**Schwarzschild Radius**

The Schwarzschild radius (*r _{s}*) 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

*r*= 1.353×10−57m._{s}

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

where *q* is the charge and *ε _{0}* is the vacuum permittivity.

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

*r*= 9.152×10−37m._{q}

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 *r _{s}* or

*r*, 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.

_{q}Read more about this topic: Black Hole Electron

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