ABSTRACT:
Re: the information paradox. When a theory contains a paradox, it is a clue that one or more premises the theory relies on are faulty. In this paper, we examine the premises, arguments and assumptions that are the foundation of black-hole physics.
All roads lead to Rome. Somewhere in Rome there is a particle trapped in a rotating potential well. Which road (or path) did the particle take to get to Rome? The Heisenberg Uncertainty Principle prevents us from simultaneously knowing the particle's position and momentum. That information could help us determine from whence the particle came.
Prior to being trapped in the potential well, the particle's momentum vector could have given us a sense of direction, but that vector is now rotated. The the information we need is lost. We can't determine the particle's previous positions and momenta, i.e., the road it took to Rome. This is one example of irreversibility that flies in the face of the claim that the state of a particle is always reversible. Yet, it is this claim or premise that leads to the black-hole information paradox.
According to the current paradigm, quantum information is conserved. With perfect knowledge of a particle's current state, it should be possible to trace it backwards and forwards in time. This principle would be violated if information were lost. When information enters a black hole we might assume the information is inside, but then black holes evaporate due to Hawking radiation, and the black hole's temperature is as follows:
As the black hole evaporates, its mass shrinks and its temperature increases. Take note that equation 1 fails to tell us what information went into the black hole, so looking at the final information (remaining mass, momentum, charge) pursuant to the no-hair theorem we can't extrapolate that data backwards and determine what information went into the black hole. It's irreversible. But as shown earlier, irreversibility is not unique to black holes.
Here is yet another example of irreversibility: take two systems, each containing various particles with either positive or negative charge. Coarse grain both systems to get a final result of negative charge for each. The final result is identical for each system; yet, what went into each system varied widely. The final information (negative charge) fails to tell us what went in. A unitary operator would erroneously give the same previous state for each system:
The premise that irreversibility can't and should never happen seems untenable.
Now let's shift our focus to Hawking radiation. If Hawking radiation does not exist, life would be easy. The second law of thermodynamics would never be violated if the black hole maintains or gains mass:
One argument used to justify the existence of Hawking radiation is, "Black holes have temperature; therefore, they radiate." Unfortunately, temperature is not the only variable that determines how much a body radiates. The Stephan-Boltzmann equation below shows that emissivity also plays a role:
The black hole's temperature is irrelevant if the emissivity is zero. And why would the emissivity be zero? Because a black hole's gravity is so strong ... nothing can escape--not even light. Of course, at the quantum scale, there are likely to be events that defy classical physics, but we don't observe them at the macro scale. Apparently, they cancel each other and the classical events are what we observe. So it is not a stretch to assert that a black hole's emissivity is zero (or negative if you count the stuff falling in).
Even if the emissivity is positive, large black holes have a lower temperature than the surrounding environment, so they won't be evaporating any time soon. Small black holes that have a higher temperature probably don't exist, since at least three solar masses are required to create a black hole. Thus, it is no surprise that Hawking radiation has not been observed.
Hawking radiation may also be untenable if the following axiom is true: a system's total mass and temperature emerge from smaller constituents. So the question arises: can a quantum particle pair emerge from parameters such as temperature and total mass? Take note of the following equations:
Equation 4 is consistent with the axiom: a sum of quantum masses make up the total black-hole mass. But at equation 5 we have a pair of radiation particles that depend on the black hole's average temperature which, in turn, depends on the black hole's mass. To sort this out, imagine a single photon at the sun's surface. It's frequency is independent of the sun's average temperature; but the average temperature depends on the photon's frequency along with countless other photons and their frequencies.
To assert that a photon's frequency depends on the temperature is to turn the axiom on its head. Put aside such an assertion and imagine each Hawking particle with its own frequency and other quantum parameters. Together they could be constituents of the black-hole temperature just like the information that entered the black hole. Thus one might be tempted to argue that, at least quantitatively, Hawking radiation preserves the information that entered the black hole. At the very minimum, if black holes evaporate, mass is conserved, so we can justify the following:
Equations 5b and 5c confirm that what leaves the black hole is equivalent to what went in. This may be the inspiration behind another premise: Information is conserved. Really? If it is proportionate to energy, yes. But it is not. It is proportionate to the imaginary surface area A of the event horizon (see equation 3).
Adding a qubit of information to a black hole is done in the following manner:
The result of equations 6 and 7 is one Planck area is added to the surface if a photon with the same wavelength as the Scharzschild radius falls into a black hole. The premise here is the Planck length is the shortest possible length; however, the change-of-Scharzschild radius is shorter than the Planck length if the Scharzschild radius is large (see equation 6). And then there's the sloppy math. Here's the math done properly:
As you can see, at equation 8, there is an extra term added to the Planck-area term, and both terms are multiplied by 8pi. At 9 and 10, The change-of-radius variable is made independent of the Scharzschild radius, and, why not? What are the odds that a particle falling into a black hole will have a wavelength equal to the the Scharzschild radius? Equation 8 shows that the amount of area the particle contributes can vary depending on the size of the Scharzschild radius. If information is proportionate to area, then the amount of information contributed will vary as well. Also, at equation 3, entropy is a function of area. Since entropy must either remain the same or increase, so must information. Information (proportionate to entropy or area) is not conserved!
In conclusion, it is not surprising there is an information paradox, but that paradox is just the tip of the iceberg. It is a clue that one or more premises are flawed. Theorists need to re-examine them. And while they're at it, check the math.
References:
1. Hossenfelder, Sabine (23 August 2019). "How do black holes destroy information and why is that a problem?". Back ReAction. Retrieved 23 November 2019.
2. Hawking, Stephen (1 August 1975). "Particle Creation by Black Holes" (PDF). Commun. Math. Phys. 43 (3): 199–220.
3. Susskind, Leonard (2008-07-07). The Black Hole War: My Battle with Stephen Hawking to Make the World Safe for Quantum Mechanics. Little, Brown and Company.
4. Black hole information paradox. Wikipedia.
5. Mathur, Samir D. 03/21/2021. The Elastic Vacuum. Gravity Research Foundation.
6. Chaisson, Eric. Astronomy Today. Englewood, NJ: Prentice Hall, 1993: 503
