Friday, April 10, 2015

The Black Hole Information Paradox...Isn't

Black holes are dead stars that have collapsed under the gravitational attraction of their own mass. The matter density at the center of a black hole is extremely high. Some say it is infinite and the center of a black hole is a singularity. The density is so high that a horizon exists outside of the black hole where the escape velocity exceeds the speed of light. Nothing that falls through the horizon can escape, even light. Thus, a black hole is  black because nothing inside can be seen from outside the horizon.

In 1974, Stephen Hawking showed that the temperature of a black hole is not absolute zero as it was previously assumed, and that black holes radiate energy like black bodies. Eventually, a black hole will evaporate. However, the temperature is inversely proportional to the mass of the black hole so that very large black holes have exceedingly low temperatures, much lower than the current temperature in space, which means they are currently absorbing energy rather than radiating it.
In 1976, Hawking concluded that the information content of whatever falls into a black hole is irretrievably lost from the universe. The Hawking radiation, he said, is purely thermal and contains no information other than temperature. The infalling matter goes down the singularity along with its information content. Leonard Susskind, Gerard t’ Hooft, and other physicists realized that this would violate the unitarity that is fundamental to quantum mechanics. They insisted that there could be no loss of information and have been trying ever since to prove that the information comes back out in the Hawking radiation. The controversy is not yet settled, although not for lack of efforts to solve the problem. A few days ago, Sabine Hossenfelder at Backreaction posted a detailed refutation of a recent paper implying that its authors had solved the problem.

It is generally agreed that getting rid of the singularity would eliminate the information loss problem, although it does not show what happens to the information. 

Spacetime Model
The problem is a result of an incorrect spacetime model. In the spacetime model I’m describing in this blog, singularities are impossible, so the black hole information loss problem does not exist. Spacetime consists of quantum entities called points, which are mixtures of fermionic and bosonic points. It is indistinguishable from a spacetime consisting of two coupled fields, one fermionic and the other bosonic. Obeying Bose-Einstein statistics, the bosonic points seek the same quantum state, dragging the coupled fermionic points with them. This is gravity. However, the fermionic points obey Fermi-Dirac statistics and cannot occupy the same state. This is degeneracy pressure. The result is a quantum lattice of fermionic points pulled together by gravity but held apart by degeneracy pressure. Each fermionic point is confined to a Planck-scale cell bounded by other fermionic points. The positions of these points are subject to quantum fluctuations. There is a ground state of lowest energy corresponding to “empty” space. If the energy of a point is above the ground state, a particle exists at that point.

Black holes form when stars exhaust their fuel and are too large to be held up by the degeneracy pressure of their electrons or neutrons. They then undergo gravitational collapse, forming a dense core surrounded by a horizon located at the point of no return for matter and energy falling in. The theory of General Relativity says that the core is a singularity, a point of zero size and infinite mass density. However, General Relativity does not apply to Planck-scale physics. In our spacetime model, the maximum theoretical mass density, assuming that all fermionic points are stationary and in an excited state, is the Planck density. A singularity cannot form as long as the fermionic points are held apart by degeneracy pressure. Degeneracy pressure for points is much stronger than for particles. Since space is undergoing an accelerating expansion, it is possible that at some time in the far distant future space may become large enough for its gravity to overcome its degeneracy pressure just as it does for black holes, in which case it will collapse to a singularity. However, unless the entire universe collapses, no isolated singularity can ever form at any point. Therefore, we can conclude that there is no singularity in any black hole, and no information is lost.

Black Hole Evaporation

If there is no singularity, the core of a black hole is simply a very dense amalgamation of all of the matter that has fallen through the horizon. It just consists of particles that are very close together. Their positions are still subject to quantum fluctuations, that is, they have position wave functions. Naturally, these position wave functions are sharply peaked in the region around the center of the black hole, but there is always some nonzero probability that any given particle could be found at any finite distance from the center, even beyond the horizon. In other words, the particles inside a black hole can tunnel through the gravitational barrier, enormous as it is, and escape from the black hole. Given enough time, some say 1068 years or so for a solar-mass black hole, a black hole will evaporate. What looks like Hawking radiation from outside the horizon looks like quantum tunneling to an inside observer. The larger the black hole, the smaller the tunneling probability and the longer it takes for the black hole to evaporate. For Hawking radiation, larger black holes have lower Hawking temperatures and therefore radiate at slower rates. Thus, with respect to evaporation, the views of observers outside black holes are consistent with those of inside observers. But the Hawking radiation is not devoid of information. It contains all of the information that has ever fallen into the black hole.
Black Hole Entropy

Jacob Bekenstein realized that black holes must have entropy and calculated that the entropy of a black hole is proportional to the area of the horizon, not to the volume enclosed by the horizon. This surprised everyone because it is counterintuitive. Entropy is a measure of information. One would expect that the amount of information contained in a lump of stuff would be proportional to the volume of the lump. In our spacetime model, there is no need to give up this intuitive notion. The mass density between the core and the horizon is near zero compared to the density of the core, so it makes sense to conjecture that the information content of a black hole is proportional to the volume of the core. Then Bekenstein’s conclusion simply means that the volume of the core must be proportional to the area of the horizon. The horizon area is equal to a certain number of Planck areas (the Planck length squared), as Bekenstein showed, and the horizon area increases by one Planck area for each bit of information falling into the black hole, that is, the number of bits of information equals the number of Planck areas on the horizon. Now, if we assume that the core contains the same number of Planck volumes (the Planck length cubed)  as the horizon contains Planck areas, then the entropy, or information content, of the black hole is proportional to both the core volume and the horizon area, so we can have our cake and eat it, too. See here for some calculations of core size and density.