Schwarzschild radius

In 1916, Karl Schwarzschild obtained the exact solution{{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 }}{{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 }} to Einstein's field equations for the gravitational field outside a non-rotating, spherically symmetric body with mass $M$ (see Schwarzschild metric). The solution contained terms of the form $1-\{r\_\backslash text\{s\}\}/r$ and $\backslash frac\{1\}\{1-\{r\_\backslash text\{s\}\}/r\}$, which become singular at $r\; =\; 0$ and $r=r\_\backslash text\{s\}$ respectively. The $r\_\backslash text\{s\}$ 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 $r\; =\; r\_\backslash text\{s\}$ is a coordinate singularity, meaning that it is an artifact of the particular system of coordinates that was used; while the one at $r=0$ is a spacetime singularity and cannot be removed.{{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]}} 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{{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 }} and Pierre-Simon Laplace.{{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 }}

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}},{{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}} whereas Earth's is approximately {{convert|9|mm|in|abbr=on}} and the Moon's is approximately {{convert|0.1|mm|in|abbr=on}}.

{{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:

:$M\; c^2\; =\; \backslash frac\{2GM^2\}\{r\}$

So, the Schwarzschild radius reads as

:$r\; =\; \backslash frac\{2GM\}\{c^2\}$

Any object whose radius is smaller than its Schwarzschild radius is called a black hole.{{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}}{{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.

{{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 × 10¹⁰)}} have been detected, such as NGC 4889.){{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}} 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.{{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.{{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]}} 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 ({{Solar mass|1.36 × 10⁸}}), 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.{{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 }}

The Schwarzschild radius of the supermassive black hole at the Galactic Center of the Milky Way is approximately 12 million kilometres.{{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 }} Its mass is about {{Solar mass|4.1 million}}.

{{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 10¹⁸ kg/m³; neutron stars 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}}

{{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,{{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:
volume = {{pi}}r²h/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}} {{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=m³⋅kg⁻¹⋅s⁻²}} × {{val|6.3715e14|u=kg}} / ({{val|299792458|u=m.s-1}})² {{=}} {{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 holes.{{cn|date=February 2025}}

When moving to the Planck scale {{nobr|$\backslash ell\_P=\backslash sqrt\{(G/c^3)\backslash ,\backslash hbar\}$ ≈ 10⁻³⁵ m}}, it is convenient to write the gravitational radius in the form $r\_s\; =\; 2\backslash ,(G/c^3)Mc$, (see also virtual black hole).[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.]

Gravitational time dilation near a large, slowly rotating, nearly spherical body, such as the Earth or Sun can be reasonably approximated as follows:{{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}}

$$\backslash frac\{t\_r\}\{t\}\; =\; \backslash sqrt\{1\; -\; \backslash frac\{r\_\backslash mathrm\{s\}\}\{r\}\}$$

where:

• {{var|t_r}} 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_s}} is the Schwarzschild radius.

The Schwarzschild radius ($2\; G\; M/c^2$) and the Compton wavelength ($2\backslash pi\backslash hbar/M\; c$) corresponding to a given mass are similar when the mass is around one Planck mass ($M=\backslash sqrt\{\backslash hbar\; c/G\}$), when both are of the same order as the Planck length ($\backslash sqrt\{\backslash hbar\; G/c^3\}$).

=== Gravitational radius and the Heisenberg Uncertainty Principle ===

:$r\_s=\backslash frac\{2GM\}\{c^2\}=\backslash frac\{2G\}\{c^3\}Mc=\backslash frac\{2G\}\{c^3\}P\_0\backslash Rightarrow\backslash frac\{2G\}\{c^3\}\backslash frac\{\backslash hbar\}\{2r\}=\backslash frac\{\backslash ell\_P^2\}\{r\}.$

Thus, $r\_s\; r\backslash sim\backslash ell^2\_P$ or $\backslash Delta\; r\_s\backslash Delta\; r\backslash ge\backslash ell^2\_P$, which is another form of the Heisenberg uncertainty principle on the Planck scale. (See also Virtual black hole).[https://philpapers.org/archive/ALXOTF.pdf Klimets A.P., Philosophy Documentation Center, Western University-Canada, 2017, pp.25-30]

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| ρ}},

:$r\_\backslash text\{s\}\; =\; \backslash sqrt\{\backslash frac\{3\; c^\{2\}\}\{8\; \backslash pi\; G\; \backslash rho\}\}.$

For example, the density of water is {{val|1000|u=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).

• Black hole, a general survey
• Chandrasekhar limit, a second requirement for black hole formation
• John Michell

Classification of black holes by type:

• Static or Schwarzschild black hole
• Rotating or Kerr 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 critical density, as a hypothetical black hole
• Virtual black hole

{{Reflist|group=note}}

{{Reflist|30em}}

{{Black holes}}

Category:Black holes

Category:1916 in science

Category:Radii