Hypothetical concept in physics
In
physics
, there is a speculative hypothesis that, if there were a
black hole
with the same mass, charge and angular momentum as an
electron
, it would share other properties of the electron. Most notably,
Brandon Carter
showed in 1968 that the
magnetic moment
of such an object would match that of an electron.
[1]
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
Gyromagnetic ratio
).
However, Carter's calculations also show that a would-be black hole with these parameters would be "
super-extremal
". Thus, unlike a true black hole, this object would display a
naked singularity
, meaning a
singularity
in spacetime not hidden behind an
event horizon
. It would also give rise to
closed timelike curves
.
Standard
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
[
edit
]
An article published in 1938 by
Albert Einstein
,
Leopold Infeld
, and
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.
[2]
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
quantum mechanics
, one can treat the electron as a
black hole
and attempt to compute its radius. The
Schwarzschild radius
r
s
of a mass
m
is the radius of the event horizon for a non-rotating uncharged
black hole
of that mass. It is given by
![{\displaystyle r_{\text{s}}={\frac {2Gm}{c^{2}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d40c2b87bc89a4a96070439b748fb4acb0bca0e5)
where
G
is the
Newtonian constant of gravitation
, and
c
is the
speed of light
. For the electron,
- m
=
9.109
×
10
?31
kg
,
so
- r
s
=
1.353
×
10
?57
m
.
Thus, if we ignore the electric charge and angular momentum of the electron and 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
![{\displaystyle \ell _{\text{P}}={\sqrt {\frac {G\hbar }{c^{3}}}}=1.616\times 10^{-35}~{\text{m}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29779b7c339cac7a558a7a8343d2cf3e234e9702)
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 "
super-extremal
", displaying a
naked singularity
, meaning a singularity not shielded by an
event horizon
.
To see that this is so, it suffices to consider the electron's charge and neglect its angular momentum. In the
Reissner?Nordstrom metric
, which describes electrically charged but non-rotating black holes, there is a quantity
r
q
, defined by
![{\displaystyle r_{q}={\sqrt {\frac {q^{2}G}{4\pi \epsilon _{0}c^{4}}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/56499bf9260972332fe04146337b6d3e854e8277)
where
q
is the electron's charge, and
ε
0
is the
vacuum permittivity
. For an electron with
q
= ?
e
=
?1.602
×
10
?19
C
, this gives a value
- r
q
=
1.3807
×
10
?36
m
.
Since this (vastly) exceeds the Schwarzschild radius, the Reissner?Nordstrom 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
ring singularity
, and spacetime also has
closed timelike curves
. The size of this ring singularity is on the order of
![{\displaystyle r_{a}={\frac {J}{mc}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19fceb726a7ba4e54a6360a7ce8a3de22591e863)
where as before
m
is the electron's mass, and
c
is the speed of light, but
J
=
![{\displaystyle \hbar /2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74e0713309590b82aed3a8247c30cfdeae9f456b)
is the
spin
angular momentum
of the electron. This gives
- r
a
=
1.9295
×
10
?13
m
,
which is much larger than the length scale
r
q
associated with the electron's charge. As noted by Carter,
[3]
this length
r
a
is on the order of the electron's
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.
[4]
See also
[
edit
]
References
[
edit
]
Further reading
[
edit
]
Popular literature
[
edit
]
|
---|
|
Types
| | |
---|
Size
| |
---|
Formation
| |
---|
Properties
| |
---|
Issues
| |
---|
Metrics
| |
---|
Alternatives
| |
---|
Analogs
| |
---|
Lists
| |
---|
Related
| |
---|
Notable
| |
---|
|