Black hole electron

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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 magnetic moment and Compton wavelength.

1 Overview
2 The concept
3 Problems
4 Schwarzschild radius
5 See also
6 References
7 Popular literature

[edit] Overview

The discussion of this idea apparently dates as far back as 1919 and was known to Sir Arthur Eddington. More recently, the suggestion has been made by Brian Greene in his book The Elegant Universe. Taken literally, the core idea is that the electron may be a micro black hole, not just resemble it. The idea is appealing for several reasons. To this date, there are no verified theories for the unification of quantum mechanics and gravitation. Furthermore, the Standard Model, although a marvelously successful theory of high-energy physics, has many fundamental particles and adjustable parameters in it. Thus, a speculative, basic-principles approach of this type could conceivably be useful.

[edit] The concept

In general relativity there is no minimum mass for a black hole and there is no gravitational time dilation limit, other than (the unrealistic) zero seconds per second. Very small-mass black holes would look like elementary particles because they would be completely defined by their mass, charge and spin. Their uncertainty of position would be defined by their diameter.

The Kerr-Newman black hole is a solution to the Einstein equations for a black hole with charge and spin. The spin of a Kerr-Newman black hole has no exact counterpart in the classical world. The extreme time dilation required at the photon capture region indicates that the electron gravitational field has a ring singularity. This ring singularity could be described as a closed-loop vibrating string. Black hole theory predicts that a black hole with charge and spin will have magnetic moment equal to the charge times angular momentum divided by mass, which is equal to the Dirac electron magnetic moment. However, the correct result for the electron magnetic moment contains a small, but very precisely measured, correction from emission and re-absorption of virtual photons.

In 1955, John Archibald Wheeler explored this concept, describing a structure he named a geon. He found that while the structure was interesting, it was unstable. He suggested that geons could provide an intermediate stage in the creation of micro black holes and, since geons are not stable, a geon might radiate away some of its energy in electron-positron pairs. Wheeler also worked on the concept that the things we call particles (electrons) can be accounted for by an inner spacetime structure.

[edit] Problems

As a description, the black hole electron theory is incomplete. The first problem is that black holes tend to merge when they meet. Therefore, a collection of black-hole electrons would be expected to become one big black hole. Also, an electron-positron collision would be expected to produce a larger neutral black hole instead of two photons as is observed. These problems reflect the non-quantum nature of general relativity theory.

A more serious issue is Hawking radiation. According to Hawking's theory, a black hole the size and mass of an electron should vanish in a shower of photons (not just two photons of a given energy) within a small fraction of a second. Again, the current incompatibility of general relativity and quantum mechanics at electron scales prevents us from understanding why this never occurs.

The Kerr-Newman metric used to represent a charged, rotating black hole in General Relativity has three specifiable parameters: the mass of the hole, M; the charge of the hole, Q; and the angular momentum per unit mass, a. This metric defines a black hole with an event horizon only when these quantities satisfy the relation

$a^2 + Q^2 \leq M^2.$

An electron's a and Q (suitably specified in geometrized units) both exceed its mass M by many orders of magnitude. Using these values in the Kerr-Newman solution yields a superextreme Kerr-Newman metric. This metric has no event horizons and thus no black hole, only a naked, spinning ring singularity. A superextreme metric has many seemingly unphysical properties, the most severe being the ring's violation of the cosmic censorship hypothesis and appearance of causality-violating closed timelike curves in the immediate vicinity of the ring.

For these reasons, this speculation is considered a 'toy model', an at best incomplete description of the nature of an electron.

[edit] Schwarzschild radius

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

r_s = 2Gm/c²

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_e = 1.353×10⁻⁵⁷m

So if the electron has a radius smaller than this, it would become a gravitational singularity.

The classical electron radius of 2.818×10⁻¹⁵ meters is substantially larger than its Schwarzschild radius. 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 $106$ GeV, or

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

[edit] See also

Quantum gravity

[edit] References

Roger Penrose, The Road to Reality: A Complete Guide to the Laws of the Universe. (2004) Jonathan Cape, London.
S. W. Hawking, Monthly Notices of the Royal Astronomical Society, 152 (1971) 75.
Abdus Salam, chapter in Quantum Gravity: an Oxford Symposium, Eds. Isham, Penrose and Sciama, Oxford University Press.
G. 't Hooft, The black hole interpretation of string theory Nuclear Physics B, 335 (1990) 138-154.
M. J. Duff, Kaluza-Klein Theory in Perspective, (1994).
A. Burinskii, The Dirac-Kerr electron, (2005) preprint.