Is the existence of hydrostatic relativistic objects a coincidence or inevitable?

Question

Neutron stars are unique in the universe because their sizes are just slightly bigger than their Schwarzschild radii. Because their sizes are comparable with their Schwarzschild radii, the nonlinear effects of general relativity are no longer negligible, which makes neutron stars a good test bed of general relativity.

I just come up with an interesting question: would such relativistic objects still exist if the physics constants (the equation of state (EOS) of matters, speed of light ($c$) or the gravitational constant ($G$)) vary a little bit? To clarify, here relativistic means the objects’ radii are no more than an order of magnitude bigger than the Schwarzschild radii. These objects should be supported by the hydrostatic pressure, which rules out black holes and collapsing stars. If they are rotating, they should maintain perfect axial symmetry and every part should rotate at the exact same rate, otherwise they will still collapse due to viscosity or gravitational wave radiation. Besides they should be stable against small perturbations. According to a simulation, if a neutron star is spinning too fast, even if it’s at hydrostatic equilibrium a slight perturbation can disrupt the star. The stiffness of matters should not exceed the theoretical limit (beyond which the speed of sound will exceed the speed of light). Taken together, is there a wide margin for such objects to exist, or is their existence a lucky coincidence?

Answer 1

The existence of stable neutron stars is not inevitable. It relies on the properties of the strong nuclear force. In our universe that allows neutron stars to exist between about 0.2 and 2.5 solar masses (both limits subject to some uncertainty). The upper mass limit is determined by the equation of state, which is hardened by the repulsive force between nucleons at close separations.

Without this repulsion, neutron stars would be limited to about 0.75 solar masses.

If this were the case, then no stable neutron stars would exist, or at least there would be no obvious way to produce them, because objects with masses below 0.75 solar masses can be supported by electron degeneracy and would have much larger radii.

The ratio of the maximum mass of an electron-degenerate white dwarf to the maximum mass of a neutron-degenerate neutron star (supported by ideal degeneracy pressure in both cases) is, to first order, independent of the physical constants $G$, $c$ and $h$. It depends on the number of mass units per particle in the gas, which is always going to be lower in a gas containing free electrons compared with a gas of neutrons and the ratio of the neutron to electron mass.

In other words, introducing a strong force nucleon repulsion appears to be an essential requirement for neutron stars to exist as a stable state at masses above the maximum mass supportable by electron degeneracy.

Of course there are other possibilities - including quark stars - that might be feasible instead of neutron stars. There are far too many free parameters to provide a comprehensive answer.

Answer 2

There is a diagram in (Livio & Rees 2018) showing the distribution of objects in equilibrium in the universe:

Objects we see around us are typically on the atomic density line or a slightly offset molecular density line to the right. They note that without gravity these objects could continue up the slope all the way to the black hole limit, but in practice gravity increases the density moving things to the left and leading to an earlier approach.

Exactly where material objects end up hitting the black hole line as their mass is increased depends on the relative strengths of the other forces of nature. We get planet and star densities from the relative strength of electromagnetism to gravity, and neutron star densities from the relative strength of the strong nuclear force. Had they had different values the locations would move around, but it is impossible to avoid hitting the slope 1 black hole line with the slope 3 lines below it.

Now, in reality there is a gap between black holes and neutron stars (or any other kind of hydrostatically stable object obeying the energy conditions) due to the Buchdahl theorem. Unless $$M< \frac{4Rc^2}{9G}$$ the core pressure diverges (and it implodes). But this condition just adds a line a tiny fraction to the right of the black hole line, and again lines of hydrostatic equilibrium objects will touch it (and definitely be in the relativistic domain).

Change the constants enough and you will not get any objects, of course. With the "right" settings (like a low $G$ or faster expansion) matter will never coalesce into stars or heavier things and just remain gas, and there would not be any such relativistic hydrostatic equilibria.