The Black Hole Density Coincidence

Could it be that the reason that the theoretical contradictions between quantum mechanics and general relativity don't arise is that the conditions where these contradictions could theoretical arise are unphysical for some other reason that is deeper than it seems?

Maybe so.

If fundamental particles have even a minimal radius, many orders of magnitude less than the Planck length, then they can't form black holes, and the relativistic effects on their behavior at Planck length or greater is trivial, although fundamental particles would be more dense than atomic nuclei if they had volumes on the order of magnitude of the Planck length.

Indeed, there is no mechanism in nature, apart from the Big Bang, that can produce black holes smaller than three stellar masses, which coincidentally (or perhaps not so coincidentally) have mass densities (assuming that their total mass is distributed evenly throughout their unobservable behind the event horizon volumes) no greater than the same order of magnitude as neutron stars and atomic nuclei. The larger a black hole, the lower its mass density. And, it isn't obvious that any relics of smaller black holes that haven't evaporated due to Hawking radiation still exist in the universe, nor is it clear that any stars more dense than neutron stars actually exist, although they have been hypothesized.

Black Holes and Neutron Stars

Everybody knows that general relativity predicts the existence of black holes, from which even massless photons moving at the speed of light cannot escape became the gravity of a black hole within its "event horizon" is overwhelming.

Less well known is the fact that in all black holes created in the usual way, through the gravitational collapse of a large star (the minimum threshold for this to happen is about 3 solar masses), the total mass of the black hole divided by the volume of the region within the event horizon is always less than or equal to the mass density of a neutron star, which are predicted to be up to about twice the mass density of a typical atomic nucleus.

The density of black holes (measured by its mass divided by the volume included in the event horizon) increases as black hole mass decreases. At about three solar masses (6*10^30 kg), the mass of a black hole divided by its event horizon volume is comparable to the density of an atomic nucleus, and larger black holes are less dense than an atomic nucleus.

Smaller black holes are theoretically possible in the equations of general relativity, but they aren't formed by gravitational collapse and have never been observed, even though they could conceivably have been created by non-gravitational forces in the Big Bang or some highly energetic controlled conditions like those in a particle accelerator.

Another thing that many people don't realize about black holes is that they do radiate and lose mass over time through what is called Hawking radiation, after celebrity physicist Stephen Hawking. The smaller the black hole, the more rapidly it loses mass.

All black holes are believed by many theorists to emit Hawking radiation at a rate inversely proportional to their mass. Since this emission further decreases their mass, black holes with very small mass would experience runaway evaporation, creating a massive burst of radiation at the final phase, equivalent to millions of one-megaton hydrogen bombs exploding. A regular black hole (of about 3 solar masses) cannot lose all of its mass within the lifetime of the universe (they would take about 10^69 years to do so, even without any matter falling in).

However, since primordial black holes [if they exist] are not formed by stellar core collapse, [theoretically] they may be of any size. A black hole with a mass [at the time it is formed] of about 10^11 kg would have a lifetime about equal to the age of the universe.

Thus, if black holes formed in the Big Bang that initially had masses of more than 10^11 kg and less than 10^30 kg, if they exist at all, and they were formed in great enough numbers, we should be able to observe some of those that are relatively nearby in our own Milky Way galaxy exploding today. If a black hole too small to be a black hole created by the collapse of a star was created around the time of the Big Bang, it could be no bigger than 10^23 kg now because it would have been losing mass to Hawking radiation over time. This is about one 10,000,000th the mass of the Sun. This is about the one-third of the mass of the planet Mercury. These small black holes could be considerably lighter, if it started out smaller. No such black holes, called "primordial black holes" have ever been observed, although astronomers are actively in engaged in the business of looking for them.

The most dense stars we are pretty sure exist (about 2,000 of them that have been observed) are neutron stars, which are estimated to be on the order of 12-15 km or less in radius:

A typical neutron star has a mass between 1.35 and about 2.0 solar masses, with a corresponding radius of about 12 km . . . . In contrast, the Sun's radius is about 60,000 times that.

Neutron stars have overall densities . . . of 3.7×10^17 to 5.9×10^17 kg/m3 (2.6×10^14 to 4.1×10^14 times the density of the Sun), which compares with the approximate density of an atomic nucleus of 3×10^17 kg/m3. The neutron star's density varies from below 1×10^9 kg/m3 in the crust, increasing with depth to above 6×10^17 or 8×10^17 kg/m3 deeper inside (denser than an atomic nucleus). . . .

In general, compact stars of less than 1.44 solar masses – the Chandrasekhar limit – are white dwarfs, and above 2 to 3 solar masses (the Tolman–Oppenheimer–Volkoff limit), a quark star might be created; however, this is uncertain. Gravitational collapse will usually occur on any compact star between 10 and 25 solar masses and produce a black hole.

No quark stars have ever been definitively observed, although three quark star candidates (a minuscule number relative to the 2,000 observed neutron stars) have been identified for further investigation. And, somewhat surprisingly, models of quark stars suggest that a quark star with 2.5 solar masses would have more volume than a neutron star with 2.0 solar masses, so the difference in mass density between the two varieties of post-supernovae stars would be modest.

The equations of general relativity imply that deep within a black hole is a singularity in a finite mass in an infinitesimal volume, but the laws of nature, in an act of cosmic censorship make it not just practically, but theoretically impossible to ever observe that singularity.

If no primordial black holes, which have no mechanism for forming after the Big Bang, exist, this would imply that there is nothing in the macroscopic universe with an observable density much greater than a neutron star or greater than twice the density of an atomic nucleus.

How Big Would Fundamental Particles Be At Atomic Densities?

The fundamental particles of quantum mechanics (quarks, leptons, photons, gluons and W bosons and Z bosons) are treated as point-like in quantum mechanics. But, given their known masses, they would not form gravitational singularities, which happens at the Schwarzschild gravitational radius = 2Gmc−2 (where m is the gravitational mass as measured by a distant observer), unless they each a a radius many orders of magnitude smaller than the Planck length ∼ 10^−33cm.

Suppose that fundamental particles had mass densities comparable to that of atomic nuclei. How big would they be?

Atomic nuclei densities provides an estimate of the size of the neutron and proton: "The diameter of the nucleus is in the range of 1.75 fm (femtometer) (1.75×10−15 m) for hydrogen (the diameter of a single proton) to about 15 fm for the heaviest atoms, such as uranium. These dimensions are much smaller than the diameter of the atom itself (nucleus + electronic cloud), by a factor of about 23,000 (uranium) to about 145,000 (hydrogen)." A nuclear radius is roughly the cube root of the number of protons and neutrons combined in the atom times 1.25 × 10−15 m +/- 0.2 fm (this +/- factor varies from atom to atom), and the shape is approximately spherical.

Protons and neutrons, of course, are made up of three quarks each bound by gluons. Bare quarks make up about 1% of the mass of a proton or neutron, so a first generation quark is about 0.3% of the mass of a proton, with the rest of its mass coming from the "glue" and thus quarks would be 14% of the size of a proton or neutron if they had an equivalent mass density, i.e. about 0.2 fm (2*10^-16 meters).

Interestingly, this coincides rather neatly with the apparent charge distribution within a neutron (which has one up quark of charge +2/3 that appears to be centrally located and two down quarks of charge -1/3 that appear to orbit the up quark): "The neutron has a positively charged core of radius ≈ 0.3 fm surrounded by a compensating negative charge of radius between 0.3 fm and 2 fm. The proton has an approximately exponentially decaying positive charge distribution with a mean square radius of about 0.8 fm." But, a 10^-16 meter order of magnitude for up and down quarks would be much larger than the Planck length that conventional wisdom would ordinarily expect.

Of course, the cosmic censorship imposed by the quark confinement implied by QCD in the case of quarks other than top quarks, and exceedingly rapid weak force decays almost exclusively to bottom quarks, in the case of top quarks, greatly constrain our capacity to directly observe the radius (if any) of a quark.

At a mass density equivalent to atomic nuclei, electrons would have about half the radius of an up or down quark, but this would seem to contradict the limits set by experiment that suggest that an electron cannot have a radius of more 10^-20 meters, a factor of 10,000 smaller. However, some of these experiments seem to set limits on electron compositeness, rather than actual electron radius in a genuinely fundamental non-composite electron with a homogeneous spherically distributed charge. Such a simple space filling electron lacks the traits of a composite electron, it would behave far more like a point particle, but does address inconsistencies between general relativity and quantum mechanics.

An electron neutrino would have a radius of about 10^-18 or 10^-19 meters if its radius were set based on it having a mass density comparable to an atomic nucleus, a radius that also seems to exceed the experimental limits on that radius by a factor of at least 10 to 100.

Thus, first generation electrons and neutrinos, at least, would seem to have a greater mass density than atomic nuclei, but would not be singularities in general relativity unless their size in space was many orders of magnitude smaller than the Planck length.