</sub>]])
|
| {{val|1.8011|e=10|u=kg/m3}}
|-
| style="text-align:left;" |Resulting [[Intermediate-mass black hole|intermediate black hole]] from [[GW190521]] merger<ref>{{Cite journal |last1=Abbott |first1=R. |last2=Abbott |first2=T. D. |last3=Abraham |first3=S. |last4=Acernese |first4=F. |last5=Ackley |first5=K. |last6=Adams |first6=C. |last7=Adhikari |first7=R. X. |last8=Adya |first8=V. B. |last9=Affeldt |first9=C. |last10=Agathos |first10=M. |last11=Agatsuma |first11=K. |date=2020-09-02 |title=Properties and Astrophysical Implications of the 150&nbsp;''M''<sub>⊙</sub> Binary Black Hole Merger GW190521 |journal=The Astrophysical Journal |language=en |volume=900 |issue=1 |pages=L13 |doi=10.3847/2041-8213/aba493 |arxiv=2009.01190 |bibcode=2020ApJ...900L..13A |s2cid=221447444 |issn=2041-8213 |doi-access=free }}</ref>
| {{val|2.823|e=32|u=kg}}
| {{val|4.189|e=5|u=m}} (0.066&nbsp;[[Earth radius|''R''<sub>🜨</sub>]])
|
| {{val|9.125|e=14|u=kg/m3}}
|-
| style="text-align:left;" | [[Sun]]
| {{val|1.99|e=30|u=kg}}
| {{val|2.95|e=3|u=m}}
| {{val|7.0|e=8|u=m}}
| {{val|1.84|e=19|u=kg/m3}}
|-
| style="text-align:left;" | [[Jupiter]]
| {{val|1.90|e=27|u=kg}}
| {{val|2.82|u=m}}
| {{val|7.0|e=7|u=m}}
| {{val|2.02|e=25|u=kg/m3}}
|-
| style="text-align:left;" | [[Saturn]]
| {{val|5.683|e=26|u=kg}}
| {{val|8.42|e=-1|u=m}}
| {{val|6.03|e=7|u=m}}
| {{val|2.27|e=26|u=kg/m3}}
|-
| style="text-align:left;" | [[Neptune]]
| 1.024{{e|26}} kg
| 1.52{{e|-1}} m
| 2.47{{e|7}} m
| 6.97{{e|27}} kg/m<sup>3</sup>
|-
| style="text-align:left;" | [[Uranus]]
| 8.681{{e|25}} kg
| 1.29{{e|-1}} m
| 2.56{{e|7}} m
| 9.68{{e|27}} kg/m<sup>3</sup>
|-
| style="text-align:left;" | [[Earth]]
| 5.97{{e|24}} kg
| 8.87{{e|-3}} m
| 6.37{{e|6}} m
| 2.04{{e|30}} kg/m<sup>3</sup>
|-
| style="text-align:left;" | [[Venus]]
| 4.867{{e|24}} kg
| 7.21{{e|-3}} m
| 6.05{{e|6}} m
| 3.10{{e|30}} kg/m<sup>3</sup>
|-
| style="text-align:left;" | [[Mars]]
| 6.39{{e|23}} kg
| 9.47{{e|-4}} m
| 3.39{{e|6}} m
| 1.80{{e|32}} kg/m<sup>3</sup>
|-
| style="text-align:left;" | [[Mercury (planet)|Mercury]]
| 3.285{{e|23}} kg
| 4.87{{e|-4}} m
| 2.44{{e|6}} m
| 6.79{{e|32}} kg/m<sup>3</sup>
|-
| style="text-align:left;" | [[Moon]]
| 7.35{{e|22}} kg
| 1.09{{e|-4}} m
| 1.74{{e|6}} m
| 1.35{{e|34}} kg/m<sup>3</sup>
|-
| style="text-align:left;" | [[Human]]
| 70&nbsp;kg
| 1.04{{e|-25}} m
| ~5{{e|-1}} m
| 1.49{{e|76}} kg/m<sup>3</sup>
|-
| style="text-align:left;" | [[Planck mass]]
| 2.18{{e|-8}} kg
| 3.23{{e|-35}} m (2&nbsp;[[Planck length|''l''<sub>P</sub>]])
| style="text-align:left;" |
| 1.54{{e|95}} kg/m<sup>3</sup>
|}

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

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: 

:<math>M c^2 = \frac{2GM^2}{r}</math>

So, the Schwarzschild radius reads as 

:<math>r = \frac{2GM}{c^2}</math>

==Black hole classification by Schwarzschild radius==
{|class="wikitable" style="float:right; margin:0 0 0.5em 1em;"
|+ Black hole classifications
|-
! Class !! Approx.<br />mass !! Approx.<br />radius
|-
|[[Supermassive black hole]] ||style="text-align: center;"|10{{sup|5}}–10{{sup|11}} [[solar mass|''M''{{sub|Sun}}]] ||style="text-align: center;"|0.002–2000 [[Astronomical unit|AU]]
|-
|[[Intermediate-mass black hole]] ||style="text-align: center;"|10{{sup|3}} ''M''{{sub|Sun}} ||style="text-align: center;"|3 x 10{{sup|3}} km ≈ [[Mars radius|''R''{{sub|Mars}}]]
|-
|[[Stellar black hole]] ||style="text-align: center;"|10 ''M''{{sub|Sun}} ||style="text-align: center;"|30&nbsp;km
|-
|[[Micro black hole]] ||style="text-align: center;"|up to ''M''{{sub|[[Moon]]}} ||style="text-align: center;"|up to 0.1&nbsp;mm
|}
Any object whose radius is smaller than its Schwarzschild radius is called a [[black hole]].<ref>{{Cite book |last=Zee |first=Anthony |title=Einstein Gravity in a Nutshell |date=2013 |publisher=Princeton University Press |isbn=978-0-691-14558-7 |edition=1 |series=In a Nutshell Series |location=Princeton}}</ref>{{rp|410}} 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.

===Supermassive black hole===
{{main|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 |s2cid=4408896}}</ref> Unlike [[Stellar black hole|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.{{cn|date=February 2025}}

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/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&nbsp;×&nbsp;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.<ref>{{Cite journal|last1=Pacucci|first1=Fabio|last2=Loeb|first2=Abraham|date=2020-06-01|title=Separating Accretion and Mergers in the Cosmic Growth of Black Holes with X-Ray and Gravitational-wave Observations|bibcode=2020ApJ...895...95P|journal=The Astrophysical Journal|volume=895|issue=2|pages=95|doi=10.3847/1538-4357/ab886e|arxiv=2004.07246|s2cid=215786268 |doi-access=free }}</ref>

The Schwarzschild radius of the [[Sagittarius A*|supermassive black hole]] at the [[Galactic Center]] of the [[Milky Way]] 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
 | s2cid=18335611 }}</ref> Its mass is about {{Solar mass|4.1&nbsp;million}}.

===Stellar black hole===
{{main|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]].{{cn|date=February 2025}}

===Micro black hole===
{{main|Micro black hole}}
{{original research|section|date=February 2025}}
A small mass has an extremely small Schwarzschild radius. A black hole of mass similar to that of [[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 |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 {{val|2750|u=kg.m-3}}. |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}}</ref> {{val|6.3715e14|u=kg}}, would have a Schwarzschild radius much smaller than a [[nanometre]].{{cn|date=February 2025}} The Schwarzschild radius would be 2 × {{val|6.6738e-11|u=m<sup>3</sup>⋅kg<sup>−1</sup>⋅s<sup>−2</sup>}} × {{val|6.3715e14|u=kg}} / ({{val|299792458|u=m.s-1}})<sup>2</sup> {{=}} {{val|9.46e-13|u=m}} {{=}} {{val|9.46e-4|u=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 of matter were extremely high. Therefore, these hypothetical miniature black holes are called [[primordial black hole]]s.{{cn|date=February 2025}}

When moving to the [[Planck length|Planck scale]] {{nobr|<math>\ell_P=\sqrt{(G/c^3)\,\hbar}</math> ≈ 10<sup>−35</sup> m}}, it is convenient to write the gravitational radius in the form <math>r_s = 2\,(G/c^3)Mc</math>, (see also [[virtual black hole]]).<ref name="klimets1">[https://www.opastpublishers.com/open-access-articles/quantum-gravity.pdf A.P. Klimets. (2023). Quantum Gravity. Current Research in Statistics & Mathematics, 2(1), 141-155.]</ref>

== 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 as follows:<ref>{{Cite book |last=Keeton |first=Charles |url=https://books.google.com/books?id=PoQpBAAAQBAJ |title=Principles of astrophysics: using gravity and stellar physics to explore the cosmos |date=2014 |publisher=Springer |isbn=978-1-4614-9236-8 |series=Undergraduate Lecture Notes in Physics |location=New York |pages=208}}</ref>
<math display="block"> \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.

=== Compton wavelength intersection ===
The Schwarzschild radius (<math>2 G M/c^2</math>) and the [[Compton wavelength]] (<math>2\pi\hbar/M c</math>) corresponding to a given mass are similar when the mass is around one [[Planck mass]] (<math display="inline">M=\sqrt{\hbar c/G}</math>), when both are of the same order as the [[Planck length]] (<math display="inline">\sqrt{\hbar G/c^3}</math>).

=== Gravitational radius and the Heisenberg Uncertainty Principle === 
:<math>r_s=\frac{2GM}{c^2}=\frac{2G}{c^3}Mc=\frac{2G}{c^3}P_0\Rightarrow\frac{2G}{c^3}\frac{\hbar}{2r}=\frac{\ell_P^2}{r}.</math>

Thus, <math>r_s r\sim\ell^2_P</math> or <math>\Delta r_s\Delta r\ge\ell^2_P</math>, which is another form of the [[Heisenberg uncertainty principle]] on the [[Planck units|Planck scale]]. (See also [[Virtual black hole]]).<ref name="klimets1"/><ref>[https://philpapers.org/archive/ALXOTF.pdf Klimets A.P., Philosophy Documentation Center, Western University-Canada, 2017, pp.25-30]</ref>

=== Calculating the maximum volume and radius possible given a density before a black hole forms ===
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 {{math| ''&rho;''}},

:<math>r_\text{s} = \sqrt{\frac{3 c^{2}}{8 \pi G \rho}}.</math>

For example, the density of water is {{val|1000|u=kg/m<sup>3</sup>}}. This means the largest amount of water you can have without forming a black hole would have a radius of 400 920 754&nbsp;km (about 2.67 [[Astronomical unit|AU]]).

==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
* [[Planck length]]
* [[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]]
* [[Virtual black hole]]

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

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

{{Black holes}}

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