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Degenerate Objects

The Inevitability of Black Holes

Not all stars end their lives as degenerate (white) dwarfs or neutron stars; the largest stars become black holes.  This outcome is inevitable under our current theories of particle physics and gravity.  No degenerate object larger than about 3 to 5 solar masses can stabilize itself against gravitational collapse.

A degenerate dwarf is stable when the electron Fermi energy is less than the rest-mass energy of the electron.  A neutron star is stable when the proton Fermi energy is less than the rest-mass energy of the proton.  These criteria are violated when the mass of the object becomes too large.  Too large for a degenerate dwarf is the Chandrasekhar limit of 1.4 solar masses, while too larger for a neutron star is roughly double this value.

What is interesting about these limits is that they are set by the mass of the proton mp.  The Chandrasekhar limit is proportional to mp−2.  The mass of the particle providing the degeneracy pressure does not set the mass limit for stability; rather, the mass of the particle providing the gravitational field sets this limit.  For this reason, the upper mass limit for a neutron star is of order the Chandrasekhar limit despite the different source of degeneracy pressure.

The source of the gravitational instability, as explained elsewhere, is that the degeneracy pressure changes more slowly with density when the thermal velocity of the degenerate particles is close to the speed of light than when it is much less than the speed of light.  This slow change in pressure with density cannot counter a gravitational force that increases with radius R as R−2, as it does under Newtonian gravity, so once such a body starts collapsing, it continues to collapse.

One point immediately becomes clear when considering this collapse: to have a degenerate object that is stable against gravitational collapse, either the particles providing the object's gravitational field must be significantly lighter than the proton, or the gravitational force must increase less rapidly than R−2 at some radius.  Either condition requires that physics differ from our current theories.

On the particle physics side of this problem, one can create a high-mass degenerate object if one can find a stable Fermion—a particle that obeys the Pauli exclusion principle—with a mass much less than the proton's.  The problem with finding such a particle is that few suitable particles are known.  Most fundamental particles are heavier than the proton, and they are unstable, so they rapidly decay away.  The muon is a Fermion that is lighter than the proton, but it is also unstable.  The pion is lighter than the proton, but it is not a Fermion, so it cannot provide degeneracy pressure; it is also unstable.  Only the quarks, which make up the protons and neutrons, are suitable alternatives to the proton and the neutron for providing mass and degeneracy pressure, but objects created out of quarks are highly speculative and are generally no more massive than neutron stars.  In fact, theorists studying compact stars composed of “deconfined” quarks (quarks no longer confined to protons and neutrons) find that they are about 20% smaller in radius than neutron stars of equivalent mass.  This smaller size implies that quark degenerate objects have a lower value for their upper mass limit than do the neutron stars because of the effects of general relativity.

The largest black hole candidates are those at the center of galaxies.  The black hole candidate at the center of the Milky Way, Sgr A*, has a mass of 3.6×106 solar masses.  For something this massive to be a stable degenerate object, it would have to be composed purely of Fermions with the mass of the electron, but the only particles meeting this criterion are the electron and the positron, and because the second is the antiparticle of the first, they would combine to produce gamma-rays rather than exist together in a stable body.

The very fact that no object of over 1 solar mass with a radius intermediate between the degenerate dwarf and the neutron star has been found supports the belief that stable degenerate objects can only be created from electrons, protons, neutrons, and quarks.  Particle physics therefore does not spare us the existence of black holes.

If black holes do not exist, the culprit would be a gravity that departs from general relativity.  The degenerate dwarf is unstable at high mass because its Newtonian gravitational field is proportional to R−2.  General relativity changes this relationship when the object's radius approaches the event horizon radius of a black hole of comparable mass, but in a way that causes the star to be more unstable to gravitational collapse.  The mass at which a neutron star is unstable under general relativity is therefore lower than that of a neutron star with a Newtonian gravitational field.

If gravity behaved differently than is predicted by general relativity, however, our universe could be free of black holes.  This would require a theory of gravity where the gravitational force exerted within the star rises less rapidly with decreasing radius than R−2. If gravity behaved in this way at some point, which would mean deviating from Newtonian gravity at some point as the object shrinks, any object, regardless of how massive it is, could be stabilized by the degeneracy pressure exerted by neutrons, protons, or deconfined quarks.

Under the currently accepted theories, there are no degenerate objects of high mass, and the very massive objects in our universe are inevitably black holes.  This means that Sgr A* is a massive black hole in our theories, and if in reality it is not a black hole, then general relativity is likely an incorrect theory of gravity.

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