# Theoritical Infinite Density of a Point Particle

1. Jul 3, 2012

### e2m2a

In the science literature I read of particles, such as electrons, that are essentially dimensionless-- a point. Now science knows that an electron has a definite rest mass. If an electron is essentially a point, then its mass/volume density would approach infinity. If so, why doesn't an electron collaspse into a mini-black hole?

2. Jul 3, 2012

### Simon Bridge

Because "essentially dimentionless" is not the same thing as "actually a point mass".

The electron is usually treated as a statistical object under the rules of quantum mechanics in any situation where it's "size" matters.

3. Jul 3, 2012

### nasimrahaman

Electrons are considered to be 'dimensionless' because it makes absolutely no sense to measure how large they really are. In quantum mechanics, you just can't measure a distance with arbitrary precision. An electron does fill a volume*, and no two electrons are allowed to fill the same volume: just the way no two identical coffee mugs are allowed to be at the same place on your desktop simultaneously.

4. Jul 3, 2012

### Andy Resnick

Classically, the electron radius is about 3 fm, which means the (classical) density is about 10^10 times higher than water (10^10 g/ml)- significantly less dense than a nucleus, and less dense than a neutron star. No worries about gravitational collapse.

5. Jul 3, 2012

### Simon Bridge

To see how that was calculated...

A femtometer (10[sup-15[/sup]) seems pretty small until you realize that the atomic nucleus is about 1.75fm across (hydrogen - ~15fm for Uranium)... which means that the classical electron is about 3x bigger than a proton ;)

Huh yeah - I was gonna say... same volume yes, but they have to have different quantum states.

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[1] pretty bad article but OK for this purpose

6. Jul 3, 2012

### bcrowell

Staff Emeritus
There are both classical and quantum-mechanical reasons why we know that electrons aren't black holes or naked singularities.

Classically, a spinning, charged black hole has constraints on its angular momentum and its charge in relation to its mass. Otherwise, there is no event horizon, and we have a naked singularity rather than a black hole. An electron violates both of these limits, but we don't observe that electrons have the properties predicted for these naked singularities. For example, naked singularities have closed timelike curves in the spacetime surrounding them, which would violate causality, but there is no evidence that electrons cause causality violation.

Quantum-mechanically it is believed that microscopic black holes would evaporate into photons, whereas electrons, for example, do not seem to. The time a black hole takes to evaporate becomes shorter as the black hole gets smaller. When the black hole has a mass equal to the Planck mass, which is about 22 micrograms, the lifetime becomes on the order of the Planck time (or a few thousand times greater). All known fundamental particles have masses many orders of magnitude less than the Planck mass, so there is no way they could have long lifetimes if they were black holes.

This establishes that they aren't GR-style singularities, but doesn't explain why they aren't. Classically, the mass-energy of a finite-radius charged sphere is not all concentrated within the sphere; some of it is carried by the energy of the electric field outside the sphere. Quantum-mechanically, QED describes a particle as being surrounded by a region of the vacuum that's full of virtual particle-antiparticle pairs, and the "dressed" particle has properties that have to be renormalized. I don't think a full description is possible without a theory of quantum gravity, which we don't have.

7. Jul 3, 2012

### Simon Bridge

Just wanted to repeat this :)