Question regarding the nature of mass inside a black hole

[PeterDonis]

The observer sees the mass-energy M fall uneventfully to the horizon. He also sees the same mass-energy M collected by the shell.

Not at the same time. First the observer sees M fall uneventfully to the horizon, Then, much later, he sees it collected by the shell. He never sees it in both places at once.

 

[PeterDonis]

In the folio F3 of the diagram, the geometry reflects both the mass M which falls to the horizon, and also the same mass M contained in the shell. This contradicts the conservation of energy.

First, there is no global conservation of energy in GR. Energy conservation is local: the covariant divergence of the stress-energy tensor is zero at every event. That is satisfied by the solution you are describing.

It is true that, because of the way the geometry works inside the horizon, you can find spacelike hypersurfaces on which the infalling matter region appears (far inside the horizon) and a region containing Hawking radiation also appears (outside the horizon). However, you can also find other spacelike hypersurfaces for which this is not the case. As the article you linked to notes, there is no way to pick out any particular set of spacelike hypersurfaces as the "correct" or "true" one.

 

[Heikki Tuuri]

the forming
horizon
______/ quantum E
_____/_/_____ F4 folio
_\_\/_/______ F3 folio
__\/\/_______ F2 folio
__/\_\_______ F1 folio
_/ B1 B2
mass M

The diagram I drew in the previous message tells us what should happen, to keep the geometry consistent for a collapsing mass M.

Let us first think about a classic non-evaporating black hole. The baryons B1, B2 fall into the singularity. There are no outcoming photons. The horizon forms at the folio F3 as a perfect straight 45 degree line to northeast.

Let us then add a quantum of energy E of Hawking radiation. The quantum is present in the folio F4. How do we modify the diagram, so that the geometry for an outside observer stays the same? Birkhoff's theorem states that the gravitational field of the collapsing spherical mass M must stay constant at all times.

If we add another quantum with energy -E falling into the horizon, then Birkhoff will be happy:

-E \/ E

A problem in this is that no one has been able to formulate a mathematical model where quanta of negative energy pop up and are swallowed by the black hole. In Hawking's 1975 paper, the quanta E gain their energy at the forming horizon. The energy E is the negative frequencies in a hypothetical wave packet which travels through the collapsing mass. There is no negative energy quantum in the calculation.

Anyway, let us assume that there is a process which sends the energy -E down to the horizon.

If E gains its energy at the intersection of the lines B2 and E, then B2 is accompanied by an energy -E falling with B2 to the horizon. Actually, this description fits ordinary radiation emitted by B2. No horizon ever forms because all the mass-energy of B2 is canceled by the Hawking quanta -E.

For a horizon to form, the quantum E must gain its energy at a folio later than F2. In Hawking's 1975 paper, the energy E is gained at the folio F3, because the horizon has just formed there. But then the negative quanta -E would fall into the horizon in one huge batch, and the horizon would disappear at that instant.

A peaceful evaporation of a black hole is possible if the quanta E gain their energy at very late folios and send the negative energy -E down to the horizon. But in Hawking's 1975 paper, we do follow the path of E back in time all the way to the distant past. The path is extremely close to the forming horizon at the folio F3. That is the natural place where the energy E is gained.

The title of this thread is "The nature of mass inside a black hole". If there is no Hawking radiation, then the mass is in a singularity and we do not know anything about the nature there.

If Hawking radiation exists, then we face many additional problems. A few of those problems are presented in this message. The information loss problem and the firewall come on top of that.

The geometry outside a horizon is not affected in any way by what happens behind the horizon, because no event behind the horizon is in the light cone of an outside observer. The observer does not need to know anything about the nature behind the horizon. Currently, there is no consensus about what happens to an observer who jumps into the horizon.

I will look at the AdS/CFT black hole. As far as I know, no one has been able to decipher there what is the backreaction to hypothetical black hole evaporation.

There is one good reason to believe that Hawking radiation exists: thermodynamics. The "derivations" of Hawking radiation are riddled with problems.

...

UPDATE: It just occurred to me that if the quanta of negative energy -E are moving outward, just like the quanta E of positive energy, then the diagram would allow the creation of a horizon. Then there would be a positive mass M behind the horizon, a negative mass -M very slowly falling into the horizon, and a positive mass M slowly escaping the black hole.

This would also solve the problem of the conservation of momentum: if Hawking quanta exist, from where do they get the momentum p to the direction out of the horizon? The momenta of -E and E would cancel each other. The black hole would swallow the momentum -p as well as the energy -E.

The diagram which describes the creation of quanta -E and E should look like this:

-E // E

The new diagram stresses that the quanta move at the speed of light upward from the horizon. The quanta -E very slowly approach the singularity. Slowly, because they are moving at the speed of light upward. The quanta -E are below the Schwarzschild radius. Light moving straight out there will end up at the singularity.

An observer falling into the black hole would encounter a huge mass -M of negative energy quanta at the horizon. Does he see them? He does not see upcoming Hawking quanta.

 

[PeterDonis]

Birkhoff's theorem states that the gravitational field of the collapsing spherical mass M must stay constant at all times.

Not in this case, it doesn't. Birkhoff's theorem applies to a spherically symmetric vacuum region of spacetime. If there is Hawking radiation present, the region of spacetime exterior to the collapsing matter is not vacuum.

If we add another quantum with energy -E falling into the horizon

Real quanta can't have negative energy.

Please review the PF rules on personal speculation, which is what you are verging on here.

 

[PeterDonis]

The OP question has been addressed. Thread closed.