Hawking Radiation Doesn't Work the Way You Think

ABSTRACT:

Hawking radiation has not been directly observed. Maybe it exists, maybe it doesn't. Even it it exists, some very fundamental physical laws prevent black hole evaporation. At the quantum level, Hawking radiation can be turned on its head. It could just as easily add mass to a black hole.

Hawking radiation has not been observed for a very good reason: it does not work the way you think. The hypothesis seems sound at first blush: Two particles pop into existence. One outside the black-hole horizon, and the other trapped inside the black hole. The outside particle escapes, and, can be deemed positive energy, since it adds energy to the outside universe. The trapped particle can be deemed negative energy since, according to Hawking, will reduce the black hole's mass. Note that energy is always conserved, since positive energy and negative energy mathematically cancel each other. The bottom line is black holes allegedly evaporate due to Hawking radiation. I say "allegedly" because there is more to the story. The Hawking-radiation hypothesis is incomplete. Let's do a more complete thought experiment and see what happens.

Imagine a star that is virtually all matter, with next to no anti-matter. The star collapes into a black hole. Its mass is still composed of virtually all matter. The stuff that falls into this black hole is virtually all matter. In the black-hole diagram below, we represent a particle of this matter with the Greek letter mu preceded by a plus sign. Ellipses before and after the plus mu's represent multiple particles that may have been crushed into a singularity.

The above diagram represents the starting mass and the state prior to the appearance of a particle (plus mu) and an antiparticle (minus mu). The next state below is where a particle-antiparticle pair appears. The antiparticle escapes the black hole's gravity, but the particle is trapped.

In the next diagram the trapped particle has no anti-particle to interact with, so it adds mass to the black hole! This particle can be deemed the positive energy. The escaped anti-particle is then deemed negative energy and has the potential to interact with any particle it encounters. Such interaction will reduce the mass of the universe outside the black hole.

Of course there is only a 50% probability the black hole will gain mass and the remaining universe will lose mass. Below we see that there is a 50% probability that the particle will escape and the antiparticle is trapped.

The antiparticle has no problem finding particles to interact with:

The black hole loses the mass it previously gained:

The escaped particle and the escaped anti-particle may annihilate each other, or, if they are too far apart, will interact with other particles and antiparticles.

The space outside the black hole returns to nothing and the black hole returns to its starting mass:

The above thought experiment can also be performed with anti-matter black holes. The main problem with Hawking's hypothesis is it has the following implicit assumption: That all black holes have fairly equal amounts of matter and anti-matter. One might ponder whether a star that precedes a black hole can have fairly equal amounts of matter and anti-matter and still exist. Assuming the answer is a resounding no, then black holes don't evaporate via Hawking radiation.

For the sake of argument, let's assume Hawking was right. There is Hawking radiation and it causes black holes to evaporate. Why should black holes have all the fun? Imagine a particle-antiparticle pair appearing above the earth's surface. One escapes earth's gravity, the other does not. When they first appeared, they each had velocity v which is less than light speed. Velocity v was an escape velocity for one but not the other--the other being too close to earth's center of mass. If Hawking was right, the trapped particle should reduce the earth's mass. Over time the earth will completely evaporate. Thus, if Hawking was correct, all planets, stars, etc. should evaporate. The counter-argument is no such evaporation has been observed.

The equations below further demonstrate why black holes, in particular, refuse to evaporate:

Since light can't escape a black hole, a black hole's emissivity is zero. Even if it has an emissivity of one, power (P) according to the Stefan-Boltzmann equation above, is less than zero. This implies there is more radiation entering a black hole than randiation escaping. The minimum mass required to make a black hole is approximately three solar masses. So much mass causes the black hole's temperature to be less than its surrounding environment: deep space. The second law of thermodynamics would be violated if the net thermal transfer favors black-hole evaporation. Black hole entropy increases when a black hole's mass increases:

On the flip side, a compelling argument in favor of black-hole evaporation is the following thought experiment: Imagine a photon-antiphoton pair. Photons and antiphotons are indistinguishable from each other. So if the photon is trapped, it could behave like an antiphoton and annihilate matter inside the black hole, reducing the black-hole's mass. However, the escaped anti-photon can also behave like an antiphoton. If the escaped antiphoton finds another photon first, it becomes the negative-energy particle and reduces the energy of the universe outside the black hole. The trapped photon (or antiphoton) will add energy or mass to the black hole.

Photon-antiphoton Hawking radiation is more likely to cause the black-hole to lose mass if the black-hole's surrounding environment is empty space with a lower temperature. Albeit, this is an ideal and unrealistic condition. The cosmic microwave background raises the temperature of the surrounding environment to approximately 2.73 Kelvin, well above the temperature of the typical black hole. Thus, the following scenario is consistent with equations 1 through 6 above: The escaped photon is more likely to find another photon to interact with. When it does, the two photons vanish. The trapped photon adds mass to the black hole.

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

7. Severino, Paul. THe Black Hole Information Paradox: A Quantum Information Perspective. 03/24/2020

8. Wilkins, Alex. Ilands Poking Out of Black Holes May Solve the Information Paradox. 01/11/2024. UC Berkeley Physics.

The Faulty Premises of Black-hole Physics

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

Resolving the Black Hole Information Paradox: How Information is Lost and Conserved

According to the current paradigm, quantum information is conserved. With perfect knowledge of the current universe it should be possible to trace the universe 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. This is known as the Black Hole Information Paradox.

Many hypotheses have been set forth that attempt to resolve this paradox. One popular one is the holographic principle (T'Hooft and Susskind). Unfortunately, it is easy to punch holes in this one. You can read about it by clicking here.

Another common proposal is the information goes inside the black hole, then through a wormhole into another universe. Personally, I don't care for this one, since it requires the establishment of another universe (good luck!). Then there's the explanation that begins with a shrug and ends with a sigh: the information is lost.

Of course I'm not without a brainstorm of my own, which is why I'm now scribbling. It occurred to me that maybe there's at least two kinds of information: information that is conserved and information that is not. This random thought popped into my head when I was working on the following math proof:

The proof starts with the absurd claim that if 'a' doesn't equal 'b' then 'a' is equal to 'b.' Let's suppose 'a' is information. At equation 2 it is defined. However, by the time we get to equation 4, 'a' becomes undefined. Zero times infinity can equal any number, so the definite information we started with appears to be lost. Although, unlike black-hole information, by the time we get to equation 7, 'a' is defined again, but this time it is defined as 'b.'

What we can take away from the proof above is specific information is not conserved, but information overall is conserved. The information changed from 'a' to undefined to 'b.' Unfortunately, even though the information is conserved, we can't tell by looking at 'b,' that it was once 'a.'

Below is another example of what I'm scribbling about. Start with two distinct binary numbers. Let's pretend they enter a fictitious binary black hole and come out identical (zeros on the left, ones on the right). Now we put them into a cosmic hat. You reach in and pull one out. Can you tell whether it used to be 1010 or 0101? I don't see how. The information is conserved however--there's still the same number of ones and zeros.

Rather than say information is lost, perhaps it is more prudent to say it is undefined. In the case of a black hole, most of the information becomes undefined. Having perfect knowledge of it doesn't help us trace it back to its defined state prior to entering a black hole.

There are many examples in everyday life where we can observe information evolving from defined to undefined. Write a message on a blackboard. Erase the message. The chock that made up the message is now smeared onto the eraser. Give the eraser to a physicist and see if he/she can tell you what your unique message was. At this point, the chalk has mass, for instance, but chances are excellent that physicist won't be able to know your unique message. That information is lost. It was defined, now it is undefined (except for some basic properties like mass, etc.)

The no-hair theorem reminds me of brown paint. Imagine some masterpiece paintings, each with a unique set of information. The paint on each painting is scraped off the canvass and mixed in a bucket of paint thinner. At the end you have several buckets of brown paint. If they are mixed up and you choose one at random, can you tell which painting it came from? Probably not. It's another case where defined information becomes less defined--so it may also be true even at the quantum scale. For example, according to quantum field theory, particles and their unique, well-defined properties are excitations of fields where the information is kind of blurry or undefined.

Imagine an electron-positron pair popping into existence. The electron is spin up, the positron is spin down. They annihilate. Is it possible to look at the resulting photon and know it was previously a spin-up electron and a spin-down positron? For all you know the electron was spin-down and the positron was spin up. Yet another case of information evolving into something where you can't know its previous state. So why should we be surprised there's an information paradox if we believe perfect knowledge of the current state of information allows us to trace it backwards and forwards in time?

Quantum Tunneling Out of the Black Hole

A particle is trapped inside a black hole. What is the probability it will escape via quantum tunneling? To figure this out, we begin by defining the variables involved:

The diagram below shows two regions: I and O. Region I is inside the event horizon, within the Schwarzshild radius (rs). Region O is outside the event horizon. The question we ask is what is the probability the particle (red dot) will reach region O? And, we could also ask what is the probability the particle will fail to reach O and remain trapped in region I? These probabilities, when added together should equal 1.

Region I covers a distance (r) from zero to rs. Region O is from rs to infinity. These will be the boundaries we will be using in the integrals below.

To find a probability in quantum mechanics we are told to square the wave-function amplitude. To see how this works, let's consider finding the probability without squaring the wave function. (See equations 1 and 2 below.)

We can model the wave function using right triangles. This is appropriate, since sine and cosine represent waves.

At the second triangle above we substitute some dummy wave functions for demonstration purposes. The trigonometric proof below shows why it is important to square a wave function to get a probability:

At equations 3 and 4 we didn't square the wave functions--they rarely add up to one, so they can't be probabilities. However once they are squared, they add up to one and could be probabilities (see equations 5 through 9). You may have noticed the wave functions have negative exponents. This feature prevents an exponential blow up to infinity.

Now, to get the right values for the probabilities we also need to include a normalization factor (A). At equations 10 to 15 below, we calculate the value of A and see why we need it. When we take the sum of all probabilities, from zero to infinity, we want the grand total to be one.

Equation 16 below is our new-and-improved wave function. Equation 17 is the one we use to calculate the probability densities of regions I and O. Equations 18 through 22 yield our desired result: equation 23, the probability density of region O; i.e., the probability that the particle will successfully escape the black hole.

Equation 24 should be the probability density for the particle's failure to escape. Let's check this:

Equation 28 confirms and matches 24. Below we use Schrodinger's equation to find the value of the wave number k:

Equation 34 shows that k is a function of V--the black hole's potential. The bigger V is, the bigger k is, the smaller the wave function and the probability that the particle will escape.

For more on the topic of quantum tunneling and QM, I highly recommend Robert Eagle's (aka: DrPhysicsA) video series: