B Crossing the Bekenstein Bound at Black Hole Event Horizon

[.Scott]

TL;DR Summary

I'm very unclear on how the Bekenstein Bound is enforced during EH crossings.

The Bekenstein Bound places a upper limit on the amount of entropy that a given volume of space may contain.
This limit was described by Jacob Bekenstein who tied it quite closely to the Black Hole Event Horizon.

Put simply, black holes hold the maximum entropy allowed for their volume. If you drop more stuff into the BH, it gets a little bigger - just enough to accommodate that additional information burden.

So I pick a black hole that's big enough to allow me to comfortably cross its event horizon while momentarily avoiding that sinking feeling of tidal spaghettification. And I ponder the mass of the black hole that remains below me.

I do not doubt that I will fall "towards the center", but as I fall "towards the center" how can the diameter of the space around me and the diameter of the object below me shrink unless I fall pass other falling mass in the process?

What is different about the interior of a black hole that allows it to ignore the Bekenstein Bound?

 

[PeterDonis]

I ponder the mass of the black hole that remains below me.

There is no such thing. Once you are inside the hole, there is no sense in which some of its mass is "above" you and some is "below" you.

how can the diameter of the space around me and the diameter of the object below me shrink

There are no such things; your ordinary concepts of "the space around me" and "the object" do not work inside a black hole. Once you are inside the hole, it has no well-defined "size" and, as above, there is no sense in which some of it is "above" you while the rest of it is "below" you.

 

[PeterDonis]

The Bekenstein Bound places a upper limit on the amount of entropy that a given volume of space may contain.

While this is a common heuristic description, it's actually not correct. The bound is actually an upper bound on the entropy that can be associated with a given amount of energy enclosed by a 2-sphere with a given area. (In the usual formula, a "radius" $R$ appears, but this is an "areal radius", i.e., it is defined by the area of a 2-sphere enclosing the object, not its volume.)

For ordinary objects the difference is not a big deal, since there is a well-defined connection between area and volume (in GR you have to allow for the non-Euclideanness of space when deriving the connection, but that's easy to do). But there is no such connection in the interior of a black hole; as soon as you are dealing with 2-spheres inside a black hole with areas smaller than the horizon area, those 2-spheres do not enclose any well-defined volume. That is the geometric reason behind the statements I made in post #2 above.

 

[.Scott]

Once you are inside the hole, there is no sense in which some of its mass is "above" you and some is "below" you.

With the help of friends, I will recast my description below and get rid of "above" and "below".

There are no such things; your ordinary concepts of "the space around me" and "the object" do not work inside a black hole. Once you are inside the hole, it has no well-defined "size"...

When I recast my description, I will still be using "distances". But I think that is fair. After all, I am supposedly closing in on the singularity (getting "closer").

The bound is actually an upper bound on the entropy that can be associated with a given amount of energy enclosed by a 2-sphere with a given area. (In the usual formula, a "radius" $R$ appears, but this is an "areal radius", i.e., it is defined by the area of a 2-sphere enclosing the object, not its volume.)

I am actually depending on this.

So here is my recast:
I get a thousand friends (or perhaps enemies) and we spread ourselves out evenly around the event horizon and allow ourselves to drop through the event horizon at the same time (from the appropriate reference frame) and at the same speed. As we all cross through the even horizon, we keep track of the distances between pairs of adjacent friends.
At the time we cross through the event horizon, that distance will work out to approximately the minimum 2-sphere area for the singularity energy, thus this distance can never get smaller until we reach the singularity. But it's unclear to me whether there would really be a singularity. If my friends are keeping their distance, why the crunch?

 

[PeterDonis]

I am supposedly closing in on the singularity (getting "closer").

Not in the sense of spatial distance. The singularity is a moment of time, not a place in space.

I get a thousand friends (or perhaps enemies) and we spread ourselves out evenly around the event horizon and allow ourselves to drop through the event horizon at the same time (from the appropriate reference frame)

What reference frame? How are you going to ensure this?

and at the same speed.

Speed relative to what?

As we all cross through the even horizon, we keep track of the distances between pairs of adjacent friends.

How are you going to keep track of these distances?

At the time we cross through the event horizon, that distance will work out to approximately the minimum 2-sphere area for the singularity energy

The mass of the black hole is not "the singularity energy". It is not "located" at the singularity. (As above, the singularity is not even a place in space.)

thus this distance can never get smaller until we reach the singularity.

You have given no valid basis for this claim.

 

[.Scott]

Not in the sense of spatial distance. The singularity is a moment of time, not a place in space.

Okay. I'm guessing it's only a moment in time after you cross the event horizon. Before that, one could say it's at the center of the event horizon, but I'm not sure how meaningful that would be.

How are you going to keep track of these distances?

I am still able to exchange information with some of these friends. For objects that are adjacent to my fall, can't I just use RADAR?

The mass of the black hole is not "the singularity energy". It is not "located" at the singularity. (As above, the singularity is not even a place in space.)

That distance thing was one area where I thought there might be a problem with my description. It seemed to me that as I cross the event horizon, my trajectory somehow transitioned away from gravity and more to time. But I think what I was missing is that there's no real transition from a gravitational pull through space to a time singularity destiny, it's just that there is no longer any impressive consequence to location and a hugely impressive consequence to elapsed time. But it isn't on Physics to track "Impressiveness".

The "singularity energy" issue is much more of a mystery to me. Whether I cross time or distance to get there, am I not somehow colliding with whatever else fell before me?

As I approach the event horizon, the BH acts as if it is a huge mass that lies before me. As I cross the EH, it should still look that way. Does it ever not look that way? I think the answer is simply that I loose reference points to judge my approach to the presumed singularity - but not really, because we presume that tidal effects are getting more severe. So I could judge my radial acceleration that way. But do tidal effects get more severe within the EH?

You have given no valid basis for this claim.

My basis for the claim only exists if we (me and my friends) are enclosing something with mass. If we are, then our mutual distances from each other can be used to compute the area of the boundary (the boundary that started out as roughly a 2-sphere). That area cannot decrease without decreasing the topologically bounded mass. I guess the crux of the issue is whether there is a way for that mass to escape our bound. If not, then either the area (and thus the distances) cannot decrease or we need an exception to the Bekenstein Bound.

 

[PeterDonis]

I'm guessing it's only a moment in time after you cross the event horizon.

It's a moment of time, period. That is a matter of spacetime geometry. You can only reach this moment of time by falling inside the horizon.

For objects that are adjacent to my fall, can't I just use RADAR?

For a certain period of time after you cross the horizon, yes. But no matter how close to you an adjacent object is, there will be some time before you reach the singularity at which you will no longer be able to exchange light signals with them.

It seemed to me that as I cross the event horizon, my trajectory somehow transitioned away from gravity and more to time.

I don't even know what this means. But there is no "transition" in your trajectory when you cross the horizon. In fact you can't even tell when you cross the horizon from local measurements.

there is no longer any impressive consequence to location and a hugely impressive consequence to elapsed time

I don't know what this means either. I think you are focusing too much on vague ordinary language and not enough on the actual physics.

Whether I cross time or distance to get there, am I not somehow colliding with whatever else fell before me?

No. The matter that fell in before you is not reachable by you.

As I approach the event horizon, the BH acts as if it is a huge mass that lies before me.

Only if "a huge mass that lies before me" is compatible with the BH being vacuum.

As I cross the EH, it should still look that way.

What does "look that way" mean? What physical observations tell you that there is "a huge mass" before you?

do tidal effects get more severe within the EH?

Yes.

My basis for the claim only exists if we (me and my friends) are enclosing something with mass.

What does "enclosing something with mass" mean?

Again, you are focusing too much on vague ordinary language and not enough on actual physics. The actual physics is that, inside the horizon, it is simply impossible--geometrically impossible--to construct 2-spheres that "enclose mass" the way they would have to for the Bekenstein Bound reasoning to even apply. No amount of hand-waving and playing around with vague ordinary language will change that. Nor will it help you to understand why that is true. You will need to look at the actual gory details of the physics.

 

[PeterDonis]

I guess the crux of the issue is whether there is a way for that mass to escape our bound.

No, that's not the crux of the issue. The crux of the issue is that spacetime geometry inside the horizon does not work the way you think it works. The way it actually works, as I said in post #7 just now, makes it geometrically impossible for 2-spheres inside the horizon to "enclose mass" the way they would have to for the Bekenstein Bound reasoning to even apply.

 

[PeterDonis]

@.Scott here is some more food for thought: your basic intuition seems to be that we ought to be able to take an object that has some finite energy, and apply the Bekenstein Bound not just to the object as a whole, but also "divided up" between parts of the object. In other words, we ought to be able to treat the entropy contained in the object as being "spread out" among the parts of the object the way the object's energy is "spread out" among its constituents.

The problem with applying this to a black hole is that we currently do not have any model of a black hole in terms of more fundamental constituents. Our current model of a black hole, the classical GR model, only tells us that the hole itself is a self-contained entity with a mass $M$ and a horizon area $A$. It does not give any way of dividing those quantities up into smaller pieces that get added together to form the hole. That means there is no way, with our current models, to divide up the Bekenstein Bound for a black hole the way you are thinking.

We could someday come up with a quantum gravity theory that gives us a model of a black hole in terms of more fundamental constituents, that would then tell us how to account for the hole's entropy in terms of the (logarithm of the) number of ways those constituents could be assembled to form a particular hole of mass $M$ and horizon area $A$. We could then use such a model to assess the Bekenstein Bound for some subset of the hole as well as for the entire hole. But nobody currently has such a model.

 

[.Scott]

The problem with applying this to a black hole is that we currently do not have any model of a black hole in terms of more fundamental constituents. Our current model of a black hole, the classical GR model, only tells us that the hole itself is a self-contained entity with a mass $M$ and a horizon area $A$. It does not give any way of dividing those quantities up into smaller pieces that get added together to form the hole. That means there is no way, with our current models, to divide up the Bekenstein Bound for a black hole the way you are thinking.

I can still described the problem. The method of demonstrating the event horizon from General Relativity uses a definition of "$r$" (radius) as the circumference over 2 pi. So, at least while I am outside the event horizon, I can use that same definition of $r$.

Working with a Schwarzchild black hole, with event horizon at $r_s$ and a reference frame at infinity:
At any $r>r_s$, we will have a sphere with the black hole centered within it. The surface of that sphere will be large enough to "holographically" describe the content (ie, the black hole). This is a hard requirement of the sphere. If at any point, the content cannot be described by the enclosing surface, then information will be lost from the universe - a QM no-no.

But from the frame of a falling object, the event horizon is uneventful. There seems to be a widespread willingness to say that once you cross the event horizon, there is an abrupt change that saves the falling object from the QM no-no, but that this abrupt change in the geometry and/or information tally will go unnoticed by the object. It's like "harmless magic happens here".

So let me propose something more sensible: As the object drops, the physical accommodations to this information crunch begins well before the hard Bekenstein limit is reached. I expect this "relief" will be similar to what is described by le Chatelier's principle. Since loosing information is not allowed, and the information-holding capacity of a surface is securely tied to Planck's Constant, it would seem that relief is only available by expanding the surface area and thus the "r". Fortunately, the universe is made of stuff that expands. But in order to couple it to the necessary relief, I need to change the rate of expansion from a cosmological constant to a function that tends to infinity as the Bekenstein bound is approached - so there would need to be an (information_density - Plank_limited_density) term in the denominator.

The result of this expansion function would be to eliminate the event horizon all together - and with it any pesky singularities.

As space is dragged into the black hole and $r$ starts to approach $r_s$, the information density will soar and the space will expand. This expansion will keep $r$ from ever reaching $r_s$ and so the object will simply fall forever. And since the rate of change to $r$ would approach zero, there will soon be no significant tidal effect.

As for the enhanced information density, the falling object (from its reference frame) would be forever passing through surfaces with near-saturated boundary entropy, potent enough to keep $r$ from changing despite strong gravity. Would there be some associated physical manifestation?

As usual, you will be doing me a great service by poking holes in this proposal - or in my description of the problem.

 

[PeterDonis]

I can still described the problem. The method of demonstrating the event horizon from General Relativity uses a definition of "$r$" (radius) as the circumference over 2 pi.

No, it doesn't. Demonstrating that an event horizon is present means demonstrating that there is a region of spacetime that is not in the causal past of future null infinity--the event horizon is then just the boundary of that region. There is no need to say anything at all about the area of the horizon, or define an "areal radius" $r$.

at least while I am outside the event horizon, I can use that same definition of $r$.

You can use that definition everywhere, not just outside (or at) the horizon. But using it doesn't make any difference to any invariants.

Working with a Schwarzchild black hole, with event horizon at $r_s$ and a reference frame at infinity:

What does this even mean?

At any $r>r_s$, we will have a sphere with the black hole centered within it.

Sort of; the hole itself has no "center". $r = 0$ is not the "center" of anything. It's a moment of time, not a place in space.

The surface of that sphere will be large enough to "holographically" describe the content (ie, the black hole).

What does this even mean? What sort of "description" are you talking about?

You need to show some math instead of just waving your hands.

But from the frame of a falling object, the event horizon is uneventful.

In classical GR, yes. Not necessarily in all quantum models. Exactly which quantum models correctly describe quantum corrections in this regime is a major open question in this area of research. And since it is an open question, you can't make any blanket statements of the sort you are making here.

There seems to be a widespread willingness to say that once you cross the event horizon, there is an abrupt change that saves the falling object from the QM no-no, but that this abrupt change in the geometry and/or information tally will go unnoticed by the object. It's like "harmless magic happens here".

Please give specific references. What you are describing might be a viewpoint that is given in the literature, but it's certainly not the only one.

So let me propose something more sensible

This is personal speculation and is off limits here.

 

[.Scott]

No, it doesn't. Demonstrating that an event horizon is present means demonstrating that there is a region of spacetime that is not in the causal past of future null infinity--the event horizon is then just the boundary of that region. There is no need to say anything at all about the area of the horizon, or define an "areal radius" $r$.

In the most common derivation of the formula for the event horizon radius ($R = 2GM/c²$), either the escape velocity is simply set to the speed of light or the singularity in the Schwarzschild Metric (equation 36 of Karl's March 1916 paper, On the gravitational field of a sphere of incompressible fluid according to Einstein's theory) is noted (as with Science Direct 2018, "Schwarzschild Radius").

So you are right. Defining (or demonstrating) the event horizon doesn't require working with an areal radius.

But you clearly get one anyway. And in all the discussions of the Bekenstein bound, area is key.

You can use that definition everywhere, not just outside (or at) the horizon. But using it doesn't make any difference to any invariants.

Fine, but I suggest we not treat distances in space as strictly invariant. We know it expands - but we don't know what determines that expansion.

What does this even mean?

I.e., "Working with a Schwarzchild black hole, with event horizon at $r_s$ and a reference frame at infinity.".
The symbol for the Schwarzchild radius is often $R$, $r$, or $r_s$. So, I'm just identifying which symbol I was using. And I was using the most common reference frame for specifying the coordinate. For example, not "proper" coordinates.
Sorry if this isn't the common way of flagging these things.

What does this even mean? What sort of "description" are you talking about?

You need to show some math instead of just waving your hands.

I am refering to the Holographic Principle which was part of common discussion in the immediate wake of the Bekenstein Bound (1981). According to the wiki article, it became associated with string theory as well.
In a nut shell, if you don't want the universe to loose entropy, then the area enclosing any content must have the capacity to hold that entropy information. (ie, "describe" it) If the area is too small or the amount of entropy is too large, then there is a QM limit (a la Planck's distance) that throttles the entropy.

Since spheres enclose regions of entropy with a certain efficiency, there is an equation for that specific boundary. That equation (since you asked) is provided in the wiki article. It is:
$S \le \frac{ 2\pi k R E}{\hbar c}$
where $S$ is the entropy, $k$ is the Boltzmann constant, $R$ is the radius of the sphere, and $E$ is the total mass-energy enclosed.

So, I was taking about the enclosing surface "describing" the entropy of its content. (as a 2D hologram describes a 3D image).

In classical GR, yes. Not necessarily in all quantum models. Exactly which quantum models correctly describe quantum corrections in this regime is a major open question in this area of research. And since it is an open question, you can't make any blanket statements of the sort you are making here.

(regarding the uneventful crossing of the event horizon in the proper reference frame)
This open question is very closely related to what I am commenting on.
My point is that there is a real QM issue and that it makes no sense to treat it as strictly a problem for what happens after you cross the event horizon. The problem develops as you approach the EH and as that $S$
approaches that $\frac{ 2\pi k R E}{\hbar c}$. And I am suggesting that we look for a resolution that shows its effects during that approach.

Please give specific references. What you are describing might be a viewpoint that is given in the literature, but it's certainly not the only one.

Here are some that are worth mentioning:

* Most recently that dark energy report (also here). This article suggests that BH generate dark energy. Given that I see the "dark energy" expansion effect as a fruitful avenue, I also see a tie-in to this article.
* Published in 2017 by Science Direct Argues that the Hawking Radiation source is within a BH "atmosphere": It's hardly an argument for adjusting the event horizon, but at least it's focusing on what happens as you approach that horizon.
* Published in 2018, "Information Preservation and Weather Forecasting for Black Holes": Hawkings actually argued for some way of addressing the information paradox by replacing the "hard" event horizon with an apparent one. I'm not saying that he had the answer. But, from my point of view, he was looking in the right direction.

This is personal speculation and is off limits here.

The only important point I was trying to make is that it should be possible to resolve the information paradox and maybe the whole event horizon geometry by looking at that Bekenstein Limit.

 

[PeterDonis]

the most common derivation of the formula for the event horizon radius

For the particular case of a Schwarzschild black hole, yes. But that is by no means the only spacetime with an event horizon. The statement I made was completely general.

I suggest we not treat distances in space as strictly invariant.

I wasn't. I didn't bring up "distances in space" at all. You did, earlier in this thread, and I explained why the way you were using the concept didn't really work.

We know it expands

I have no idea what you are talking about here.

I was using the most common reference frame for specifying the coordinate.

You didn't give a reference frame at all. And note that the "areal radius" $r$ can be specified as an invariant, independent of any choice of coordinates. So you don't even have to specify a reference frame to make use of it.

My point is that there is a real QM issue

Perhaps, but if so, nobody knows exactly what it is or exactly how to address it. As I already pointed out earlier in this thread (in post #9, which was back in 2023), we don't have a model of a black hole in terms of more fundamental constituents, whether those are some kind of elementary quantum systems or anything else. So any claims about "a real QM issue" are just speculation at this point. And we're at the point in this thread where speculation is pointless. Yes, we know we don't understand everything about black hole entropy. No, we can't have a useful discussion about it because all we have are different speculative models and no way to usefully test any of them in actual experiments.

This article suggests that BH generate dark energy.

Speculation.

Published in 2017 by Science Direct Argues that the Hawking Radiation source is within a BH "atmosphere":

Speculation.

Hawkings actually argued for some way of addressing the information paradox by replacing the "hard" event horizon with an apparent one.

Speculation.

The only important point I was trying to make is that it should be possible to resolve the information paradox and maybe the whole event horizon geometry by looking at that Bekenstein Limit.

Sure, go right ahead. When you get your solution published in a peer-reviewed journal (which IMO will be pretty tough for someone who marked this thread as "B" level, but you can try), then we can discuss it here.

In the meantime, this thread is closed.