# Black Hole Electron - Schwarzschild Radius

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

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

rs = 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 rq is defined to be

$r_{q} = sqrt{frac{q^{2}G}{4piepsilon_{0} c^{4}}}$

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

rq = 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 GeV, or

.

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.