There is something I am not understanding about black hole singularities (I fully expect to be corrected - I can’t believe nobody has thought of this).
It’s my understanding that a black hole forms when gravity predominates over all other forces so that there is nothing that prevents matter in the black hole from collapsing to a singularity. This assumes that quarks (the most basic particle we currently know of) can themselves be compressed. Do we know that they can be? Why do we not postulate that matter gets compressed down to “solid quarks” (lousy terminology I’m sure, but I’m sure you know what I am getting at) and that’s as far as it goes?
Quarks, like all other fundamental particles (electrons, neutrinos, photons, Higgses, etc.), are presumed to be point particles, so there’s nothing to compress. It’s possible that they do have a nonzero size, but if so, then it’s so incredibly small that nobody’s ever been able to measure it.
Composite particles like protons, neutrons, mesons, atoms, planets, etc. do all have nonzero size, but the size of all such things are governed by the various forces of nature holding the fundamental point particles that make them up some distance from each other. As you noted, none of these forces can stand up to gravity, in black hole conditions.
Note also that “singularity” is really a math term, and the usage related to black holes means “the equations of general relativity have a singularity here” not necessarily “there actually is a real object called a singularity here”.
It is likely that GR is not a good approximation to reality at the centre of a black hole and a more complete theory would not have a different solution with a finite (but probably large) density there.
Really, the best description of the singularity in a black hole is “there is no there there”. Assuming that the Schwarzschild metric continues to hold all the way in, r=0 is simply not a point in the spacetime.
Though, of course, you’re right that the Schwarzschild metric probably does break down in some way extremely close to the center-- That’s a job for quantum gravity.
In general relativity there’s simply no such thing as an incompressible object and infact due to what are known as singularity theorums the actual fine details of how for example quarks behave under such extreme conditions don’t matter so much; the application of general relativity to certain situations will necessarily lead to gravitational singularities. Of course as general relativity itself does not take in to account quantum physics and in a theory of quantum gravity graviational singualrities might not be a factor However within the general relativity paradigm graviational singualrities are an unavoidable fact of certain realistic solutions.
Gravitational singualrities are defined by “geodesic imcompleteness” which ‘basically’ means that the trajectories of some objects cannot be extended beyond some point in their finite future (and by “basically” I mean that there are certain cavaets that I’ve delibrately left out). As Chronos points out there are actually problems in general relativity as viewing a singualrity as a place in space or an event in spacetime.
IIRC there is a requirement that the local speed of sound would have to be greater than c in any material whose modulus was high enough to resist collapse near a black hole singularity. I think this poses an independent upper limit on the density that real matter can have.
I’m not sure about that, ou can’t ignore relativistic corrections to the equation for the speed of sound in such a situation. Overall there is no relativistic limit on the density of a material.
The proof of existance of singularities in GR solutions comes from imposing certian conditions on spacetime and then seeing what sort of limits that imposes on the geometry and topology of the spacetime. Often such proofs are on the non-constructive side. I.e. they quite often don’t tell you much about the actual nature of the singularity.