Black hole electron

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 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 motion as part of general relativity.

Problems

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.

Schwarzschild radius

The Schwarzschild radius (r_s) of any mass is calculated using the following formula:

$r_{s}={\frac {2Gm}{c^{2}}}$

For an electron,

G is Newton's gravitational constant,

m is the mass of the electron = 9.109×10⁻³¹kg, and

c is the speed of light.

This gives a value

r_s = 1.353×10⁻⁵⁷m.

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_q is defined to be

$r_{q}={\sqrt {\frac {q^{2}G}{4\pi \epsilon _{0}c^{4}}}}$

where q is the charge and ε₀ is the vacuum permittivity.

For an electron with q = -e = −1.602×10⁻¹⁹C, this gives a value

r_q = 9.152×10⁻³⁷m.

This value suggests that an electron black hole would be 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 $10^{6}$ GeV, or

$r\approx {\frac {\alpha \hbar c}{10^{6}GeV}}\approx 10^{-24}m$.

No proposed experiment would be capable of probing r to values as low as r_s or r_q, 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.

See also

• Quantum gravity
• Extremal black hole
• Geon (physics)
• Ring singularity

References

• Burinskii, A., (2005) "The Dirac-Kerr electron."
• Burinskii, A., (2007) "Kerr Geometry as Space-Time Structure of the Dirac Electron."
• Michael Duff (1994) "Kaluza-Klein Theory in Perspective."
• Stephen Hawking (1971), " ," Monthly Notices of the Royal Astronomical Society 152: 75.
• Roger Penrose (2004) The Road to Reality: A Complete Guide to the Laws of the Universe. London: Jonathan Cape.
• Abdus Salam, chapter in Isham, Penrose, and Sciama, eds., Quantum Gravity: an Oxford Symposium. Oxford University Press.
• G. 't Hooft (1990) "The black hole interpretation of string theory," Nuclear Physics B 335: 138-154.
• Williamson, J. G., and Van der Mark, M. B. (1997) "Is the electron a photon with toroidal topology?" Annales de la Fondation Louis de Broglie 22(2): 133.

Popular literature

• Brian Greene, 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)

External links

• The Geometry of the Torus Universe, which is related to "Hierarchical Cantor set in the large scale structure 3 with torus geometry".