As shown on the previous page, there is only one way to eliminate a black hole from this world according to the existing theory.
Now, can we really eliminate a black hole using this method?
We will confirm this below. (Note 1)
Recipe for eliminating a black hole (= procedure)
The unpleasant thing about eliminating a black hole is that it possesses two types of momentum, namely, momentum and angular momentum, just before its disappearance.
In this state, the momentum, angular momentum, and energy must all be reduced to zero in the final Hawking radiation.
It is impossible to achieve such a miraculous thing in reality.
Therefore, it is desired that the angular momentum of the black hole has already been reduced to zero just before its disappearance.
If this can be achieved, the situation where the virtual particle that jumps into the black hole at the final stage of its elimination only needs to reduce the momentum and energy of the black hole to zero will be created.
By the way, in this case, energy must of course be relative energy, and this requires setting a coordinate system. In this case, the coordinate system is set to the reference inertial system (= CMB rest frame). (Note 2)
So if that is the case, we just need to set the angular momentum of the black hole to zero relative to the reference inertial system.
First, let's assume a black hole that emits Hawking radiation randomly and decreases in mass from over 100 grams to about 100 grams. Then let's say that the black hole continues to emit Hawking radiation randomly and its mass decreases to 10 grams.
Thanks to the randomly generated Hawking radiation, the black hole now has non-zero momentum and angular momentum. (Note 3)
So when the mass of the black hole is reduced from 10 grams to 1 gram, we precisely control the virtual particles that jump into the black hole to stop its rotation. In other words, we inject the virtual particles in the opposite direction of the black hole's rotation.
By doing so, the angular momentum of the black hole could be reduced to zero when it reaches 1 gram. From then on, the virtual particles that are injected into the black hole must be precisely aimed at the center of the black hole. Otherwise, the angular momentum of the black hole, which has just been reduced to zero, will be increased again.
Now, we need to reduce the mass of the black hole from 1 gram to the Planck mass (= mp). (Note 4) The key here is to avoid increasing the momentum of the black hole as much as possible. This means injecting virtual particles into the black hole in the opposite direction to its motion.
In this way, we can reduce the mass of the black hole while minimizing the increase in its momentum. (Note 5)
When the mass of the black hole is reduced to nearly 1 Planck mass, its diameter becomes 4 Planck lengths. (Note 6)
Now, the Schwarzschild radius Rs is proportional to the mass of the black hole, so when the mass of the black hole decreases to 0.25 Planck mass, its diameter becomes 1 Planck length. Assuming the size of a particle to be about 1 Planck length, the point at which the diameter of the black hole reaches 1 Planck length is the smallest size at which Hawking radiation can still occur, as there are no virtual particles that can enter the black hole once its size becomes smaller than 1 Planck length. (Note 7)
Let PBH denote the momentum of the black hole at this point. Then, the momentum of the virtual particles that need to be injected into the black hole should also be PBH, in the opposite direction to the black hole's motion.
The remaining problem is the relativistic energy of the black hole. The relativistic energy EBH of the black hole at this point can be expressed as follows:
EBH = sqrt(PBH^2C^2 + (0.25mp)^2*C^4) ... equation (1)
Therefore, the relativistic energy E (virtual particle) of the virtual particle that needs to be injected into the black hole must also be:
E (virtual particle) = sqrt(PBH^2C^2 + (0.25mp)^2*C^4) ... equation (2)
At this point, the momentum of the virtual particle that needs to be injected into the black hole is the same as the relativistic energy of the black hole, as shown in equation (1), but in the opposite direction to the black hole's motion.
The problem here is the size of the rest mass of the virtual particle that needs to be injected into the black hole.
As shown in equation (2) above, the rest mass of the virtual particle at this point must be (0.25mp), which is 0.00545 milligrams.
However, there is no elementary particle with such a huge mass. (Note 8)
Therefore, even with such precision-controlled Hawking radiation, it is impossible to make the black hole disappear in the end. (Note 9)
So here is the conclusion.
Randomly occurring Hawking radiation cannot make a black hole disappear.
Even with precision-controlled Hawking radiation, a black hole cannot be eliminated.
Thus, it is confirmed that "the only remaining method for black hole disappearance is actually impossible to achieve" = "Black Hole (BH) Annihilation Impossibility theorem is strictly valid."
Note 1: Let's put aside the fact that we know "no matter how much we try, it is impossible", which was discussed so far.
Also, we will assume that there is no claim that we cannot control the production of virtual particle pairs.
Now, under this assumption, the question becomes "Is it possible to achieve Hawking radiation, which is the only way to eliminate a black hole?" = "Is Black Hole (BH) Annihilation Impossibility theorem invalid?"
However, in this discussion, we use the condition that "elementary particles have a finite size including virtual particles" and "when the size of the black hole becomes smaller than the size of elementary particles, Hawking radiation stops".
By the way, if we remove this condition and ask whether the black hole disappears, another restriction of "elementary particles have a rest mass" is imposed, so the situation regarding the disappearance of the black hole is not so advantageous. (This will be explained later.)
Note 2: It is an objectively existing stationary system in the place where the target black hole exists.
Note 3: In order to emit Hawking radiation, virtual particles must first jump into the black hole.
The virtual particles bring momentum and angular momentum to the black hole.
By the way, the angular momentum is brought to the black hole by the virtual particle that jumps into the black hole and flies towards a position away from the center of the black hole.
Note 4: According to "Planck Mass (mp)": https://archive.md/sRXXo, 1 Planck mass is 0.0218 milligrams.
"1 Planck mass is about the mass of a piece of copy paper (weight per unit area: 64 g/m2) cut to a size of 1mm x 0.3mm."
It is the weight of visible-sized paper scraps or garbage.
Note 5: I will not go into detail about the reason here (I will explain it later), but not moving the BH as much as possible makes it easier for the BH to emit Hawking radiation from then on.
Conversely, "a BH that is moving around emits Hawking radiation less frequently."
Note 6: According to "Planck length": https://archive.md/IOCb9, the Schwarzschild radius rs at Planck mass mp is rs = 2*lp.
Here, lp is the Planck length.
In other words, the diameter of a BH with Planck mass is 4*lp, which is four times the size of Planck length.
Note 7: In this discussion, we introduce the assumption that the virtual particles produced have a finite size, and Hawking radiation stops when the size of the BH becomes smaller than that of the virtual particles.
Note 8: 0.00545 milligrams is 3x10^18 times the mass of a proton.
We can confidently say that such particles with this mass do not exist in the universe we live in.
This is the next wall that prevents us from eliminating a BH, even if someday humanity could precisely control the generation of virtual particle pairs.
Note 9: As a result of attempting to eliminate a BH according to this recipe, the remaining BH can be described as having zero momentum and angular momentum, meaning it can be considered "at rest relative to the inertial frame of reference", and has a mass of approximately 0.00545 milligrams (= 0.25 mp).
This BH will no longer emit Hawking radiation and will remain stable in the universe.
Postscript: The above discussion assumes a particle size of about one Planck length, but even if the size is 0.5 Planck lengths or 0.1 Planck lengths, the discussion can be modified slightly and still be essentially the same.
In other words, "as long as particles have finite size, BHs cannot be eliminated through Hawking radiation."
Furthermore, to eliminate a BH in the last remaining Hawking radiation event, the virtual particle mass required must be the same as the mass of the BH to be eliminated. However, particles with such enormous mass do not exist, and therefore, virtual particles with such enormous mass do not exist either. Therefore, the conclusion is that it is impossible to eliminate this BH.
Postscript 2: The process by which a primordial black hole (PBH) generated beyond the Planck scale becomes dark matter.
This PBH emits Hawking radiation from the moment it is born, gradually decreasing its mass to the Planck level.
When its mass approaches 0.25 mp, it waits for the next Hawking radiation event with the appropriate momentum and angular momentum, which will be the last such event for the BH.
By this random event, the BH's mass will be reduced to below 0.25 mp.
However, this last Hawking radiation event does not reduce the momentum and angular momentum of the BH to zero.
Therefore, the remaining BH has a mass slightly below 0.25 mp and non-zero momentum and angular momentum at the Planck level.
This BH will no longer emit Hawking radiation and will remain stable in this state until the end of the universe.
Moreover, this stable PBH at the Planck level is the true identity of dark matter.
By the way, if a PBH with a mass below 0.25 mp is born at the beginning of the universe, it will become dark matter without emitting any Hawking radiation.
Thus, a PBH that does not emit Hawking radiation from the beginning is the optimal candidate for cold dark matter (CDM).
According to inflationary cosmology, the lower the mass of the primordial black hole, the more frequently it is generated, which is a favorable situation for the claim that PBHs beyond the Planck scale are CDM.
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