# 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.

Black Hole - History - General Relativity
... Only a few months later, Karl Schwarzschild found a solution to Einstein field equations, which describes the gravitational field of a point mass and a spherical mass ... A few months after Schwarzschild, Johannes Droste, a student of Hendrik Lorentz, independently gave the same solution for the point mass and wrote more ... had a peculiar behaviour at what is now called the Schwarzschild radius, where it became singular, meaning that some of the terms in the Einstein equations became infinite ...
Spaghettification - Inside or Outside The Event Horizon
... For small black holes whose Schwarzschild radius is much closer to the singularity, the tidal forces would kill even before the astronaut reaches the event horizon ... It will break at a distance of 320 km, well outside the Schwarzschild radius of 30 km ... will break at a distance of 3200 km, well inside the Schwarzschild radius of 30,000 km ...
Other Uses For The Schwarzschild Radius - Relativistic Circular Orbits and The Photon Sphere
... object orbiting at the speed of light would have an orbital radius of 1.5 times the Schwarzschild radius ...
Two-body Problem In General Relativity - Schwarzschild Solution
... See also Schwarzschild solution An exact solution to the Einstein field equations is the Schwarzschild metric, which corresponds to the external gravitational field of a stationary, uncharged, non-rot ... It is characterized by a length scale rs, known as the Schwarzschild radius, which is defined by the formula where G is the gravitational constant ... For example, the Schwarzschild radius rs of the Earth is roughly 9 mm (3⁄8 inch) at the surface of the Earth, the corrections to Newtonian gravity are only one part in a billion ...