Editing Black hole (section)

=== General relativity ===
{{See also|History of general relativity}}
{{General relativity sidebar|phenomena}}

In 1915, [[Albert Einstein]] developed his theory of [[general relativity]], having earlier shown that gravity does influence light's motion. Only a few months later, [[Karl Schwarzschild]] found a [[Schwarzschild metric|solution]] to the [[Einstein field equations]] that describes the [[gravitational field]] of a [[point mass]] and a spherical mass.<ref name="Levy">{{cite journal |last1=Levy |first1=Adam |title=How black holes morphed from theory to reality |journal=Knowable Magazine |date=11 January 2021 |doi=10.1146/knowable-010921-1 |doi-access=free |url=https://knowablemagazine.org/article/physical-world/2021/how-black-holes-morphed-theory-reality |access-date=25 March 2022}}</ref><ref name="Schwarzschild1916">{{Cite journal |last=Schwarzschild |first=K. |author-link1=Karl Schwarzschild |date=1916 |title=Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie |url=https://archive.org/stream/sitzungsberichte1916deutsch#page/188/mode/2up |journal=Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften |volume=7 |pages=189–196|bibcode=1916SPAW.......189S}}
:* Translation: {{cite arXiv |last1=Antoci |first1=S. |last2=Loinger |first2=A. |year=1999 |title=On the gravitational field of a mass point according to Einstein's theory |eprint=physics/9905030}} and {{Cite journal |last=Schwarzschild |first=K. |author-link1=Karl Schwarzschild |date=1916 |title=Über das Gravitationsfeld einer Kugel aus inkompressibler Flüssigkeit nach der Einsteinschen Theorie |url=https://archive.org/stream/sitzungsberichte1916deutsch#page/424/mode/2up |journal=Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften |volume=18 |pages=424–434|bibcode=1916skpa.conf..424S}}
:* Translation: {{cite arXiv |last1=Antoci |first1=S. |year=1999 |title=On the gravitational field of a sphere of incompressible fluid according to Einstein's theory |eprint=physics/9912033}}</ref> A few months after Schwarzschild, [[Johannes Droste]], a student of [[Hendrik Lorentz]], independently gave the same solution for the point mass and wrote more extensively about its properties.<ref>{{Cite journal |last=Droste |first=J. |title=On the field of a single centre in Einstein's theory of gravitation, and the motion of a particle in that field |journal=Proceedings Royal Academy Amsterdam |date=1917 |volume=19 |issue=1 |pages=197–215 |url=http://www.dwc.knaw.nl/DL/publications/PU00012325.pdf|archive-url=https://web.archive.org/web/20130518034708/http://www.dwc.knaw.nl/DL/publications/PU00012325.pdf |archive-date=18 May 2013 |access-date=16 September 2012 }}</ref><ref>{{cite book |title=Studies in the history of general relativity |editor1-last=Eisenstaedt |editor1-first=Jean |editor2-last=Kox |editor2-first=A. J. |isbn=978-0-8176-3479-7 |date=1992 |publisher=Birkhäuser |chapter=General Relativity in the Netherlands: 1915–1920 |last=Kox |first=A. J. |chapter-url=https://books.google.com/books?id=vDHCF_3vIhUC&pg=PA41 |page=41 |access-date=23 February 2016 |archive-date=10 August 2016 |archive-url=https://web.archive.org/web/20160810215219/https://books.google.com/books?id=vDHCF_3vIhUC&pg=PA41 |url-status=live }}</ref> This solution had a peculiar behaviour at what is now called the [[Schwarzschild radius]], where it became [[singularity (mathematics)|singular]], meaning that some of the terms in the Einstein equations became infinite. The nature of this surface was not quite understood at the time.

In 1924, [[Arthur Eddington]] showed that the singularity disappeared after a change of coordinates. In 1933, [[Georges Lemaître]] realised that this meant the singularity at the Schwarzschild radius was a non-physical [[coordinate singularity]].<ref name="HooftHist">{{Cite web |last='t Hooft |first=G. |author-link1=Gerard 't Hooft |date=2009 |title=Introduction to the Theory of Black Holes |url=http://www.phys.uu.nl/~thooft/lectures/blackholes/BH_lecturenotes.pdf |publisher=Institute for Theoretical Physics / Spinoza Institute |pages=47–48|archive-url=https://web.archive.org/web/20090521082736/http://www.phys.uu.nl/~thooft/lectures/blackholes/BH_lecturenotes.pdf |archive-date=21 May 2009 |access-date=24 June 2010 }}</ref> Arthur Eddington commented on the possibility of a star with mass compressed to the Schwarzschild radius in a 1926 book, noting that Einstein's theory allows us to rule out overly large densities for visible stars like [[Betelgeuse]] because "a star of 250 million km radius could not possibly have so high a density as the Sun. Firstly, the force of gravitation would be so great that light would be unable to escape from it, the rays falling back to the star like a stone to the earth. Secondly, the red shift of the spectral lines would be so great that the spectrum would be shifted out of existence. Thirdly, the mass would produce so much curvature of the spacetime metric that space would close up around the star, leaving us outside (i.e., nowhere)."<ref name="eddington1926">{{cite book |first=Arthur |last=Eddington |author-link=Arthur Eddington |date=1926 |title=The Internal Constitution of the Stars |series=Science |volume=52 |issue=1341 |pages=233–40 |url=https://books.google.com/books?id=RjC9DpnWFbkC&pg=PA6 |publisher=Cambridge University Press |isbn=978-0-521-33708-3 |url-status=live |archive-url=https://web.archive.org/web/20160811034409/https://books.google.com/books?id=RjC9DpnWFbkC&lpg=PP1&pg=PA6 |archive-date=11 August 2016 |pmid=17747682 |doi=10.1126/science.52.1341.233|bibcode=1920Sci....52..233E }}</ref><ref>{{cite book |last1=Thorne |first1=Kip S. |url=https://archive.org/details/blackholestimewa0000thor |title=Black Holes and Time Warps: Einstein's Outrageous Legacy |last2=Hawking |first2=Stephen |date=1994 |publisher=W. W. Norton & Company |isbn=978-0-393-31276-8 |editor-last=Agrawal |editor-first=Milan |edition=1st |pages=[https://archive.org/details/blackholestimewa0000thor/page/134 134]–135 |quote=The first conclusion was the Newtonian version of light not escaping; the second was a semi-accurate, relativistic description; and the third was typical Eddingtonian hyperbole{{nbsp}}... when a star is as small as the critical circumference, the curvature is strong but not infinite, and space is definitely not wrapped around the star. Eddington may have known this, but his description made a good story, and it captured in a whimsical way the spirit of Schwarzschild's spacetime curvature." |access-date=12 April 2019 |url-access=registration}}</ref>

In 1931, [[Subrahmanyan Chandrasekhar]] calculated, using special relativity, that a non-rotating body of [[electron-degenerate matter]] above a certain limiting mass (now called the [[Chandrasekhar limit]] at {{Solar mass|1.4}}) has no stable solutions.<ref name="venkataraman92">{{cite book |first=G. |last=Venkataraman |title=Chandrasekhar and his limit |page=89 |publisher=Universities Press |date=1992 |url=https://books.google.com/books?id=HNSdDFOJ4wkC&pg=PA89 |isbn=978-81-7371-035-3 |url-status=live |archive-url=https://web.archive.org/web/20160811035848/https://books.google.com/books?id=HNSdDFOJ4wkC&pg=PA89 |archive-date=11 August 2016}}</ref> His arguments were opposed by many of his contemporaries like Eddington and [[Lev Landau]], who argued that some yet unknown mechanism would stop the collapse.<ref>{{Cite journal |last=Detweiler |first=S. |date=1981 |title=Resource letter BH-1: Black holes |journal=[[American Journal of Physics]] |volume=49 |issue=5 |pages=394–400 |doi=10.1119/1.12686|bibcode=1981AmJPh..49..394D}}</ref> They were partly correct: a [[white dwarf]] slightly more massive than the Chandrasekhar limit will collapse into a [[neutron star]],<ref>{{cite book |last1=Harpaz |first1=A. |date=1994 |title=Stellar evolution |url=https://books.google.com/books?id=kd4VEZv8oo0C&pg=PA105 |publisher=[[A K Peters, Ltd.|A K Peters]] |page=105 |isbn=978-1-56881-012-6 |url-status=live |archive-url=https://web.archive.org/web/20160811025449/https://books.google.com/books?id=kd4VEZv8oo0C&pg=PA105 |archive-date=11 August 2016}}</ref> which is itself stable.

In 1939, [[Robert Oppenheimer]] and others predicted that neutron stars above another limit, the [[Tolman–Oppenheimer–Volkoff limit]], would collapse further for the reasons presented by Chandrasekhar, and concluded that no law of physics was likely to intervene and stop at least some stars from collapsing to black holes.<ref name="OV1939">{{Cite journal |last1=Oppenheimer |first1=J. R. |author-link1=J. Robert Oppenheimer |last2=Volkoff |first2=G. M. |author-link2=George Volkoff |date=1939 |title=On Massive Neutron Cores |journal=[[Physical Review]] |volume=55 |issue=4 |pages=374–381 |doi=10.1103/PhysRev.55.374|bibcode=1939PhRv...55..374O}}</ref> Their original calculations, based on the [[Pauli exclusion principle]], gave it as {{Solar mass|0.7}}. Subsequent consideration of neutron-neutron repulsion mediated by the strong force raised the estimate to approximately {{Solar mass|1.5}} to {{Solar mass|3.0}}.<ref name="Bombaci">{{cite journal |first=I. |last=Bombaci |date=1996 |title=The Maximum Mass of a Neutron Star |journal=[[Astronomy and Astrophysics]] |volume=305 |pages=871–877 |bibcode=1996A&A...305..871B}}</ref> Observations of the neutron star merger [[GW170817]], which is thought to have generated a black hole shortly afterward, have refined the TOV limit estimate to ~{{Solar mass|2.17}}.<ref name="Cho2018">{{cite journal |last=Cho |first=A. |title=A weight limit emerges for neutron stars |journal=Science |volume=359 |issue=6377 |pages=724–725 |date=16 February 2018 |doi=10.1126/science.359.6377.724 |pmid=29449468 |bibcode=2018Sci...359..724C}}</ref><ref name="Margalit2017">{{cite journal |last1=Margalit |first1=B. |last2=Metzger |first2=B. D. |title=Constraining the Maximum Mass of Neutron Stars from Multi-messenger Observations of GW170817 |journal=The Astrophysical Journal |volume=850 |issue=2 |date=1 December 2017 |page=L19 |doi=10.3847/2041-8213/aa991c |arxiv=1710.05938 |bibcode=2017ApJ...850L..19M|s2cid=119342447 |doi-access=free }}</ref><ref name="Shibata2017">{{cite journal |last1=Shibata |first1=M. |last2=Fujibayashi |first2=S. |last3=Hotokezaka |first3=K. |last4=Kiuchi |first4=K. |last5=Kyutoku |first5=K. |last6=Sekiguchi |first6=Y. |last7=Tanaka |first7=M. |title=Modeling GW170817 based on numerical relativity and its implications |journal=Physical Review D |volume=96 |issue=12 |page=123012 |date=22 December 2017 |doi=10.1103/PhysRevD.96.123012 |arxiv=1710.07579 |bibcode=2017PhRvD..96l3012S|s2cid=119206732 }}</ref><ref name="Ruiz2018">{{cite journal |last1=Ruiz |first1=M. |last2=Shapiro |first2=S. L. |last3=Tsokaros |first3=A. |title=GW170817, general relativistic magnetohydrodynamic simulations, and the neutron star maximum mass |journal=Physical Review D |volume=97 |issue=2 |page=021501 |date=11 January 2018 |doi=10.1103/PhysRevD.97.021501 |pmid=30003183 |pmc=6036631 |arxiv=1711.00473 |bibcode=2018PhRvD..97b1501R}}</ref><ref name="Rezzolla2018">{{cite journal |last1=Rezzolla |first1=L. |last2=Most |first2=E. R. |last3=Weih |first3=L. R. |title=Using Gravitational-wave Observations and Quasi-universal Relations to Constrain the Maximum Mass of Neutron Stars |journal=Astrophysical Journal |volume=852 |issue=2 |date=9 January 2018 |page=L25 |doi=10.3847/2041-8213/aaa401 |arxiv=1711.00314 |bibcode=2018ApJ...852L..25R|s2cid=119359694 |doi-access=free }}</ref>

Oppenheimer and his co-authors interpreted the singularity at the boundary of the Schwarzschild radius as indicating that this was the boundary of a bubble in which time stopped.  This is a valid point of view for external observers, but not for infalling observers.  The hypothetical collapsed stars were called "frozen stars", because an outside observer would see the surface of the star frozen in time at the instant where its collapse takes it to the Schwarzschild radius.<ref>{{Cite journal |last1=Ruffini |first1=R. |author-link1=Remo Ruffini |last2=Wheeler |first2=J. A. |author-link2=John Archibald Wheeler |date=1971 |title=Introducing the black hole |url=http://authors.library.caltech.edu/14972/1/Ruffini2009p1645Phys_Today.pdf |journal=[[Physics Today]] |volume=24 |issue=1 |pages=30–41 |doi=10.1063/1.3022513|bibcode=1971PhT....24a..30R |archive-url=https://web.archive.org/web/20110725133758/http://authors.library.caltech.edu/14972/1/Ruffini2009p1645Phys_Today.pdf |archive-date=25 July 2011 |access-date=5 December 2009 }}</ref>

Also in 1939, Einstein attempted to prove that black holes were impossible in his publication "On a Stationary System with Spherical Symmetry Consisting of Many Gravitating Masses", using his theory of general relativity to defend his argument.<ref name=":0">{{Cite journal |last=Bernstein |first=Jeremy |title=The Reluctant Father of Black Holes |url=https://www.scientificamerican.com/article/the-reluctant-father-of-black-holes-2007-04/ |access-date=3 August 2023 |journal=[[Scientific American]] |date=2007 |volume=17 |pages=4–11 |doi=10.1038/scientificamerican0407-4sp |language=en}}</ref> Months later, Oppenheimer and his student [[Hartland Snyder]] provided the [[Oppenheimer–Snyder model]] in their paper "On Continued Gravitational Contraction",<ref>{{cite journal |last1=Oppenheimer |first1=J.R. |last2=Snyder |first2=H. |author-link2=Hartland Snyder |year=1939 |title=On Continued Gravitational Contraction |journal=[[Physical Review]] |volume=56 |issue=5 |pages=455–459 |bibcode=1939PhRv...56..455O |doi=10.1103/PhysRev.56.455 |doi-access=free}}</ref> which predicted the existence of black holes. In the paper, which made no reference to Einstein's recent publication, Oppenheimer and Snyder used Einstein's own theory of general relativity to show the conditions on how a black hole could develop, for the first time in contemporary physics.<ref name=":0" />

==== Golden age ====
In 1958, [[David Finkelstein]] identified the Schwarzschild surface as an event horizon, "a perfect unidirectional membrane: causal influences can cross it in only one direction".<ref>{{Cite journal |last=Finkelstein |first=D. |author-link1=David Finkelstein |date=1958 |title=Past-Future Asymmetry of the Gravitational Field of a Point Particle |journal=[[Physical Review]] |volume=110 |issue=4 |pages=965–967 |doi=10.1103/PhysRev.110.965|bibcode=1958PhRv..110..965F}}</ref> This did not strictly contradict Oppenheimer's results, but extended them to include the point of view of infalling observers. Finkelstein's solution extended the Schwarzschild solution for the future of observers falling into a black hole. A [[Kruskal–Szekeres coordinates|complete extension]] had already been found by [[Martin Kruskal]], who was urged to publish it.<ref>{{cite journal |last1=Kruskal |first1=M. |author-link1=Martin Kruskal |date=1960 |title=Maximal Extension of Schwarzschild Metric |journal=[[Physical Review]] |volume=119 |issue=5 |page=1743 |doi=10.1103/PhysRev.119.1743 |bibcode=1960PhRv..119.1743K}}</ref>

These results came at the beginning of the [[golden age of general relativity]], which was marked by general relativity and black holes becoming mainstream subjects of research. This process was helped by the discovery of [[pulsar]]s by [[Jocelyn Bell Burnell]] in 1967,<ref>{{Cite journal |last1=Hewish |first1=A. |author-link1=Antony Hewish |last2=Bell |first2=S. J. |author-link2=Jocelyn Bell Burnell |last3=Pilkington |first3=J. D. H. |last4=Scott |first4=P. F. |last5=Collins |first5=R. A. |display-authors=1 |date=1968 |title=Observation of a Rapidly Pulsating Radio Source |journal=[[Nature (journal)|Nature]] |volume=217 |issue=5130 |pages=709–713 |doi=10.1038/217709a0|bibcode=1968Natur.217..709H|s2cid=4277613 }}</ref><ref>{{Cite journal |last1=Pilkington |first1=J. D. H. |last2=Hewish |first2=A. |author-link2=Antony Hewish |last3=Bell |first3=S. J. |author-link3=Jocelyn Bell Burnell |last4=Cole |first4=T. W. |display-authors=1 |date=1968 |title=Observations of some further Pulsed Radio Sources |journal=[[Nature (journal)|Nature]] |volume=218 |issue=5137 |pages=126–129 |doi=10.1038/218126a0|bibcode=1968Natur.218..126P|s2cid=4253103 }}</ref> which, by 1969, were shown to be rapidly rotating neutron stars.<ref name="araa8_265">{{cite journal |last=Hewish |first=A. |author-link1=Antony Hewish |date=1970 |title=Pulsars |journal=[[Annual Review of Astronomy and Astrophysics]] |volume=8 |issue=1 |pages=265–296 |bibcode=1970ARA&A...8..265H |doi=10.1146/annurev.aa.08.090170.001405}}</ref> Until that time, neutron stars, like black holes, were regarded as just theoretical curiosities; but the discovery of pulsars showed their physical relevance and spurred a further interest in all types of compact objects that might be formed by gravitational collapse.<ref>{{Cite web |url=https://www.smithsonianmag.com/science-nature/Fifty-years-ago-grad-students-discovery-changed-course-astrophysics-180968288/ |title=Fifty Years Ago, a Grad Student's Discovery Changed the Course of Astrophysics |date=28 February 2018 |access-date=22 December 2023 |website=Smithsonian Magazine |last=Boissoneault |first=Lorraine}}</ref>

In this period more general black hole solutions were found. In 1963, [[Roy Kerr]] found [[Kerr metric|the exact solution]] for a [[rotating black hole]]. Two years later, [[Ezra T. Newman|Ezra Newman]] found the [[axisymmetric]] solution for a black hole that is both rotating and [[electrically charged]].<ref>{{Cite journal |last1=Newman |first1=E. T. |author-link1=Ezra T. Newman |last2=Couch |first2=E. |last3=Chinnapared |first3=K. |last4=Exton |first4=A. |last5=Prakash |first5=A. |last6=Torrence |first6=R. |display-authors=1 |date=1965 |title=Metric of a Rotating, Charged Mass |journal=[[Journal of Mathematical Physics]] |volume=6 |issue=6 |page=918 |doi=10.1063/1.1704351 |bibcode=1965JMP.....6..918N}}</ref> Through the work of [[Werner Israel]],<ref>{{cite journal |last=Israel |first=W. |date=1967 |title=Event Horizons in Static Vacuum Space-Times |journal=[[Physical Review]] |volume=164 |issue=5 |page=1776 |doi=10.1103/PhysRev.164.1776 |bibcode=1967PhRv..164.1776I}}</ref> [[Brandon Carter]],<ref>{{cite journal |last=Carter |first=B. |author-link1=Brandon Carter |date=1971 |title=Axisymmetric Black Hole Has Only Two Degrees of Freedom |journal=[[Physical Review Letters]] |volume=26 |issue=6 |page=331 |doi=10.1103/PhysRevLett.26.331 |bibcode=1971PhRvL..26..331C}}</ref><ref>{{cite book |last=Carter |first=B. |author-link1=Brandon Carter |date=1977 |chapter=The vacuum black hole uniqueness theorem and its conceivable generalisations |title=Proceedings of the 1st Marcel Grossmann meeting on general relativity |pages=243–254 }}</ref> and David Robinson<ref>{{cite journal |last1=Robinson |first1=D. |date=1975 |title=Uniqueness of the Kerr Black Hole |journal=[[Physical Review Letters]] |volume=34 |issue=14 |page=905 |doi=10.1103/PhysRevLett.34.905 |bibcode=1975PhRvL..34..905R}}</ref> the [[no-hair theorem]] emerged, stating that a stationary black hole solution is completely described by the three parameters of the [[Kerr–Newman metric]]: [[mass]], [[angular momentum]], and electric charge.<ref name="HeuslerNoHair" />

At first, it was suspected that the strange features of the black hole solutions were pathological artefacts from the symmetry conditions imposed, and that the singularities would not appear in generic situations. This view was held in particular by [[Vladimir A. Belinsky|Vladimir Belinsky]], [[Isaak Markovich Khalatnikov|Isaak Khalatnikov]], and [[Evgeny Lifshitz]], who tried to prove that no singularities appear in generic solutions. However, in the late 1960s [[Roger Penrose]]<ref name="penrose1965">{{cite journal |last1=Penrose |first1=R. |author-link1=Roger Penrose |date=1965 |title=Gravitational Collapse and Space-Time Singularities |journal=[[Physical Review Letters]] |volume=14 |issue=3 |page=57 |doi=10.1103/PhysRevLett.14.57 |bibcode=1965PhRvL..14...57P |s2cid=116755736|url=http://pdfs.semanticscholar.org/faad/1f4358fddf70df2e00c0a290b7e4501c27de.pdf |archive-url=https://web.archive.org/web/20201011221750/http://pdfs.semanticscholar.org/faad/1f4358fddf70df2e00c0a290b7e4501c27de.pdf |archive-date=11 October 2020 }}</ref> and [[Stephen Hawking]] used global techniques to prove that singularities appear generically.<ref>{{cite journal |last1=Ford |first1=L. H. |date=2003 |title=The Classical Singularity Theorems and Their Quantum Loopholes |journal=[[International Journal of Theoretical Physics]] |volume=42 |issue=6 |pages=1219–1227 |doi=10.1023/A:1025754515197 |arxiv=gr-qc/0301045 |bibcode=2003gr.qc.....1045F |s2cid=14404560}}</ref> For this work, Penrose received half of the 2020 [[Nobel Prize in Physics]], Hawking having died in 2018.<ref>{{Cite web|title=The Nobel Prize in Physics 2020|url=https://www.nobelprize.org/prizes/physics/2020/summary/|access-date=8 October 2020|website=NobelPrize.org|archive-date=24 April 2021|archive-url=https://web.archive.org/web/20210424115309/https://www.nobelprize.org/prizes/physics/2020/summary/|url-status=live}}</ref> Based on observations in [[Royal Greenwich Observatory|Greenwich]] and [[David Dunlap Observatory|Toronto]] in the early 1970s, [[Cygnus X-1]], a galactic [[X-ray]] source discovered in 1964, became the first astronomical object commonly accepted to be a black hole.<ref>{{citation | last=Rolston | first=Bruce | date=10 November 1997 | url=http://news.utoronto.ca/bin/bulletin/nov10_97/art4.htm | title=The First Black Hole | publisher=University of Toronto | access-date=11 March 2008 | archive-url = https://web.archive.org/web/20080307181205/http://www.news.utoronto.ca/bin/bulletin/nov10_97/art4.htm | archive-date = 7 March 2008 }}</ref><ref name="Shipman1975">{{citation | last1=Shipman | first1=H. L. | date=1975 | title=The implausible history of triple star models for Cygnus&nbsp;X-1 Evidence for a black hole | journal=Astrophysical Letters | volume=16 | issue=1 | pages=9–12 | bibcode=1975ApL....16....9S | doi=10.1016/S0304-8853(99)00384-4 | last2=Yu | first2=Z | last3=Du | first3=Y.W }}</ref>

Work by [[James Bardeen]], [[Jacob Bekenstein]], Carter, and Hawking in the early 1970s led to the formulation of [[black hole thermodynamics]].<ref>{{Cite journal |last1=Bardeen |first1=J. M. |author-link1=James M. Bardeen |last2=Carter |first2=B. |author-link2=Brandon Carter |last3=Hawking |first3=S. W. |author-link3=Stephen Hawking |date=1973 |title=The four laws of black hole mechanics |journal=[[Communications in Mathematical Physics]] |volume=31 |issue=2 |pages=161–170 |doi=10.1007/BF01645742 |mr=0334798 |zbl=1125.83309 |bibcode=1973CMaPh..31..161B |s2cid=54690354 |url=http://projecteuclid.org/euclid.cmp/1103858973 |access-date=4 June 2021 |archive-date=16 May 2020 |archive-url=https://web.archive.org/web/20200516211604/https://projecteuclid.org/euclid.cmp/1103858973 |url-status=live }}</ref> These laws describe the behaviour of a black hole in close analogy to the [[laws of thermodynamics]] by relating mass to energy, area to [[entropy]], and [[surface gravity]] to [[temperature]]. The analogy was completed when Hawking, in 1974, showed that [[quantum field theory]] implies that black holes should radiate like a [[black body]] with a temperature proportional to the surface gravity of the black hole, predicting the effect now known as [[Hawking radiation]].<ref name=Hawking1974 />