What does the equation C = 12 J mean for black holes?

[joemorin]

TL;DR Summary

Saw this formula on documentary about black holes. What does it mean … really mean ... in lay terms as much as possible.

Just saw a documentary about resolving Hawking's "information paradox". In my own lay terms the physicists appear to theorize resolving the paradox with with their proposed C = 12 J . C is the central charge (which I don't fully understand) and J is the total angular momentum of the black hole. 12 is an interesting and elegant coefficient (IMHO) having something to do with geometry, vectors and spheres … perhaps

In lay terms, the physicists seem to be saying that this simple formula demonstrates that the event horizon (EH) of a black hole has the capacity to absorb and retain information about all matter and energy that has fallen into the black hole by means of holding or releasing the information somehow through a quantum mechanism or structure called "soft hair" that radiates information out and away from the EH. Thus the information paradox could be resolved if C = 12 J is valid.

My question(s) are how does some value for total angular momentum relate to a capacity for a black hole system (through its soft hair) to preserve information? What is the central charge (in simple terms) in this context? Why 12 ??

 

[anorlunda]

Summary:: Saw this formula on documentary about black holes. What does it mean … really mean ... in lay terms as much as possible.

Just saw a documentary about resolving Hawking's "information paradox".

Please provide a link to what you saw so that we know what you are asking about.

 

[joemorin]

The documentary's focus is more on the first image of a black hole in 2018 or so but delves into Hawking's work on the paradox up to his death. Here is documentary : https://www.netflix.com/title/81343342

 

[joemorin]

Hawking prompts his team (six months or so before his death) with "How many conformal killing vectors on the two sphere. An infinite number?" The team guesses at first (3)? Then (6) perhaps?

Hawking also asks "Would diffeomorphism give all the entropy?"

 

[Ibix]

I'm afraid that black hole thermodynamics is beyond me. I've suggested the thread be moved to the relativity forum where you may get more help. I have to warn you that I suspect that the answers will boil down to "you need to learn the maths for this to make any sense".

One thing I do know is that the angular momentum of the hole affects the area of its event horizon, and it's the event horizon area that's related to the entropy of the hole.

 

[Doc Al]

Here's the paper that explains how they derived that equation for central charges: Black Hole Entropy and Soft Hair

It's rather technical, so it may not help -- as @Ibix has warned. (I won't pretend to understand it.)

 

[Ibix]

I note that the paper comments that they choose to work in units where $c=G=k=\hbar=1$, so in more usual unit systems that 12 is probably going to become $12c^\alpha G^\beta k^\gamma \hbar^\delta$ where the Greek symbols represent numbers chosen to make units consistent.

 

[PeterDonis]

Just saw a documentary about resolving Hawking's "information paradox".

This would be better described as a proposed resolution. This entire area of research is purely theoretical; there is no way to do experiments to test any of the proposals. Partisans of various proposals like to claim that they are the final answer, but without experimental confirmation, such claims should be taken with a huge helping of salt.

 

[PeterDonis]

I've suggested the thread be moved to the relativity forum where you may get more help.

This is not really classical relativity; it's quantum gravity research. I have moved the thread to the Beyond the Standard Model forum.

 

[PeterDonis]

C = 12 J . C is the central charge

Actually, it's not, it's just a constant in the formula (called $c_R$ or $c_L$ in the paper @Doc Al linked to). A better representation of the full central charge would be the full coefficient in $K_{m,n}$ in that paper, which turns out to be $J m^3$. The factor of $12$ only appears in the formulas $c_R = 12 J$ and $c_L = 12 J$ because a factor of $1/12$ appears in a previous formula for $K_{m,n}$. So I'm not sure the factor of $12$ has any direct physical significance; the key result of physical significance seems to me to be the factor $J m^3$.

 

[joemorin]

In Feynman style, might you be able to simplify for lay persons the general nature of this proposed resolution? Does it show, for example, some relationship between surface area of event horizon and entropy as Ibix suggests. Any other ways to help non-quantum physicists conceptualize this resolution would be appreciated.

Also, is it in any way legitimate to suggest that all matter falling into a black hole may essentially reach the speed of light with that matter becoming all kinetic energy with total of mc ⌃2 .

 

[Ibix]

Also, is it in any way legitimate to suggest that all matter falling into a black hole may essentially reach the speed of light with that matter becoming all kinetic energy with total of mc ⌃2 .

No, definitely not. Nothing with mass can reach the speed of light under any circumstances and it cannot "become kinetic energy". Energy is a property of a particle or field, not a thing, so "becoming kinetic energy" makes no sense, I'm afraid.

 

[anorlunda]

Any other ways to help non-quantum physicists conceptualize this resolution would be appreciated.

You might try this book by Leonard Susskind. It is written for laymen.
https://en.wikipedia.org/wiki/The_Black_Hole_War

If I remember correctly, it does discuss how the surface area of the event horizon is proportional to the information contend.

Susskind has gone beyond that now suggesting that information complexity theory might be use to unite quantum physics with general relativity. There are numerous videos of his lectures on YouTube. Search for "ER=EPR". But it is all cutting edge speculative research, not layman stuff.

 

[joemorin]

No, definitely not. Nothing with mass can reach the speed of light under any circumstances and it cannot "become kinetic energy". Energy is a property of a particle or field, not a thing, so "becoming kinetic energy" makes no sense, I'm afraid.

I've read recently some measurements or approximations have indicted that some black holes rotate at nearly the speed of light, they say for example NGC 1365 is turning at 84% the speed of light…. Thus it's reasonable to speculate that matter drawn into the hole does at least approach the speed of light and that much of the matter's inertial? mass would consist of kinetic energy — just a guess that's probably wrong. So if matter is not becoming energy (like kinetic energy) at least in part, what does the matter that "falls" into a black hole actually become?

 

[PeterDonis]

just a guess that's probably wrong

Please stop guessing. Personal speculation is off limits here at PF.

This topic is a very advanced one, and I have changed the level of the thread to "A". The paper itself is aimed at experts who have that level of background knowledge of the topic; you should not expect to be able to understand it if you don't have that background knowledge. I can't follow a lot of it myself since my knowledge of the subject area is pretty basic.

In Feynman style, might you be able to simplify for lay persons the general nature of this proposed resolution?

First you have to understand what they are trying to "resolve". They are not trying to "resolve" whether a black hole's entropy is proportional to the area of its horizon. That's already agreed on by everybody.

All they are trying to do is to see if they can get a particular kind of quantum gravity model they are interested into predict that the entropy of a black hole is proportional to the area of its horizon. In other words, they are trying to see if this kind of model is a viable candidate for a quantum gravity model of a black hole. "Soft hair" is just a name for the particular quantum gravity degrees of freedom that this model is using.

how does some value for total angular momentum relate to a capacity for a black hole system (through its soft hair) to preserve information?

Because the total angular momentum (and the total mass) affects the area of the hole's horizon, which in turn is (proportional to) the hole's entropy. This is not anything particular to the model being investigated in the paper; it's a known property of black holes in classical General Relativity.

What is the central charge (in simple terms) in this context?

"Central charge" is a particular property that particular kinds of algebras have (the model in the paper is using a kind of algebra called a "Virasoro algebra"). I don't know of any simple way to explain it if you don't already have a good understanding of what algebras are, what kinds of algebras there are, and how their properties differ. Possibly other experts here can do a better job of that.

 

[Ibix]

Thus it's reasonable to speculate that matter drawn into the hole does at least approach the speed of light

There's no such thing as speed unless you specify speed relative to something, so you need to be rather careful how you think about it. That said, it's reasonable to suppose that matter orbiting a black hole might approach the speed of light in some sense - but it can never reach it.

much of the matter's inertial? mass would consist of kinetic energy — just a guess that's probably wrong.

Better to say that defining mass in general relativity is complicated. Inertial mass isn't a particularly helpful concept outside Newtonian physics, since the relationship between force and acceleration is more complex even for point particles.

So if matter is not becoming energy (like kinetic energy) at least in part, what does the matter that "falls" into a black hole actually become?

It doesn't become anything. It just falls in and hits the singularity. Classical general relativity doesn't know what happens there. Quantum gravity should have an answer but, as Peter notes there are a lot of theories and we don't know which one (if any) is right.

 

[DrDick1954]

My background is colloquially entitled "Applied Physics", but this is an entertaining topic to me. I have recently seen the original subject, addressed within the documentary "Black Holes: The Edge of Everything We Know". After reading the comments here, I am going to take a stab at a simplifying explanation. Here goes:

If the area of the EH represents the total entropy of the contents of a BH, the conjecture is based upon the premise that an object heading into a BH, has an initial descriptive entropy, which we also call "information". Once this object falls into/through the EH, the 'new area' of the event horizon will be exactly what the old one was, plus a delta of exactly the addition of the object's information, its initial entropy. If all the esoteric math comes to a conclusion that this is the case, then no information/entropy, is lost or gained.

 

[joemorin]

Thank you Doctor D. That is the kind of Feynman-like simplification I've been asking for. I also appreciate Ibix and PeterDonis addressing the significance (or non-significance) and origin of the coefficient 12. A humbling, but useful, experience for a guy in need of more "maths" to appreciate some of this.

 

[geshel]

My background is colloquially entitled "Applied Physics", but this is an entertaining topic to me. I have recently seen the original subject, addressed within the documentary "Black Holes: The Edge of Everything We Know". After reading the comments here, I am going to take a stab at a simplifying explanation. Here goes:

If the area of the EH represents the total entropy of the contents of a BH, the conjecture is based upon the premise that an object heading into a BH, has an initial descriptive entropy, which we also call "information". Once this object falls into/through the EH, the 'new area' of the event horizon will be exactly what the old one was, plus a delta of exactly the addition of the object's information, its initial entropy. If all the esoteric math comes to a conclusion that this is the case, then no information/entropy, is lost or gained.

I believe the area of the BH must grow by at least the amount equivalent to the entropy of the stuff added. However, it most likely grows by much more than that. A BH of a certain mass has an area corresponding to the maximum entropy such a mass could have (this is for a simple Schwarzschild - non-charged, non-rotating - black hole. There's a similar bound otherwise but I believe stating it is more complicated). It is often far larger than the actual entropy of the original matter in the state it was in when "added" to the BH (and what "adding" things to black holes means is a whole other subject).

 

[joemorin]

I believe the area of the BH must grow by at least the amount equivalent to the entropy of the stuff added. However, it most likely grows by much more than that. A BH of a certain mass has an area corresponding to the maximum entropy such a mass could have (this is for a simple Schwarzschild - non-charged, non-rotating - black hole. There's a similar bound otherwise but I believe stating it is more complicated). It is often far larger than the actual entropy of the original matter in the state it was in when "added" to the BH (and what "adding" things to black holes means is a whole other subject).

Do you have an opinion on the origin or significance of the coefficient 12 ?

 

[geshel]

Sorry I don't know anything about that equation yet.

 

[PeterDonis]

If the area of the EH represents the total entropy of the contents of a BH, the conjecture is based upon the premise that an object heading into a BH, has an initial descriptive entropy, which we also call "information". Once this object falls into/through the EH, the 'new area' of the event horizon will be exactly what the old one was, plus a delta of exactly the addition of the object's information, its initial entropy. If all the esoteric math comes to a conclusion that this is the case, then no information/entropy, is lost or gained.

I believe the area of the BH must grow by at least the amount equivalent to the entropy of the stuff added. However, it most likely grows by much more than that. A BH of a certain mass has an area corresponding to the maximum entropy such a mass could have (this is for a simple Schwarzschild - non-charged, non-rotating - black hole. There's a similar bound otherwise but I believe stating it is more complicated). It is often far larger than the actual entropy of the original matter in the state it was in when "added" to the BH (and what "adding" things to black holes means is a whole other subject).

Both of these posts are misleading because both of you are missing an essential point: entropy is not a property of a piece of matter that the piece of matter carries around with it. Entropy is a property of a system in a certain macroscopic state.

In the "object falls into a black hole" case that both of you are discussing, the total system is the object plus the black hole: more precisely, the total system is all of the physical degrees of freedom that are manifested as the object and the black hole. The entropy of the total system is (the logarithm of) the phase space volume corresponding to the macroscopic state of the system; roughly speaking, it's (the logarithm of) the number of ways you could assemble all of those degrees of freedom so that they look the same macroscopically.

Before the object falls into the black hole, the macroscopic state of the total system is "object of mass m with such-and-such other properties (color, shape, etc., etc.) and black hole of mass M". There is some number of ways of assembling the degrees of freedom in the total system that look like that macroscopic state; call that number n.

After the object falls into the black hole, the macroscopic state of the total system is "black hole of mass m + M". There is some number of ways of assembling the degrees of freedom in the total system that look like that macroscopic state: call that number N.

For pretty much any ordinary object falling into a black hole, we expect N to be astronomically larger than n. Why? Because once the object falls into the hole, we lose all of the macroscopic information about it that
distinguishes it from all of the other possible objects of mass m: its color, shape, etc., etc. All we know is the mass m that it added to the hole, and there are an astronomically large number of ways of making an object that would add that mass m to the hole.

This also illustrates why a black hole of a given mass, say mass m, saturates the entropy bound (called the "Bekenstein bound" after Jacob Bekenstein, who first discovered it) for an object of that mass: because a black hole of mass m could have been made from any object (or combination of objects) with that total mass; we know nothing about what made it or how it got made, except its mass. With any other object of mass m, we know a lot more about how it got made and it has a lot more distinguishable macroscopic properties that tell us about how it got made.

 

[geshel]

Thanks for the clarification PeterDonis.

 

[DrDick1954]

Both of these posts are misleading because both of you are missing an essential point: entropy is not a property of a piece of matter that the piece of matter carries around with it. Entropy is a property of a system in a certain macroscopic state.

In the "object falls into a black hole" case that both of you are discussing, the total system is the object plus the black hole: more precisely, the total system is all of the physical degrees of freedom that are manifested as the object and the black hole. The entropy of the total system is (the logarithm of) the phase space volume corresponding to the macroscopic state of the system; roughly speaking, it's (the logarithm of) the number of ways you could assemble all of those degrees of freedom so that they look the same macroscopically.

Before the object falls into the black hole, the macroscopic state of the total system is "object of mass m with such-and-such other properties (color, shape, etc., etc.) and black hole of mass M". There is some number of ways of assembling the degrees of freedom in the total system that look like that macroscopic state; call that number n.

After the object falls into the black hole, the macroscopic state of the total system is "black hole of mass m + M". There is some number of ways of assembling the degrees of freedom in the total system that look like that macroscopic state: call that number N.

For pretty much any ordinary object falling into a black hole, we expect N to be astronomically larger than n. Why? Because once the object falls into the hole, we lose all of the macroscopic information about it that
distinguishes it from all of the other possible objects of mass m: its color, shape, etc., etc. All we know is the mass m that it added to the hole, and there are an astronomically large number of ways of making an object that would add that mass m to the hole.

This also illustrates why a black hole of a given mass, say mass m, saturates the entropy bound (called the "Bekenstein bound" after Jacob Bekenstein, who first discovered it) for an object of that mass: because a black hole of mass m could have been made from any object (or combination of objects) with that total mass; we know nothing about what made it or how it got made, except its mass. With any other object of mass m, we know a lot more about how it got made and it has a lot more distinguishable macroscopic properties that tell us about how it got made.

But it is exactly that NON-loss of information (regarding the newly added mass "m") that seems to be the hopefully proven result of the derivation C = 12 J . What am I not understanding about this proof regarding information retention? Intuition would support what you are stating, that any macroscopic definition/information about m is going to be lost as it is rendered asunder and becomes a dense plasma as it approaches the event horizon. If all we are discussing is conservation of mass(es), it would seem to be a lot of effort to state or prove the obvious. OR, does this proof mean that black holes cannot be the oft-touted "wormhole", such that m pops out someplace else, rather than being "absorbed" into the totality of the BH?

 

[anorlunda]

Information versus entropy. Ugh, there are many definitions, very confusing because you're never sure which definition your source is thinking of.

It is true that many sources equate entropy and information, but In the Insights article, How to Better Define Information in Physics, I said:

The point is that entropy (and thus knowledge) can vary with the state of the system, whereas the number of microstates (information) does not vary with state.

Enter a BH and the macrostate can be changed (as @PeterDonis said), but if information is conserved then the number of microstates remains unchanged. I suspect that much of the confusion about BHs comes because people use the words information, knowledge, and entropy as if they are synonyms, or use the word information with several possible definitions.

 

[PeterDonis]

it is exactly that NON-loss of information (regarding the newly added mass "m") that seems to be the hopefully proven result of the derivation C = 12 J

I don't see any such result in the paper; in fact, the paper proposes a quantum gravitational way of proving that there is loss of information whenever something falls into a black hole, since that is what the area law for black hole entropy means.

Intuition would support what you are stating, that any macroscopic definition/information about m is going to be lost as it is rendered asunder and becomes a dense plasma as it approaches the event horizon.

For a large enough black hole, no, the object would remain perfectly intact as it crossed the horizon. That's not why the information in the object gets lost when it falls into the hole. The information gets lost because once the object has fallen across the horizon, there is no way to tell, by looking at the hole, what object fell. All you can observe from the outside is the hole's mass.

 

[ajr]

Amount of information can be quantized to bits. Area of EH 4*Pi*R^2 divided by plank area then log2 of result.

Any delta M falling into BH will lead to increased delta R which leads to delta bits. In this way you could weigh the equivalents bits of 1 KG.

I leave the calculations to the reader.

 

[geshel]

Amount of information can be quantized to bits. Area of EH 4*Pi*R^2 divided by plank area then log2 of result.

Any delta M falling into BH will lead to increased delta R which leads to delta bits. In this way you could weigh the equivalents bits of 1 KG.

I leave the calculations to the reader.

Not really, because it depends on the initial size of the black hole.

 

[jartsa]

Also, is it in any way legitimate to suggest that all matter falling into a black hole may essentially reach the speed of light with that matter becoming all kinetic energy with total of mc ⌃2 .

No. Drop a small 0.1 kg potato towards a large black hole. At the lowest possible position take the kinetic energy away from the potato. Then drop the potato to the black hole. The mass of the black hole increases by 0.000000000001 kg. This shows us that the whole energy of the potato did not become kinetic energy.I know that the mass increase is small but nonzero, because I learned it by reading this: http://www.scholarpedia.org/article/Bekenstein_bound
That article does not talk about dropping and then stopping an object, instead it talks about lowering an object. I am assuming those two are equivalent.
On the other hand, a light beam that is not perfectly collimated has a rest mass. That rest mass can not become all kinetic energy, when dropped towards a black hole.

So my conclusion is that neither a light pulse sent towards a black hole, nor a potato dropped towards a black hole become all kinetic energy.

 

[jartsa]

So my conclusion is that neither a light pulse sent towards a black hole, nor a potato dropped towards a black hole become all kinetic energy.

Or maybe we could say that potato and light pulse both become almost essentially kinetic energy when falling towards a black hole. Or maybe not. Never mind.

I found this:
https://physics.aps.org/featured-article-pdf/10.1103/PhysRevLett.116.231301

I like the section called "Quantum hair implants". Planck-sized black holes are dropped into a big black hole, on the event horizon of which they excite Planck-sized soft photons, this transfers information that is unknown to us, entropy, to the hair consisting of said soft photons.