— How exactly is a singularity (and electrons) a point with no volume? Space is in 3 dimensions, so everything has to has some thickness, right?

In: 8

Singularities are not technically proven to exist yet. We don’t actually know what’s going on inside a black hole. It may not be a true singularity. E.g., it may have volume, even if it’s the minimum possible volume (planck sized).

The true nature of point particles is also complicated, and how we interpret them depends on which view of the physics you’re taking. Right now, quantum field theories are the thing. So point particles aren’t classical balls or points. They’re excitations of an underlying electron field or photons are excitations of the electromagnetic field, etc.

The point-like properties manifest as the location of maximum energy and in how they interact with other fields and objects. The wavelike properties are viewable when the conditions don’t force the excitation to behave like a point.

Imagine the curve `1/x`. As x approaches 0, this curve shoots off to infinity. We say that the function `1/x` has a *singularity* at `x=0`. It truly is a point (`x=0`) with no width, because even for something e.g. very close like `x=0.0000001` you still get a definite answer: `1/0.0000001 = 10000000`. Only the actual point `x=0` itself is undefined.

This is basically like what the mathematics of general relativity predicts should happen inside a black hole – the gravitational acceleration and mass-energy density shoot off to infinity and the coordinate system “collapses in” on itself. (It literally *is* a division by zero, in the mathematics)

Indeed, you don’t even need general relativity to get singularities – even newton’s law of universal gravitation predicts a singularity around every point mass – the gravitational force approaches infinity as two objects get closer and closer. If two true point masses were to fall in on each other, newtonian physics predicts that they’d release an infinite amount of potential energy in doing so.

Note that electrons aren’t point masses – they’re quantum objects, which are spread out over a volume. And we don’t have a theory of quantum gravity so we have no idea what the correct equation is for describing their gravitational attraction.

In a quantum mechanical sense, electrons necessarily have volume.

They exist as a sort of cloud of probability in an atom, and the volume of that “cloud” can be worked out.

You can find the electron as a pointlike particle at a random location in that “cloud”, but the key word here is pointlike. It displays some some behavior as if it was a point, but if you measure its properties you will always find a length of some sort.

You will run into the wavelength of the electron, something called the de Broglie wavelength. This wavelength is equal to planck’s constant divided by the momentum of the particle.

If you think about it, and if you know a bit about quantum uncertainty, you might realize that trying to measure the electron smaller than that quickly becomes fruitless.

The more accurately you try to pinpoint the position of the electron, (in order to confirm it has no volume and is indeed a point) the less you know about its momentum. And the less you know about its momentum, the less you know about its de Broglie wavelength.

The fundamental limit as far as quantum mechanics goes is what’s called the planck length, or planck volume for volumes. Smaller than that and we end up with 100% quantum uncertainties. So it’s impossible to measure the wavelength of an electron to be any smaller.

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