[Dave Lev (OP)]

In the case of a positron-electron pair, yes, one is negatively-charged and one is positively-charged.

OK
So do you finally agree that we discuss on Negative charged mass and not about Negative mass?

Stephen Hawking himself, for one. He mentions it in his book A Brief History of Time. The reason one particle has positive mass and the other negative mass is specifically because the first law of thermodynamics has to be obeyed. The particles were pulled out of a zero-energy state, so their total mass has to add up to zero. By one being positive and the other negative, this is accomplished.

I really don't understand why do you keep pushing this none realistic idea of Negative mass.
If you still want to hope that Stephen Hawking himself mentioned in his book A Brief History of Time that it is about negative mass (and not about negative charged mass), than please offer a direct web link to his statement.

We've already discussed all of this before, don't you remember? We even discussed why the negative mass particle has to be the one that falls into the hole.

As I have already stated, my intention was about Negative chraged mass.

[Kryptid]

So do you finally agree that we discuss on Negative charged mass and not about Negative mass?

In the case of a positron-electron pair formed during the Hawking process, one is negatively-charged and the other is positively-charged. One, however, must also have a negative mass while the other has a positive mass.

If you still want to hope that Stephen Hawking himself mentioned in his book A Brief History of Time that it is about negative mass (and not about negative charged mass), than please offer a direct web link to his statement.

I own A Brief History of Time and I will post the quote from the book once I get back to my house (that's where it is right now). If I'm not mistaken, Black Holes & Time Warps by physicist Kip S. Thorne says the same thing and I'm pretty sure I quoted it earlier in this thread (if I can find it...).

[Dave Lev (OP)]

In the case of a positron-electron pair formed during the Hawking process, one is negatively-charged and the other is positively-charged.

That is correct

One, however, must also have a negative mass while the other has a positive mass.

This is imagination!!!
In the article it is stated:
https://en.wikipedia.org/wiki/Antiparticle
"particle and antiparticle must have
the same mass m
the same spin state J
opposite electric charges q and -q."
Therefore, both have a positive mass.
Hence, would you kindly stop that none realistic idea of Negative mass, or offer a real article which supports this imagination...

[Kryptid]

This is imagination!!!
In the article it is stated:
https://en.wikipedia.org/wiki/Antiparticle
"particle and antiparticle must have
the same mass m
the same spin state J
opposite electric charges q and -q."
Therefore, both have a positive mass.

Under normal circumstances, you are correct. All normal matter has positive mass and pair production that occurs in places like particle accelerators has positive mass. However, the circumstances of the Hawking process are different because of the extreme nature of black holes. It's also only a relative thing. The particle only looks negative to a distant observer (anyone outside the event horizon).

Hence, would you kindly stop that none realistic idea of Negative mass, or offer a real article which supports this imagination...

Later tonight, when I have access to Hawking's book, I'll quote it.

This article from LiveScience does mention it, though: https://www.livescience.com/65683-sonic-black-hole-spews-hawking-radiation.html

Normally, after a pair of virtual particles appears, they immediately annihilate each other. Next to a black hole, however, the extreme forces of gravity instead pull the particles apart, with one particle absorbed by the black hole as the other shoots off into space. The absorbed particle has negative energy, which reduces the black hole's energy and mass. Swallow enough of these virtual particles, and the black hole eventually evaporates. The escaping particle becomes known as Hawking radiation.

Here is another web page describing the process: https://www.asc.ohio-state.edu/mathur.16/infopublic/info2.3.html

There is also something else I want to emphasize. The idea that virtual particle pairs pop out of the vacuum and are then split by tidal forces at the black hole's event horizon is only an analogy that is used by books in an attempt to illustrate the process to average people who are not experts in the physics involved. This page explains: http://math.ucr.edu/home/baez/physics/Relativity/BlackHoles/hawking.html

How does this work?  Well, you'll find Hawking radiation explained this way in a lot of "pop-science" treatments:

Virtual particle pairs are constantly being created near the horizon of the black hole, as they are everywhere.  Normally, they are created as a particle-antiparticle pair and they quickly annihilate each other.  But near the horizon of a black hole, it's possible for one to fall in before the annihilation can happen, in which case the other one escapes as Hawking radiation.

In fact this argument also does not correspond in any clear way to the actual computation.  Or at least I've never seen how the standard computation can be transmuted into one involving virtual particles sneaking over the horizon, and in the last talk I was at on this it was emphasized that nobody has ever worked out a "local" description of Hawking radiation in terms of stuff like this happening at the horizon.  I'd gladly be corrected by any experts out there...  Note: I wouldn't be surprised if this heuristic picture turned out to be accurate, but I don't see how you get that picture from the usual computation.

Honestly, I'm struggling to understand how the Hawking process actually works myself. Virtual particles don't seem to be a necessary component to the calculations at all. Instead, it seems to have something to do with the local positive and negative frequencies present in the vacuum around the event horizon: https://en.wikipedia.org/wiki/Unruh_effect

Regardless of the specifics of how it all works, there is one thing that is certain: the net result is that conservation of mass and energy are not violated. That is the most important thing to take home from all of this.

[Dave Lev (OP)]

The absorbed particle has negative energy,

Negative Energy means Negative energy charge.
It doesn't mean negative mass as you might hope for.
Any Energy - positive or Negative - must have real positive mass.
The idea of negative mass is none realistic in my point of view.

[Kryptid]

Negative Energy means Negative energy charge.
It doesn't mean negative mass as you might hope for.
Any Energy - positive or Negative - must have real positive mass.
The idea of negative mass is none realistic in my point of view.

It does mean negative mass/energy. Here is the relevant quote from Stephen Hawking's book about it:

Because energy cannot be created out of nothing, one of the partners in a particle/antiparticle pair will have positive energy, and the other partner negative energy. The one with negative energy is condemned to be a short-lived virtual particle because real particles always have positive energy in normal situations. It must therefore seek out its partner and annihilate with it. However, a real particle close to a massive body has less energy than if it were far away, because it would take less energy to lift it far away against the gravitational attraction of the body.

Normally, the energy of the particle is still positive, but the gravitational field inside a black hole is so strong that even a real particle can have negative energy there. It is therefore possible, if a black hole is present, for the virtual particle with negative energy to fall into the black hole and become a real particle or antiparticle. In this case it no longer has to annihilate with its partner. Its forsaken partner may fall into the black hole as well. Or, having positive energy, it might also escape from the vicinity of the black hole as a real particle or antiparticle (Fig. 7.8 ). To an observer at a distance, it will appear to have been emitted from the black hole. The smaller the black hole, the shorter the distance the particle with negative energy will have to go before it becomes a real particle, and thus the greater rate of emission, and the apparent temperature, of the black hole.

The positive energy of the outgoing radiation would be balanced by a flow of negative energy particles into the black hole. By Einstein's equation E = mc² (where E is energy, m is mass, and c is the speed of light), energy is proportional to mass. A flow of negative energy into the black hole therefore reduces its mass.

That last sentence makes it unambiguously clear that Hawking is talking about negative energy and not negative charge. This is only further reinforced by the fact that Hawking radiation can be composed of neutral particles like photons, neutrinos and gravitons that don't have any charge at all.

[Halc]

Negative Energy means Negative energy charge.

No it doesn't.  If you mean negative charge, then say negative charge.  Charge and energy are different things.
Gravitational potential energy is the obvious example of negative energy/mass, so it is quite real.

Any Energy - positive or Negative - must have real positive mass.

It would violate conservation of mass if this were true.

[Bogie_smiles]

As a member who knows nothing "for sure", I would submit that gravity expresses itself in waves traveling through space, and  they affect the speed of light based on the density of those waves in space. LIGO evidence shows that massive objects emit gravitational waves, and logic tells me that massive objects absorb gravitational waves. Any elucidation would be appreciated.

[Dave Lev (OP)]

No it doesn't.  If you mean negative charge, then say negative charge.  Charge and energy are different things.
Gravitational potential energy is the obvious example of negative energy/mass, so it is quite real.

I have found myself very confused with the meaning of Negative energy/mass.
Based on the following formula:
E = mc^2
A negative Energy means a negative mass (as c^2 is always positive).
However, in all/most of the articles that I have read - The moment of the pair creation is described as pair particles
One with Positive electrical charge while the other one has a Negative electrical charge.
So, I was wondering what the source is for that Negative Energy and what is the real difference between Negative electrical charge to Negative Energy.
In order to answer this question I have looked again on the following article:

This article from LiveScience does mention it, though: https://www.livescience.com/65683-sonic-black-hole-spews-hawking-radiation.html

In this article it is stated:
"(Antimatter particles have the same mass as their matter counterparts, but opposite electrical charge.)"
So, it is clear to me that when we discuss on Matter and Antimatter, or Particle and antiparticle we actually discuss on:
"the same mass as their matter counterparts, but opposite electrical charge"
However, if during the moment of creation we get Particle and Antiparticle (both with positive mass) then how suddenly we get the Negative energy out that?
This article gives the answer for this question:
"Normally, after a pair of virtual particles appears, they immediately annihilate each other. Next to a black hole, however, the extreme forces of gravity instead pull the particles apart, with one particle absorbed by the black hole as the other shoots off into space. The absorbed particle has negative energy, which reduces the black hole's energy and mass. Swallow enough of these virtual particles, and the black hole eventually evaporates. The escaping particle becomes known as Hawking radiation."
So, if I understand it correctly:
We consider the Negative energy only when the Negative electrical charge is falling into the BH:
"The absorbed particle has negative energy, which reduces the black hole's energy and mass."
Hence, as long as the antiparticle is out of the BH, It is considered as a negative electrical charged particle
However, when this antiparticle (with its negative electrical charge) falls into the BH it actually reduces the total energy/mass of the BH. Therefore our scientists consider the in falling Antiparticle as it has a Negative energy or Negative mass.
Do you agree with that?
Do you have better explanation?
Why in all/most of the articles when it comes to moment of the creation of pair production they clearly discuss on Positive/Negative electrical charged and not Positive/Negative Energy?

[Halc]

However, in all/most of the articles that I have read - The moment of the pair creation is described as pair particles
One with Positive electrical charge while the other one has a Negative electrical charge.

You misinterpret almost every sentence you read then.  Most created pairs are not charged at all.
Sgr-A (or any stellar black hole for that matter) is incapable of producing charged particles via Hawking radiation.  The energy required to create such massive particles is not there.
What happens more often is that the hot accretion disk (not the black hole) radiates light (reducing the mass/energy of the disk), and those photons gain energy (blue shift) as they get close to the black hole.  A pair of photons with sufficient positive mass/energy (over 1 MeV) might interact, resulting in this electron/positron pair, which has the same positive mass as the photons.  Yes, that's a combined positive energy like you say, but the energy was already there, having been removed from the accretion ring.  No new mass was created or destroyed.  It's just being moved around.
One of the pair might escape the black hole, and on average, as many positrons as electrons escape.  The positrons will interact with a different electron in the disk where they annihilate,  becoming photons again, for a net loss of mass to the accretion disk.

So, I was wondering what the source is for that Negative Energy and what is the real difference between Negative electrical charge to Negative Energy.

Charge is an EM property of any particle/system.  Most particles don't have charge, but all have mass/energy.  The two are completely different things.

"(Antimatter particles have the same mass as their matter counterparts, but opposite electrical charge.)"
So, it is clear to me that when we discuss on Matter and Antimatter, or Particle and antiparticle we actually discuss on:
"the same mass as their matter counterparts, but opposite electrical charge"

And since most/all the particles from Hawking radiation have zero charge, the opposite charge is also zero.  You're mistakenly taking a statement that the particles have opposite charge to mean that they have charge at all.  Hawking radiation from any known black hole is confined to photons and gravitons and such.  It involves virtual particles, and I am no expert on the mathematics involved, but there are good books/articles that explain it.  I don't think you are up to understanding those articles, given your consistent misrepresentation of even the most trivial physics.  I don't understand the articles, and they are not backed by a unified field theory, so their conclusions are in question.

However, if during the moment of creation we get Particle and Antiparticle (both with positive mass) then how suddenly we get the Negative energy out that?

You don't. Antimatter reactions go kablooey.  Lots of positive energy out of that.  No net change.

This article gives the answer for this question:
"Normally, after a pair of virtual particles appears, they immediately annihilate each other. Next to a black hole, however, the extreme forces of gravity instead pull the particles apart, with one particle absorbed by the black hole as the other shoots off into space. The absorbed particle has negative energy, which reduces the black hole's energy and mass. Swallow enough of these virtual particles, and the black hole eventually evaporates. The escaping particle becomes known as Hawking radiation."
So, if I understand it correctly:
We consider the Negative energy only when the Negative electrical charge is falling into the BH:

Energy and charge are completly different things.  The bit you quote above doesn't even mention charge.
It is in fact talking about virtual particles, not physical particles.  The former might have a combined mass of zero, which is why they can appear out of a zero energy state.

"The absorbed particle has negative energy, which reduces the black hole's energy and mass."
Hence, as long as the antiparticle is out of the BH, It is considered as a negative electrical charged particle

This is what I mean by you not being able to read any sentence without twisting it.  It says the negative energy particle falls in, not the anti-particle.  The positive mass particle that escapes might be matter or antimatter.  There should be no predominance of one over the other.

However, when this antiparticle (with its negative electrical charge)

There's no mention of an antiparticle or one with charge.  It says the negative energy particle falls in.  Remember: Energy and charge are completely different things.

Therefore our scientists consider the in falling Antiparticle as it has a Negative energy or Negative mass.
Do you agree with that?

About as completely wrong as possible.

Do you have better explanation?

Yes.  The quotes you gave is the better explanation.  Read them instead of replacing all the words with ones of your choice.

[Kryptid]

I have found myself very confused with the meaning of Negative energy/mass.

Think of units of energy like electron-volts, joules or calories. Instead of, say, 10 joules you would have -10 joules.

So, if I understand it correctly:
We consider the Negative energy only when the Negative electrical charge is falling into the BH:

You don't understand correctly. The particle that falls into the black hole could have positive charge, negative charge or no charge at all. The energy is what is negative, not the electric charge. The absorption of electric charge wouldn't reduce something's mass because mass and charge are different things.

Hence, as long as the antiparticle is out of the BH, It is considered as a negative electrical charged particle

How many times do I have to tell you that antimatter isn't always negatively-charge? Positrons are positively-charged.

Why in all/most of the articles when it comes to moment of the creation of pair production they clearly discuss on Positive/Negative electrical charged and not Positive/Negative Energy?

Because most articles talking about pair production talk about electron-positron pairs. Conservation of charge is just as important as conservation of energy.

In this article it is stated:
"(Antimatter particles have the same mass as their matter counterparts, but opposite electrical charge.)"
So, it is clear to me that when we discuss on Matter and Antimatter, or Particle and antiparticle we actually discuss on:
"the same mass as their matter counterparts, but opposite electrical charge" However, if during the moment of creation we get Particle and Antiparticle (both with positive mass) then how suddenly we get the Negative energy out that?

They are talking about "normal" circumstances. Both matter and antimatter have positive mass/energy under normal circumstances. Stephen Hawking already explained one circumstance that can make that normally positive mass/energy be negative:

Normally, the energy of the particle is still positive, but the gravitational field inside a black hole is so strong that even a real particle can have negative energy there.

[Dave Lev (OP)]

In this article it is stated:
"(Antimatter particles have the same mass as their matter counterparts, but opposite electrical charge.)"
So, it is clear to me that when we discuss on Matter and Antimatter, or Particle and antiparticle we actually discuss on:
"the same mass as their matter counterparts, but opposite electrical charge" However, if during the moment of creation we get Particle and Antiparticle (both with positive mass) then how suddenly we get the Negative energy out that?

So, those scientists are talking about "normal" circumstances. In this case, both matter and antimatter have positive mass/energy under normal circumstances.

As they are talking about "normal" circumstances, why I also can't talk about "normal" circumstances?
Why do you push that discussion into the direction of none normal circumstances?
I would like to remind you that in my theory both matter and antimatter have to have positive mass/energy under normal circumstances.
So, why those scientists can claim that Both matter and antimatter have positive mass/energy under normal circumstances while you both insist that it is forbidden for me to use the same "normal circumstances"?

[Halc]

In this case, both matter and antimatter have positive mass/energy under normal circumstances.

As they are talking about "normal" circumstances, why I also can't talk about "normal" circumstances?
Why do you push that discussion into the direction of none normal circumstances?

You're talking about the behavior of physics near an event horizon, a singularity.  Physics doesn't act normally at or across event horizons.

[Kryptid]

Why do you push that discussion into the direction of none normal circumstances?

The conditions inside of a black hole's event horizon are not normal.

So, why those scientists can claim that Both matter and antimatter have positive mass/energy under normal circumstances while you both insist that it is forbidden for me to use the same "normal circumstances"?

It's not forbidden so long as you are actually talking about normal physics conditions. By merely discussing black holes, you are entering "abnormal" territory.

[Dave Lev (OP)]

The conditions inside of a black hole's event horizon are not normal.

OK
Based on that, the conditions outside of a black hole's event horizon are normal.
Is it correct?

[Kryptid]

OK
Based on that, the conditions outside of a black hole's event horizon are normal.
Is it correct?

It depends on how near the horizon you are. In Black Holes & Time Warps, it is stated:

Then, in 1974, came a great surprise: Hawking inferred as a by-product of his discovery of black-hole evaporation (Chapter 12) that vacuum fluctuations near a hole's horizon are exotic: They have negative average energy density as seen by outgoing light beams near the hole's horizon. In fact, it is this exotic property of the vacuum fluctuations that permits the hole's horizon to shrink as the hole evaporates, in violation of Hawking's area-increase theorem. Because exotic material is so important for physics, I shall explain this in greater detail.

Recall the origin and nature of vacuum fluctuations, as discussed in Box 12.4: When one tries to remove all electric and magnetic fields from some region of space, that is, when one tries to create a perfect vacuum, there always remain a plethora of random, unpredictable electromagnetic oscillations- oscillations caused by a tug-of-war between the fields in adjacent regions of space. The fields "here" borrow energy from fields "there," leaving the fields there with a deficit of energy, that is, leaving them momentarily with negative energy. The fields there then quickly grab the energy back and with it a little excess, driving their energy momentarily positive, and so it goes, onward and onward.

Under normal circumstances on Earth, the average energy of these vacuum fluctuations is zero. They spend equal amounts of time with energy deficits and energy excesses, and the average of deficit and excess vanishes. Not so near the horizon of an evaporating black hole, Hawking's 1974 calculations suggested. Near a horizon the average energy must be negative, at least as measured by light beams, which means that the vacuum fluctuations are exotic.

I'm not sure where the cut-off point is. That is, I'm not sure how far away you have to be from the horizon before the physics goes back to "normal". The fact of the matter is this, however: Hawking radiation must obviously be formed in a region of space close enough to the horizon where negative energy can exist. That's the only way that the first law of thermodynamics can be preserved. The energy of both particles must add up to zero. The only way to do that if one particle has positive energy would be if the other has an equal amount of negative energy.

[Dave Lev (OP)]

I'm quite confused:
In one hand you claim that

Hawking radiation must obviously be formed in a region of space close enough to the horizon where negative energy can exist. That's the only way that the first law of thermodynamics can be preserved. The energy of both particles must add up to zero. The only way to do that if one particle has positive energy would be if the other has an equal amount of negative energy.

Hence, if "The energy of both particles must add up to zero. The only way to do that if one particle has positive energy would be if the other has an equal amount of negative energy" than you actually claim that at any pair production there must be a negative energy.
On the other hand you claim that the Normal condition depends on:

It depends on how near the horizon you are.

Would you kindly advice clearly when can we get the Normal condition?
Do you agree that in order to get the Normal conditions it must be close enough to the innermost side of the accretion disc (As further away there is no feasibility for the pair production activity)
If you don't agree with that, than please specify exactly the location for Antimatter particles that have the same mass as their matter counterparts, but opposite electrical charge.

[Halc]

Would you kindly advice clearly when can we get the Normal condition?

For one, the normal condition requires the energy to already be there.  So there is say one or two photons with the same mass as a pair of electrons.

Do you agree that in order to get the Normal conditions it must be close enough to the innermost side of the accretion disc (As further away there is no feasibility for the pair production activity)

Pair production happens on Earth, so obviously it doesn't need to be close to a disk. It can occur anywhere where the energy required is present. But yes, the inner side of the disk has high enough energy that the high-energy photons are emitted.  This removes the mass of those photons from the disk, as does any radiation.  Our sun is similarly continuously getting less massive due to it radiating away its mass.

If you don't agree with that, than please specify exactly the location for Antimatter particles that have the same mass as their matter counterparts, but opposite electrical charge.

The one article says the typical pair production occurs 'near' the black hole, which in the case of Sgr-A is perhaps 100,000 km from the event horizon.  Photons gain energy as they fall into the gravity well and apparently reach sufficient energy at that distance.

Virtual particle pair production can (and apparently does) happen just about anywhere since the sum of the masses is zero and hence has no requirement for the presence of high energy photons or anything.  Such pairs are almost immediately annihilated as fast as they are produced, with no resulting 'bang' like you'd get with a positive-mass antimatter collision.

[Dave Lev (OP)]

For one, the normal condition requires the energy to already be there.

Thanks
So, you agree that if there is energy, we will get Antimatter particles that have the same mass as their matter counterparts, but opposite electrical charge.
That is perfect
The question is: What is the source for this energy?
If there is no energy than it is quite clear that it must come from something.
However, If there is an energy, than we don't need to deduct any mass from any nearby object.
The gravitational Energy is an excellent energy source for the new creation:
https://en.wikipedia.org/wiki/Hawking_radiation
 "the particle–antiparticle pair was produced by the black hole's gravitational energy"
Hence, the Gravitational energy is already there.
So, why can't we assume a Normal circumstances outside the event horizon while the energy is coming from the gravitational energy?
Please be aware that it is also stated:
https://en.wikipedia.org/wiki/Pair_production
"the created particles shall have opposite values of each other. For instance, if one particle has electric charge of +1 the other must have electric charge of −1, or if one particle has strangeness of +1 then another one must have strangeness of −1."
https://en.wikipedia.org/wiki/Positron
"The positron or antielectron is the antiparticle or the antimatter counterpart of the electron. The positron has an electric charge of +1 e, a spin of 1/2 (same as electron), and has the same mass as an electron. "
So, there is a possibility for the  pair-production with the same positive mass at each particle, but with opposite electrical charge.
They don't say even one word about negative mass or deduct the mass/energy from the BH.
In ALL the articles that I have found (except of the one that you have offered) our scientists do not claim for Negative Energy/mass.
So, why do you consider that the scientists which wrote about the negative energy/mass are much more cleaver than all the others?
With regards to Negative mass:
https://en.wikipedia.org/wiki/Negative_mass
"In theoretical physics, negative mass is matter whose mass is of opposite sign to the mass of normal matter, e.g. −1 kg."
It is stated: "In theoretical physics..."
So does it mean that we have never verified a Negative mass?
Do we have any real observation for Negative mass?
Actually, it seems to me that even negative energy should have some sort of mass.
If it has mass, it must be a positive mass.
Did we ever found a Negative mass that sets a negative gravity?
If we can't see that Negative mass, why do you push in that direction?

[Halc]

So, you agree that if there is energy, we will get Antimatter particles that have the same mass as their matter counterparts, but opposite electrical charge.

You make it sound definite.  There's a probability to it.  It might transform into matter from non-matter.  If it does, it will create matter and antimatter in equal amounts, and if it has charge, the net charge will be zero.  It does not always create charged particles.

The question is: What is the source for this energy?

The photon(s) I mention, and that the articles mention.  The photon might come from the accretion disk, in which case it is carrying mass away from the disk.

However, If there is an energy, than we don't need to deduct any mass from any nearby object.

The energy needs to have come from somewhere, so you deduct the energy from that thing.

The gravitational Energy is an excellent energy source for the new creation
https://en.wikipedia.org/wiki/Hawking_radiation

No known black hole is capable of radiating charged particles.  They're fair more massive than the faint Hawking radiation of even the smallest known black holes.
So it generates some low energy photons and some gravitons and such, which don't even have antimatter equivalents.  In doing so, the mass of the black hole is reduced by the energy of the radiation.

This radiation is theoretical since it is far too weak to be seen, even without the relatively blinding light from all the material falling in the opposite direction.

Hence, the Gravitational energy is already there.

Strictly speaking, gravity is not energy.  It is an acceleration field.  Earth gravity is expressed as acceleration (m/sec˛), not in joules.

Please be aware that it is also stated:
https://en.wikipedia.org/wiki/Pair_production
"the created particles shall have opposite values of each other. For instance, if one particle has electric charge of +1 the other must have electric charge of −1, or if one particle has strangeness of +1 then another one must have strangeness of −1."
https://en.wikipedia.org/wiki/Positron
"The positron or antielectron is the antiparticle or the antimatter counterpart of the electron. The positron has an electric charge of +1 e, a spin of 1/2 (same as electron), and has the same mass as an electron. "
So, there is a possibility for the  pair-production with the same positive mass at each particle, but with opposite electrical charge.

Yes, but not from Hawking radiation, which has far too little energy to create such massive things.

They don't say even one word about negative mass or deduct the mass/energy from the BH.

Go find an article about Hawking radiation then.  An article about pair production isn't even talking about black holes.

With regards to Negative mass:
https://en.wikipedia.org/wiki/Negative_mass
"In theoretical physics, negative mass is matter whose mass is of opposite sign to the mass of normal matter, e.g. −1 kg."
It is stated: "In theoretical physics..."
So does it mean that we have never verified a Negative mass?

We've never found a real object with negative rest mass, no.  Virtual particles are not real objects, but the mathematics of virtual particles has been verified.

Do we have any real observation for Negative mass?

Not real, no.  No observation of Hawking radiation either.

Actually, it seems to me that even negative energy should have some sort of mass.

It has negative energy, yes.  Gravitational potential energy for instance is negative.  There is very much real observation of negative energy.  It just has never been manifested as matter with negative mass.