Is the Entropy of Black Holes Proportional to Their Size?

[brcooke]

Hi, I'm not a physicist so the answer to this question may be elementary but I can't figure it out for myself. I posted this elsewhere before finding this particular part of the forum.

Black holes are famous because (almost) nothing can escape from them. Once an observer has crossed the event horizon, no more contact may be made with him or her. Thus it will forever be impossible to know what the interior of a black hole is like, because observers that go inside will never be able to communicate with us.

Everyone knows that the entropy of a black hole is proportional to its area, and that there are only 3 parameters necessary to describe a black hole: its area, charge, and spin (right?). All of the books I've read about the subject say that astronomical-scale black holes have very high entropy (the exception being very small black holes created in the lab).

Now, my understanding of entropy is that it is proportional to the logarithm of the number of microstates that could underlie a single macrostate. My question is: Since it is impossible to know anything about the interior of a black hole, and thus anything about the microstates within it, how can black holes have entropy at all? And contrary to the popular assumption that the entropy of black holes is very large, given how easy it is to define a black hole, shouldn't that indicate that their entropy is in fact very low?

Thanks for your help!

Brad.

 

[Kevin_Axion]

Black Holes emit Hawking Radiation: http://en.wikipedia.org/wiki/Hawking_radiation.

 

[brcooke]

I know they do. That is why I qualified my statement by saying "(almost) nothing can escape".

 

[Kevin_Axion]

So doesn't the thermal emission constitute it having entropy?

 

[brcooke]

That's a good point. My understanding is that Hawking radiation increases in intensity as the black hole evaporates. So the dissipation of energy via this radiation constitutes an increase in entropy within our observable universe, but we still don't know how much there is within the black hole.

 

[Kevin_Axion]

Well, I think the Holographic Principle may answer your questions: http://en.wikipedia.org/wiki/Holographic_principle.

 

[Borek]

Have you seen explanation in wikipedia?

http://en.wikipedia.org/wiki/Black_hole_thermodynamics

 

[brcooke]

No, I haven't, but I'm deep into the holographic principle as suggested. I will check that next. Thanks!

 

[S.Mukherjee]

I think that entropy of black hole is very high due to its capability of absorbing everything. Talking at macroscopic level, if we consider black hole as a system, then mass as well as heat in form of radiations are going into the system and not coming outside.
So, if we use the relation of entropy balance:
S in - S out + S gen = dS sys
S out=0 and S gen is positive
Thus Change in entropy of system(R.H.S) is positive.
Therefore we can say that the entropy of black holes is increasing continuously at a high rate meaning that it has very high entropy

 

[Chronos]

The consensus on black hole entropy is it is represented by the surface area of its event horizon. That makes sense given its interior is unobservable.

 

[brcooke]

Thanks to all for their contributions.

My question is obviously the Black Hole Paradox, although I didn't realize it at the time. What led me to it is the idea that in the distant future, the universe will be dominated by black holes (of course the expansion of the universe could prevent that). If the universe is eternal, then there will be another beginning... but it will have to initiate from a low entropy state.

How to get from black holes (high entropy) to the low entropy state necessary to spawn another universe? I guess the answer must lay in the fact that the black holes will themselves eventually dissipate completely ... entropy still increasing ... but I'm puzzled: so then how to get from there to low entropy again?

Maybe black holes really have very low entropy -- its not impossible, since we cannot observe their internal state, and the Big Crunch implied by their very high gravity would suggest something akin to a neutron star only much much smaller. Isn't a neutron star ultimately very simple? And if this makes any sense at all, then a universe dominated by black holes could eventually become the Singularity that spawns another universe...?

I suppose it makes no sense for a black hole to be a entropy-eating machine as that would violate the 2nd law. But since they're forever inaccessible, and contain a singularity where the laws of physics are undefined, it seems at least *possible* that black holes to reduce entropy altogether.

 

[foolosophy]

Maybe black holes really have very low entropy -- its not impossible, since we cannot observe their internal state, .

For a body to possesses a value for entropy it must have a value for T, ie temperature.

The Hawking radiation emission occurs at the event horizon where quantum uncertainty effects can allow for one particle of a "particle pair" to exceed the speed of light and escape the gravitational pull of the black hole (singularity).

 

[twofish-quant]

Now, my understanding of entropy is that it is proportional to the logarithm of the number of microstates that could underlie a single macrostate. My question is: Since it is impossible to know anything about the interior of a black hole, and thus anything about the microstates within it, how can black holes have entropy at all?

You can take the mass of the black hole, use Planck's constant, and then calculate the possible microstates within the black hole. Since you have no idea what the states are, this gives you the entropy.

 

[foolosophy]

dS = dQ/T

 

[stevebd1]

Another interesting aspect of entropy for black holes within the current model is that it appears you can have zero temp/surface gravity but none zero entropy.

Conventional equations for BH temperature and entropy-

$$T=\frac{\kappa_+}{2\pi k_{\text{B}}}$$

$$S=\frac{k_{\text{B}}A_+}{4}=\pi (r_+^2+a^2)k_{\text{B}}$$

where

$$\kappa_+= \frac{r_+-r_-}{2(r_+^2+a^2)}$$

$$A_+=4\pi(r_+^2+a^2)$$

$$r_\pm=M\pm\sqrt{M^2-Q^2-a^2}$$

While $\kappa=0$ is against the third law of BH thermodynamics, the model still implies that while temp/surface gravity tends to zero, the entropy does not. This is addressed to some extent in this paper that takes into account temp/surface gravity of the inner horizon also-

'Entropy of Kerr-Newman Black Hole Continuously Goes to Zero when the Hole Changes from Nonextreme Case to Extreme Case' by ZHAO Zheng, ZHU Jian-yang & LIU Wen-biao
http://cpl.iphy.ac.cn/qikan/manage/wenzhang/0160698.pdfEDIT:
Some other papers covering the same subject-

'Black Holes, Entropy and the Third Law' by A. J. Meyer, II
http://arxiv.org/abs/physics/0608080

'A Proposed Absolute Entropy of Near Extremal Kerr-Newman Black Hole' by Hai Lin
http://arxiv.org/abs/gr-qc/0104098

 

[thedeester1]

Ok I only understand Entropy as the Thermo dynamic of decay...Im a bit of a book reader but not a Proffesor...Ok as the black hole gathers mass its event horizon loses density...though the mass has massive amounts of gravity its event horizon deminishes. it loses the abbility to trap close mass...its gowth is then stunded. eventually the density of the event horizon will be less than the mass of the black hole its self. once the density collapses then the black hole can no longer be a black hole. I think you will end up with a massive mass with no event Horizon...

 

[GeorgeRaetz]

It's easy to understand that black holes have very high entropy.
To describe a black hole, as you point out requres only a few macrostate variables: mass, area maybe charge. But the number of microstate variables that are consistent with the macrostate is enormous: That's because the microstate description encompasses the detailed world lines of every little atom, nucleus etc. that got sucked into the black hole, and all the world lines consistent with that macrostate description;clearly an enormous amount of information which is lost when they are absorbed into the black hole.
The key is to understand that entropy increases as matter falls into a black hole (or any other gravitating object).

Our universe is still in a state of low entropy(low enough that we can be thinking and writing about this stuff anyway), and there remains plenty of low entropy matter to feed the high entropy black holes and that's what drives the process.

A brief,simple introduction: section 27.7 of The Road to Reality, by Roger Penrose.

 

[bcrowell]

Ok I only understand Entropy as the Thermo dynamic of decay...Im a bit of a book reader but not a Proffesor...Ok as the black hole gathers mass its event horizon loses density...though the mass has massive amounts of gravity its event horizon deminishes. it loses the abbility to trap close mass...its gowth is then stunded. eventually the density of the event horizon will be less than the mass of the black hole its self. once the density collapses then the black hole can no longer be a black hole. I think you will end up with a massive mass with no event Horizon...

The event horizon doesn't have mass.

 

[thedeester1]

The event horizon doesn't have mass.

No it has density

 

[stonecoldgen]

im no physicist but if black holes had no entropy it would violate the second law of thermodynamics when it sucks things as more entropy would be gained than the entropy lost

and because entropy is proportional to the volume of the black hole, i think that by simple logic we can assume that to its area, therefore to its radius, and we know how to measure the radii of black holes, so its not 100% speculation as you suggest

and i repeat, I am no physisists, I am just a 16 year old who has read a little bit

 

[stonecoldgen]

No it has density

i haven't read about this, but is the event horizon actually something?

ive always thought of it as an area where the black hole can suck stuff, but just a vacum