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| In [[physics]], there is a '''speculative''' notion that if there were a [[black hole]] with the same mass and charge as an [[electron]], it would share many of the properties of the electron including the [[electron magnetic dipole moment|magnetic moment]] and [[Compton wavelength]]. This idea is substantiated within a series of papers published by Albert Einstein between 1927 and 1949. In them, he showed that if elementary particles were treated as singularities in spacetime, it was unnecessary to postulate [[Geodesic (general relativity)|geodesic]] motion as part of general relativity.<ref>{{cite journal | last1=Einstein | first1=A. |authorlink1=Albert Einstein | last2=Infeld | first2=L. | authorlink2=Leopold Infeld | last3=Hoffmann | first3=B. | authorlink3=Banesh Hoffmann | title=The Gravitational Equations and the Problem of Motion | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | volume=39 | issue=1 |date=January 1938 | pages=65–100 | jstor=1968714 |doi=10.2307/1968714}}</ref>
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| ==Problems==
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| Quantum mechanics permits superluminal speeds for an object with as small a mass as the electron over distance scales larger than the Schwarzschild radius of the electron.{{Citation needed|date=August 2011}}
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| ==Schwarzschild radius==
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| The [[Schwarzschild radius]] (''r<sub>s</sub>'') of any mass is calculated using the following formula:
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| :<math>r_s = \frac{2Gm}{c^2}</math>
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| For an electron,
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| :''G'' is Newton's [[gravitational constant]],
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| :''m'' is the mass of the [[electron]] = {{val|9.109|e=-31|u=[[kilogram|kg]]}}, and
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| :''c'' is the [[speed of light]].
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| This gives a value
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| :''r<sub>s</sub>'' = {{val|1.353|e=-57|u=m}}.
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| So if the electron has a radius as small as this, it would become a gravitational singularity. It would then have a number of properties in common with black holes. In the [[Reissner–Nordström metric]], which describes electrically charged black holes, an analogous quantity ''r<sub>q</sub>'' is defined to be
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| :<math> | |
| r_{q} = \sqrt{\frac{q^{2}G}{4\pi\epsilon_{0} c^{4}}}
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| </math>
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| where ''q'' is the charge and ''ε''<sub>0</sub> is the [[vacuum permittivity]].
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| For an electron with ''q'' = −[[elementary charge|''e'']] = {{val|-1.602|e=-19|u=[[coulomb|C]]}}, this gives a value
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| :''r<sub>q</sub>'' = {{val|9.152|e=-37|u=m}}. | |
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| This value suggests that an electron black hole would be [[extremal black hole|super-extremal]] and have a [[naked singularity]]. Standard [[quantum electrodynamics]] (QED) theory treats the electron as a point particle, a view completely supported by experiment. Practically, though, particle experiments cannot probe arbitrarily large energy scales, and so QED-based experiments bound the electron radius to a value smaller than the [[Compton wavelength]] of a large mass, on the order of {{val|e=6|u=GeV}}, or
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| :<math>r \approx \frac{\alpha \hbar c}{10^6 GeV} \approx 10^{-24} m</math>.
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| No proposed experiment would be capable of probing ''r'' to values as low as ''r<sub>s</sub>'' or ''r<sub>q</sub>'', both of which are smaller than the [[Planck length]]. Super-extremal black holes are generally believed to be unstable. Furthermore, any physics smaller than the [[Planck length]] probably requires a consistent theory of [[quantum gravity]].
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| ==See also==
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| * [[Quantum gravity]]
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| * [[Extremal black hole]]
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| * [[Geon (physics)]]
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| * [[Ring singularity]]
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| ==References==
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| {{reflist}}
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| ==Further reading==
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| {{refbegin}}
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| * {{cite paper | last=Burinskii | first=A. | year=2005 | arxiv=hep-th/0507109 | title=The Dirac–Kerr electron|bibcode = 2008GrCo...14..109B |doi = 10.1134/S0202289308020011 }}
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| * {{cite paper | last=Burinskii | first=A. | year=2007 | arxiv=0712.0577 | title=Kerr Geometry as Space–Time Structure of the Dirac Electron|bibcode = 2007arXiv0712.0577B }}
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| * {{cite paper | last=Duff | first=Michael | authorlink = Michael Duff (physicist) | year=1994 | arxiv=hep-th/9410046| title=Kaluza–Klein Theory in Perspective|bibcode = 1995okml.book...22D }}
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| * {{cite journal | last=Hawking | first=Stephen | authorlink=Stephen Hawking | year=1971 | title=Gravitationally collapsed objects of very low mass | journal=[[Monthly Notices of the Royal Astronomical Society]] | volume=152 | page=75 | bibcode=1971MNRAS.152...75H}}
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| * {{cite book | last=Penrose | first=Roger | authorlink=Roger Penrose | year=2004 | title=[[The Road to Reality: A Complete Guide to the Laws of the Universe]] | location=London | publisher=Jonathan Cape}}
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| * {{cite book | last=Salam | first=Abdus | authorlink=Abdus Salam | chapter=Impact of Quantum Gravity Theory on Particle Physics | editor1-last=Isham | editor1-first=C. J. | editor2-last=Penrose | editor2-first=Roger | editor3-last=Sciama | editor3-first=Dennis William | title=Quantum Gravity: an Oxford Symposium | publisher=Oxford University Press}}
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| * {{cite journal | last='t Hooft | first=Gerard | authorlink=Gerard 't Hooft | year=1990 | doi=10.1016/0550-3213(90)90174-C | title=The black hole interpretation of string theory | journal=[[Nuclear Physics B]] | volume=335 | pages=138–154|bibcode = 1990NuPhB.335..138T }}
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| * {{cite paper | last=Murdzek | first=R. | journal=[[International Journal of Modern Physics D]] | volume=16 | issue=4 | pages=681–686 | year=2007 | doi=10.1142/S0218271807009826 | bibcode=2007IJMPD..16..681M | title=The Geometry of the Torus Universe | postscript=, }} which is related to "Hierarchical Cantor set in the large scale structure 3 with torus geometry".
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| {{refend}}
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| === Popular literature ===
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| * [[Brian Greene]], ''[[The Elegant Universe|The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory]]'' (1999), ''(See chapter 13)''
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| * [[John A. Wheeler]], ''[[Geons, Black Holes & Quantum Foam]]'' (1998), ''(See chapter 10)''
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| {{black holes}}
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| [[Category:Black holes]]
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| [[Category:Quantum gravity]]
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| [[Category:Hypothetical elementary particles]] | |
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