— 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?
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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.
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