Why isn't an electron a black hole?

Question

My late night thoughts brought me to a question: If electrons are point particles, but have mass, why aren't they black holes?

Answer 1

The Schwarzschild radius of an electron is \begin{equation} r_{s,e}=\frac{2 G m_e}{c^2} = 1.4 \times 10^{-55}\ {\rm cm} \end{equation} This is actually almost two dozen orders of magnitude smaller than the Planck length, $1.6 \times 10^{-33}\ {\rm cm}$, which is the length scale at which quantum fluctuations in spacetime itself become so large that we need a quantum theory of gravity to really make sense of what's going on, but let's ignore that for the purposes of this question.

Because of the uncertainty principle, in quantum mechanics it isn't possible to localize a particle very precisely, even a point particle. The Compton wavelength of an electron gives the typical length scale over which an electron at rest will "spread out" due to quantum effects. (More precisely it is the wavelength associated with the wave function of an electron at rest, and also if you try to localize the electron to a length scale smaller than its quantum wavelength, it will have enough energy due to the uncertainty principle to start generating electron-positron pairs and therefore stops being "an electron"). The Compton wavelength of the electron is \begin{equation} \lambda_e = \frac{h}{m_e c} = 2.4 \times 10^{-10}\ {\rm cm} \gg r_{s,e} \end{equation} Therefore quantum fluctuations "fuzz out" the position of the electron at rest on a length scale approximately $10^{45}$ times larger than the Schwarzschild radius of the electron. Since a black hole only forms if the mass is concentrated inside the Schwarzschild radius, the uncertainty principle protects the electron from collapse. In fact long before you have to worry about an isolated electron collapsing into a black hole, you would need to worry about almost inconceivably high energy experiments where two scattering electrons would have enough kinetic energy packed into a small space that the pair would form a black hole.

The meaning of the electron being "point-like" in particle physics has actually very little to do with how well we can localize it. Rather, it is a statement about the kinds of interactions an electron has at high energies. Point-like particles have a simple and characteristic scattering behavior at high energies. However for composite particles in high-energy scattering experiments, you will excite internal states of a composite particle, leading to behavior which differs from the simple "point-like" behavior. if you are interested in learning more about how this kind of thinking led to the discovery that the proton (and other nuclear particles) are made of quarks, you might be interested in reading about deep inelastic scattering.

Answer 2

The description of the gravity of an electron requires quantum gravity, a theory which does not yet exist. Any statements on gravity at subatomic scale are speculation. See also https://en.wikipedia.org/wiki/Black_hole_electron.