# 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.[1]

## Contents

The Schwarzschild radius (rs) of any mass is given by:

$r_s = \frac{2Gm}{c^2}$

For an electron,

G is Newton's gravitational constant,
m is the mass of the electron = 9.109×10−31 kg, and
c is the speed of light.

This gives a value

rs = 1.353×10−57 m.

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 rq 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 = −1.602×10−19 C, this gives a value

rq = 9.152×10−37 m.

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 106 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 rs or rq, 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.