View source for Astronomy:Black hole electron

{{Short description|Hypothetical concept in physics}}
In [[Physics:Physics|physics]], there is a speculative hypothesis that, if there were a [[Astronomy:Black hole|black hole]] with the same mass, charge and angular momentum as an [[Physics:Electron|electron]], it would share other properties of the electron. Most notably, Brandon Carter showed in 1968 that the [[Physics:Electron magnetic moment|magnetic moment]] of such an object would match that of an electron.<ref>{{cite journal |last1=Carter |first1=B. |title=Global structure of the Kerr family of gravitational fields |journal=[[Physics:Physical Review|Physical Review]] |volume=174 |issue=5 |date=25 October 1968 |pages=1559–1571 |doi=10.1103/physrev.174.1559 |bibcode=1968PhRv..174.1559C }}</ref> This is interesting because calculations ignoring special relativity and treating the electron as a small rotating sphere of charge give a magnetic moment roughly half the experimental value (see [[Physics:Gyromagnetic ratio|Gyromagnetic ratio]]).

However, Carter's calculations also show that a would-be black hole with these parameters would be "[[Astronomy:Extremal black hole|super-extremal]]". Thus, unlike a true black hole, this object would display a [[Astronomy:Naked singularity|naked singularity]], meaning a [[Gravitational singularity|singularity]] in spacetime not hidden behind an [[Astronomy:Event horizon|event horizon]]. It would also give rise to closed timelike curves.

Standard [[Physics:Quantum electrodynamics|quantum electrodynamics]] (QED), currently the most comprehensive theory of particles, treats the electron as a point particle. There is no evidence that the electron is a black hole (or naked singularity) or not. Furthermore, since the electron is quantum-mechanical in nature, any description purely in terms of general relativity is paradoxical until a better model based on understanding of quantum nature of blackholes and gravitational behaviour of quantum particles is developed by research. Hence, the idea of a black hole electron remains strictly hypothetical.

==Details==
An article published in 1938 by [[Biography:Albert Einstein|Albert Einstein]], [[Biography:Leopold Infeld|Leopold Infeld]], and [[Biography:Banesh Hoffmann|Banesh Hoffmann]] showed that if elementary particles are treated as singularities in spacetime it is unnecessary to postulate geodesic motion as part of general relativity.<ref>{{cite journal | last1=Einstein | first1=A. | last2=Infeld | first2=L. |  last3=Hoffmann | first3=B. | authorlink3=Banesh Hoffmann | title=The gravitational equations and the problem of motion | journal=[[Annals of Mathematics]] |series=Second Series | volume=39 | issue=1 |date=January 1938 | pages=65–100 | jstor=1968714 |doi=10.2307/1968714 |bibcode=1938AnMat..39...65E }}</ref> The electron may be treated as such a singularity.

If one ignores the electron's angular momentum and charge, as well as the effects of [[Physics:Quantum mechanics|quantum mechanics]], one can treat the electron as a [[Astronomy:Black hole|black hole]] and attempt to compute its radius. The [[Astronomy:Schwarzschild radius|Schwarzschild radius]] {{math|''r''<sub>s</sub>}} of a mass {{mvar|m}} is the radius of the event horizon for a non-rotating uncharged [[Astronomy:Black hole|black hole]] of that mass. It is given by
<math display=block>r_\text{s} = \frac{2Gm}{c^2},</math>
where {{mvar|G}} is the Newtonian constant of gravitation, and {{mvar|c}} is the [[Physics:Speed of light|speed of light]]. For the electron,
: {{mvar|m}} = {{val|9.109|e=-31|ul=kg}},
so
: {{math|''r''<sub>s</sub>}} = {{val|1.353|e=-57|u=m}}.

Thus, if we ignore the electric charge and angular momentum of the electron and naively apply general relativity on this very small length scale without taking quantum theory into account, a black hole of the electron's mass would have this radius.

In reality, physicists expect quantum-gravity effects to become significant even at much larger length scales, comparable to the [[Planck length]]
<math display=block>\ell_\text{P} = \sqrt{\frac{G\hbar}{c^3}} = 1.616 \times 10^{-35}~\text{m}.</math>

So, the above purely classical calculation cannot be trusted. Furthermore, even classically, electric charge and angular momentum affect the properties of a black hole. To take them into account, while still ignoring quantum effects, one should use the [[Kerr–Newman metric]]. If we do, we find that the angular momentum and charge of the electron are too large for a black hole of the electron's mass: a Kerr–Newman object with such a large angular momentum and charge would instead be "[[Astronomy:Extremal black hole|super-extremal]]", displaying a [[Astronomy:Naked singularity|naked singularity]], meaning a singularity not shielded by an [[Astronomy:Event horizon|event horizon]].

To see that this is so, it suffices to consider the electron's charge and neglect its angular momentum. In the [[Astronomy:Reissner–Nordström metric|Reissner–Nordström metric]], which describes electrically charged but non-rotating black holes, there is a quantity {{mvar|r<sub>q</sub>}}, defined by
<math display=block>
r_q = \sqrt{\frac{q^2 G}{4\pi\epsilon_0 c^4}},
</math>
where {{mvar|q}} is the electron's charge, and {{math|''ε''<sub>0</sub>}} is the [[Physics:Vacuum permittivity|vacuum permittivity]]. For an electron with {{mvar|q}} = −{{mvar|[[Physics:Elementary charge|e]]}} = {{val|-1.602|e=-19|ul=C}}, this gives a value
: {{mvar|r<sub>q</sub>}} = {{val|1.3807|e=-36|u=m}}.

Since this (vastly) exceeds the Schwarzschild radius, the Reissner–Nordström metric has a naked singularity.

If we include the effects of the electron's rotation using the [[Kerr–Newman metric]], there is still a naked singularity, which is now a [[Astronomy:Ring singularity|ring singularity]], and spacetime also has [[Closed timelike curve|closed timelike curve]]s. The size of this ring singularity is on the order of
<math display=block>
r_a = \frac{J}{m c},
</math>
where as before {{mvar|m}} is the electron's mass, and {{mvar|c}} is the speed of light, but {{mvar|J}} = <math>\hbar/2</math> is the [[Physics:Spin|spin]] [[Physics:Angular momentum|angular momentum]] of the electron. This gives
: {{mvar|r<sub>a</sub>}} = {{val|1.9295|e=-13|u=m}},
which is much larger than the length scale {{mvar|r<sub>q</sub>}} associated with the electron's charge. As noted by Carter,<ref>{{cite journal | last=Carter| first=B.|  title=Global structure of the Kerr family of gravitational fields | journal=[[Physics:Physical Review|Physical Review]] | volume=174 | issue=5 |date=25 October 1968 | pages=1559–1571 |doi=10.1103/physrev.174.1559 |bibcode = 1968PhRv..174.1559C }}</ref> this length {{mvar|r<sub>a</sub>}} is on the order of the electron's [[Physics:Compton wavelength|Compton wavelength]]. Unlike the Compton wavelength, it is not quantum-mechanical in nature.

More recently, Alexander Burinskii has pursued the idea of treating the electron as a Kerr–Newman naked singularity.<ref>{{cite journal |last=Burinskii |first=Alexander |arxiv=hep-th/0507109 |title=The Dirac-Kerr-Newman electron |journal=Gravitation and Cosmology |volume=14 |issue=2 |pages=109–122 |date=April 2008 |bibcode=2008GrCo...14..109B |doi=10.1134/S0202289308020011 |s2cid=119084073 }}</ref>

==See also==
{{div col}}
* [[Physics:Quantum gravity|Quantum gravity]]
* [[Physics:Abraham–Lorentz force|Abraham–Lorentz force]]
* [[Astronomy:Black hole thermodynamics|Black hole thermodynamics]]
* [[Physics:Entropic force|Entropic force]]
* [[Physics:Hawking radiation|Hawking radiation]]
* [[Physics:List of quantum gravity researchers|List of quantum gravity researchers]]
* [[Chemistry:Ideal chain#Entropic elasticity of an ideal chain|Entropic elasticity of an ideal chain]]
* Gravitation
* [[Induced gravity]]
* [[Physics:Geon|Geon]]
* [[Astronomy:Micro black hole|Micro black hole]]
* [[Physics:Geometrodynamics|Geometrodynamics]]
{{div col end}}

==References==
{{Reflist}}

==Further reading==
{{Refbegin}}
* {{cite book | last=Duff | first=Michael |  year=1994 | arxiv=hep-th/9410046| title=Kaluza–Klein theory in perspective|bibcode =  1995okml.book...22D}}
* {{cite journal | last=Hawking | first=Stephen |  year=1971 | title=Gravitationally collapsed objects of very low mass | journal=[[Organization:Monthly Notices of the Royal Astronomical Society|Monthly Notices of the Royal Astronomical Society]] | volume=152 | page=75 | bibcode=1971MNRAS.152...75H | doi=10.1093/mnras/152.1.75| doi-access=free }}
* {{cite book | last=Penrose | first=Roger |  year=2004 | title=The Road to Reality: A Complete Guide to the Laws of the Universe | location=London | publisher=Jonathan Cape| title-link=The Road to Reality: A Complete Guide to the Laws of the Universe }}
* {{cite book | last=Salam | first=Abdus |  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=[[Organization:Oxford University Press|Oxford University Press]]}}
* {{cite journal | last='t Hooft | first=Gerard |  year=1990 | doi=10.1016/0550-3213(90)90174-C | title=The black hole interpretation of string theory | journal=Nuclear Physics B | volume=335 | issue=1 | pages=138–154|bibcode = 1990NuPhB.335..138T }}
{{refend}}

===Popular literature===
* [[Biography:Brian Greene|Brian Greene]], ''[[Astronomy:The Elegant Universe|The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory]]'' (1999), ''(See chapter 13)''
* John A. Wheeler, ''Geons, Black Holes & Quantum Foam'' (1998), ''(See chapter 10)''

{{Black holes}}

[[Category:Black holes]]
[[Category:Quantum gravity]]
[[Category:Hypothetical elementary particles]]

{{Sourceattribution|Black hole electron}}