Why Black Holes Don't Need Singularities

ABSTRACT:

According to R.P. Kerr, a black hole need not contain a singularity. Such an assertion prompted this author to explore alternate black-hole models where a singularity is unnecessary. The models presented here show that a sufficiently large black hole could contain a universe; that an average black hole could contain a neutron star. This paper also shows why micro-black holes are untenable and why a typical black hole has a mass of at least approximately three solar masses. As an additional bonus, the models presented conserve quantum information inside the black holes, and, are consistent with black-hole entropy and temperature.

Alice lives in a universe similar to ours. It has an isotropic mass density--its total mass is mostly proportionate to its total volume as well as its Schwarzschild radius. Her universe also has a cosmological horizon which limits how far out Alice can make observations. Her universe is expanding and she observes galaxies receding away from her and they eventually disappear when they cross the cosmological horizon. Below is the relevant mathematics that describe her universe:

As her universe's volume grows, its mass grows proportionately. Equation 4 above shows how the universe's radius r grows as mass m increases. Equation 5 does likewise for the universe's Schwarzschild radius. However, note that the Schwarzschild radius grows exponentially faster than r. At some point in time, the Schwarzschild radius will be greater than her universe's radius. The diagram below assumes such a point in time:

The inner red circle represents Alice's cosmological horizon. The blue circle represents the physical extent of her universe. The black circle is the extent of the Schwarzschild radius. Beyond the black circle lives Bob. When Bob looks at Alice's universe, he sees a black hole. It has gravity consistent with equation 6 above. Alice, on the other hand, believes she lives in a fairly normal expanding universe. Its expansion velocity v is consistent with equation 7.

Bob believes that the black hole has a singularity where there is infinite gravity and infinite mass density. It is where the laws of physics break down. He knows this is true because he watches the Discovery Channel. By contrast, R. P. Kerr (see reference below) suggests that black holes can exist without singularities. He also argues that no one has conclusively proven that black holes must have singularities. I will argue that the above thought experiment, involving Alice and Bob, supports his assertions.

Caveat: The above thought experiment may only prove that extremely large black holes can exist without a singularity. What about smaller black holes? According to the Hawking-Penrose theorem, geodesic paths must meet at a single point. The theorem is based on General Relativity. However, the General Relativity theory preceded quantum mechanics, so its solutions don't take into account quantum mechanics. So perhaps the same can be said about the Hawking-Penrose theorem:

Equation 8 above demonstrates that infinite energy (E) is required to have a singularity where distance x equals zero. Since black holes don't generally have infinite energy, we can infer that the stuff inside must take up some space. The question is ... how much space?

If energy E at equation 8 equals the Planck energy, then x equals the Planck length, but then so does the Schwarzschild radius. To have a black hole we need energy that is equal to or greater than the Planck energy. It seems if the particles invovled are squeezed into the smallest space, x will be less than or equal to the Planck length. Unfortunately, the particles can't be fermions, since fermions must comply with the Pauli Exclusion Principle -- two or more of the same type of fermion can't occupy the same position in space. A single fermion has a physical extent, according to equation 8, where x easily exceeds the Planck length. Multiple fermions take up even more space.

Unlike fermions, bosons don't have to comply with the Pauli Exclusion Principle. It is possible for multiple photons, for example, to occupy the same position in space. However, if a lot of photon energy is concentrated in such a small space they will produce fermion pairs which must take up more space thanks to the Exclusion Principle. If the goal is to pack the most particles in the smallest space possible, gluons seem like the best option. The strong force is weaker when they are closer together, so there is less likelyhood for fermion-pair production. Unfortunately, gluons are hopelessly wedded to quarks due to confinement. Separating gluons from quarks increases the strong force attraction between them (and increases quark-antiquark pairs) which makes the possibility of such separation untenable. Thus, if we want to create a black hole, we are essentially stuck with space-hogging fermions, so it is unlikely we will be creating or observing micro black holes.

The mass of a typical natural black hole is a minimum of around three solar masses or approximately 6E30 kilograms. This is not surprising when you take into account the previous paragraph. The stuff inside a black hole must consist of fermions which take up a lot of space (unless the Pauli Exclusion Principle is violated).

If we use equations 4 and 5 above and use the mass density of, say, a neutron's mass density in place of the universe's mass density, and have m represent mass in general, we find that when we plug into m at least 6E30 kilograms, we get a radius r that is less than or equal to the Schwarzschild radius.

Why use a neutron's mass density? Because the core of a collapsing star has a plasma consisting of electrons and protons that undergo electron capture:

The above Feynman diagram shows how each electron and proton yields a neutron and neutrino. The neutrinos escape, so that leaves the neutrons behind to form a neutron star. The star's neutrons have neutron degeneracy pressure which arises from the Pauli Exclusion Principle. This pressure resists further collapse.

Now, imagine, if you will, a neutron star with a mass that exceeds three solar masses. Its radius is less than its Schwarzschild radius. To an outside observer named Bob, it is indistinguishable from a black hole. To an inside observer named Alice, it is just a neutron star, nothing exotic or mysterious. If we do an accounting of trapped and escaped particles, etc., quantum information is conserved.

One can calculate its temperature and entropy in the usual manner, since neither of these is a function of how much space the stuff inside occupies. The surface area is a function of mass or the Schwarzschild radius, not the radius r--so r doesn't have to be a singularity or zero distance.

References:

1. Kerr, R.P.. 12/05/2023. Do Black Holes Have Singularities? University of Canterbury.

2. Hawking, S.W., Penrose, R. 04/30/1969. The Singularities of Gravitational Collapse and Cosmology.

3. Pecina-Cruz, Jose, N. On the Collapse of Neutron Stars.