I Question regarding the nature of mass inside a black hole

[Heikki Tuuri]

PAllen, thank you for the link to the

https://arxiv.org/abs/1709.00115

Arderucio-Costa & Unruh paper. I had not seen that paper. The paper is very technical and I did not yet check the math in it.

Let us think about the backreaction. A consensus is that a freely falling observer sees no drama at the forming horizon. The energy flux of hypothetical Hawking radiation is almost invisible for the freely falling observer.

On the other hand, static observers close to the horizon do see the Hawking energy flux. According to the 1975 paper by Hawking, the energy flux comes from negative frequencies which are generated by the rapidly changing gravitational field in a hypothetical wave packet which has passed through the star just before the formation of the horizon.

Static observers do see the entire mass M of the collapsing star converted to Hawking radiation. The radiation is moving up just above the forming horizon. It may take a photon 10^67 years to climb up. That is the reason why the evaporation is so slow.

A freely falling observer, on the other hand, sees almost no Hawking radiation. He sees the entire mass M falling ahead of him, first toward the forming horizon, and then toward the singularity.

Now we have a problem because static observers do see the falling mass M. It never disappears from their incoming light cone. Static observers also see the upcoming Hawking radiation. It contains the mass M, too. The total mass they see is 2M, which makes no sense.

Some of the mass M is converted to outgoing radiation through conventional processes in the accretion disk: friction and collisions. There is no paradox in that radiation. The paradox is in the bulk of matter M which seems to become duplicated at the horizon.

Leonard Susskind has tried to fix the paradox with something he calls black hole complementarity. The firewall paradox of the AMPS paper was a serious blow to complementarity.

My current view is that Hawking radiation does not exist. That would solve most paradoxes. I need to check what implications that would have for AdS/CFT and black hole thermodynamics.

 

[PeterDonis]

A consensus is that a freely falling observer sees no drama at the forming horizon.

Except for proponents of firewall models.

static observers close to the horizon do see the Hawking energy flux.

But they interpret this as Unruh radiation; the black hole horizon is also the Rindler horizon for these observers. They have no way of knowing, locally, that this radiation will over a very long period of time decrease the mass of the hole; for all they know it could be entirely due to their own acceleration, as Unruh radiation is.

static observers do see the falling mass M. It never disappears from their incoming light cone.

This is not correct. Once the falling mass is inside the horizon, it is outside the past light cone of any observer (static or not) outside the horizon.

The total mass they see is 2M

This is not correct. See above.

What the static observer actually sees is an infalling mass M disappearing from his past light cone, and then, much, much later, the same total amount of mass appearing over a long, long time as Hawking radiation. There is no issue with energy conservation at all.

 

[Heikki Tuuri]

PeterDonis,

https://phys.libretexts.org/Bookshelves/Relativity/Book:_General_Relativity_(Crowell)/7:_Symmetries/7.3:_Penrose_Diagrams_and_Causality

if you draw the Penrose diagram of a collapsing star, you notice that the falling matter remains in the light cone of a far-away observer. That is, for every future time t of the outside observer, there is a straight line 45 degrees up to the right, such that the line intersects the worldline of the observer at his proper time t. The line, of course, is outside the forming horizon. This means that the mass remains in his light cone.

If the mass inside the horizon keeps decreasing, then the line is not straight, but it nevertheless is a geodesic, a path of light.

The information problem of black holes is this paradox: how can the same matter and information fall behind the horizon in the Penrose diagram and be duplicated in the hypothetical Hawking radiation outside the horizon?

The "backreaction" to Hawking radiation should explain how the mass-energy of the black hole decreases. I will look at the AdS/CFT black hole and try to decipher the backreaction there.

UPDATE: In the Penrose diagram, let us draw the worldlines of baryons B1 and B2 falling toward the horizon. The lines go up left at an angle of, say, 50 degrees from the horizontal line.

horizon
______/ photon
_____/_/_____ F4 folio
_\_\/_/______ F3 folio
__\/\/_______ F2 folio
__/\_\_______ F1 folio
_/ B1 B2

If the baryons happen to collide, then a free falling observer sees them emit a photon and he sees the combined mass-energy of B1 & B2 decrease. A late time observer sees the same thing: he receives the photon and sees the system B1 & B2 lose some kinetic energy or heat.

The geometry of spacetime at the observer is determined from the information within his light cone. The geometry correctly reflects the fact that energy was conserved in the system B1 & B2 & photon.

If B1 and B2 fall uneventfully to the horizon, then the geometry is the classical black hole.

Let us add an energy flux of Hawking radiation. We may imagine a shell around the black hole close to the horizon. The shell collects all positive energy from Hawking radiation.

The observer sees the mass-energy M fall uneventfully to the horizon. He also sees the same mass-energy M collected by the shell. Free falling observers see these same things: the mass M is in the falling matter and the mass M is also in the shell.

General relativity calculates the geometry in the next folio of spacetime from the information in the previous folio. The input is the geometry in the previous folio plus the energy density at each point, measured by a free falling observer.

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.

 

[PeterDonis]

for every future time t of the outside observer, there is a straight line 45 degrees up to the right, such that the line intersects the worldline of the observer at his proper time t. The line, of course, is outside the forming horizon. This means that the mass remains in his light cone.

This is correct, but it does not mean what you claim it means.

Here is what it does mean: (1) For every event on the worldline of the outside observer, there is infalling matter present somewhere in the past light cone of that event (though you might have to go back very, very far in time to reach it).

Here is what it does not mean: (2) For every event in the spacetime region occupied by the infalling matter, there is an outgoing null worldline that originates at that event and reaches the worldline of the outside observer. This is obviously false for the portion of the infalling matter region that is at or below the horizon.

You are pointing out that statement #1 above is true (which it is), but the argument you are trying to make requires statement #2 above to be true (and it isn't).

If the mass inside the horizon keeps decreasing

This is not a correct description of what is happening. The region occupied by the infalling matter inside the horizon is inside the horizon; in a Penrose diagram of a spacetime with a black hole that evaporates by Hawking radiation (such as the one appearing in Hawking's original paper), statement #2 above is still false, and the Hawking radiation therefore cannot be coming from the region occupied by infalling matter inside the horizon. That matter region ends at the future singularity, just as it does in the Penrose diagrams shown in the article you linked to.

However, the "mass" of the black hole at times long after the original gravitational collapse is not obtained by looking at the infalling matter region. It's obtained by looking at the vacuum region outside the horizon and measuring its spacetime geometry. In other words, "mass" is a global property of the geometry. And so is "decreasing mass due to Hawking radiation" for that case.

The information problem of black holes is this paradox: how can the same matter and information fall behind the horizon in the Penrose diagram and be duplicated in the hypothetical Hawking radiation outside the horizon?

And one answer (the answer Hawking originally gave, although his opinion changed much later on) is that the Hawking radiation does not contain the information in the infalling matter: that the information in the infalling matter is lost when it hits the singularity. Many physicists don't like this answer because it violates unitarity, but that just means they prefer one counterintuitive thing (figuring out how unitarity is preserved) to another (figuring out how unitarity can be violated).

 

[PeterDonis]

General relativity calculates the geometry in the next folio of spacetime from the information in the previous folio.

This is one way to find a global solution in GR, but by no means the only one.

 

[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.