> Neutron stars weigh in at about 1.4 solar masses
Observed neutron stars actually have a range of masses, from the 1.4 solar mass point (which is the maximum mass for a white dwarf, so one would expect to see neutron stars of about that mass that were just over the white dwarf limit) up to, IIRC, almost 3 solar masses for the largest one that has been observed.
> A neutron star can be viewed as a mass-just-shy-of-a-black-hole
Not in general, no. A neutron star just under the maximum mass limit for neutron stars, which is believed to be about 3 solar masses, could be sort of viewed this way, but even then it's not really correct, since there is nothing that forbids a black hole with a mass smaller than that from existing. It's just extremely unlikely that such a black hole could be formed by the collapse of a star, since such a collapse would be expected to stop at the neutron star stage (or at the white dwarf stage if the star was less than 1.4 solar masses).
On the one hand that's interesting, on the other hand eponyms are generally a bad idea. If you mention "Tolman-Oppenheimer-Volkoff limit" the chances of being understood outside of a very specialized audience are far lower than if you mention "neutron star mass limit".
Excellent points! Galaxies have more mass than either, but behave very differently. I would correct my statement by saying it's the concentration of mass (density) that defines neutron stars and black holes.
> it's the concentration of mass (density) that defines neutron stars and black holes.
Not really, no. What defines a neutron star is that it is in hydrostatic equilibrium supported by neutron degeneracy pressure. If we include the quark-gluon plasma phase, we can just amend that to being supported by quark degeneracy pressure (and noting that "neutron" is just a special case of "quark" for this purpose). Whereas a normal star is in hydrostatic equilibrium supported by thermal pressure (with fusion reactions providing the heat source). And a galaxy is not in hydrostatic equilibrium at all, it's composed of stars in free-fall orbits. The average densities of the objects in all three of these cases are consequences of the above.
A black hole is defined by having an event horizon and being vacuum. There is no well-defined concept of "density" for a black hole, nor is there "concentration of mass" in the sense of the hole being made of matter; it's not, it's vacuum. The "mass" of a black hole is a global feature of its spacetime geometry.
Right, but are those phenomenon not also consequences of density? If we compressed stellar or even terrestrial material down the density of a neutron star, would we not get neutronium as a result?
I agree that the density of a black hole will depend on choice of observer, but nothing stops us from picking a reasonable one, such as a Schwarzschild observer. Mass (as you've mentioned) and volume (Schwarzschild radius) are both we enough for us to define a global average density. It does not behave intuitively though, increasing mass actually decreases density, which does seems to contradict my earlier point...
> are those phenomenon not also consequences of density?
No, density is a consequence of the object's structure. Hydrostatic equilibrium supported by degeneracy pressure leads to higher density than hydrostatic equilibrium supported by thermal pressure.
> I agree that the density of a black hole will depend on choice of observer
That's not what I said. I said there is no well-defined density of a black hole at all. That's because the hole is vacuum and has no well-defined spatial volume.
> nothing stops us from picking a reasonable one, such as a Schwarzschild observer
There are no Schwarzschild observers inside a black hole. Schwarzschild coordinates in the interior of a black hole give an infinite answer for the "spatial volume", which, of course, is not well-defined.
> volume (Schwarzschild radius)
The Schwarzschild radius is derived from the surface area of the hole's horizon. It does not mean the hole has a well-defined volume. It doesn't.
> No, density is a consequence of the object's structure.
Which is a consequence of the gravitational pressure, no? Quark degenerate matter without a confining potential would not be at equilibrium.
> There are no Schwarzschild observers inside a black hole.
Of course not, but the definition of the Schwarzschild radius does not require a black hole, just mass. For example, we define the Schwarzschild radius of the Sun to be about 3km. Outside of black hole dynamics, this defines a volume of space, and we call an object whose radius is smaller than its Schwarzschild radius a black hole. I understand that black holes are fundamentally dynamic processes, but can we not define the behavior at the limit?
> Which is a consequence of the gravitational pressure, no?
Not just gravity alone, no. (And "gravitational pressure" is not a good description of gravity's effect.) Gravity plus the fact that the object does not have a heat source inside it (fusion) to provide thermal pressure. With thermal pressure an object with the same number of particles in it would be much larger--an ordinary star--and its average density would be much smaller.
> the definition of the Schwarzschild radius does not require a black hole
But if the object is not a black hole the Schwarzschild radius is physically meaningless. It only has physical meaning for a black hole, and its physical meaning, as I said, has to do with the area of the hole's horizon, and only that.
> this defines a volume of space, and we call an object whose radius is smaller than its Schwarzschild radius a black hole
No, this is not correct. We call an object a black hole if it has an event horizon. A black hole doesn't have a well-defined "radius" in space any more than it has a well-defined volume. A black hole is certainly not just an object "made of stuff", like a neutron star, that just happens to have collapsed further. It is a different kind of thing from any ordinary object.
> I understand that black holes are fundamentally dynamic processes, but can we not define the behavior at the limit?
What you are describing is not a "limit" of anything. It's simply a wrong description of the physics of a black hole. If by "dynamic" you are talking about the process of forming a black hole from the collapse of a massive object, you're still not correctly describing the end point of that process.
A neutron star can be viewed as a mass-just-shy-of-a-black-hole, so I'd expect it to be relatively similar in scale.