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
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.
Read more about this topic: Black Hole Electron
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