A must read post on black hole entropy by physicist (Perimeter Institute) and blogger Sabine Hossenfelder! ;-)

Now that she mentions it, "Entropy" is a cool name for a kid (superhero?), maybe even cooler than "Max Talmud" :-)

These days, everybody is talking about entropy. In fact, there is so much talk about entropy I am waiting for a Hollywood starlet to name her daughter after it. To help that case, today a contribution about the entropy of black holes.

To begin with let us recall what entropy is. It's a measure for the number of micro-states compatible with a given macro-state. The macro-state could for example be given by one billion particles with a total energy E in a bag of size V. You then have plenty of possibilities to place the particles in the bag and to assign a velocity to them. Each of these possibilities is a micro-state. The entropy then is the logarithm of that number. Don't worry if you don't know what a logarithm is, it's not so relevant for the following. The one thing you should know about the total entropy of a system is that it can't decrease in time. That's the second law of thermodynamics.

It is generally believed that black holes carry entropy. The need for that isn't hard to understand: if you throw something into a black hole, its entropy shouldn't just vanish since this would violate the second law. So an entropy must be assigned to the black hole. More precisely, the entropy is proportional to the surface area of the black holes, since this can be shown to be a quantity which only increases if black holes join, and this is also in agreement with the entropy one derives for a black hole from Hawking radiation. So, black holes have an entropy. But what does that mean? What are the microstates of the black hole? Or where are they? And why doesn't the entropy depend on what was thrown into the black hole?

While virtually nobody in his right mind doubts black hole have an entropy, the interpretation of that entropy is less clear. There are two camps: On the one side those who believe the black hole entropy counts indeed the number of micro-states inside the black hole. I guess you will find most string theorists on this side, since this point of view is supported by their approach. On the other side are those who believe the black hole entropy counts the number of states that can interact with the surrounding. And since the defining feature of black holes is that the interior is causally disconnected from the exterior, these are thus the states that are assigned to the horizon itself. These both interpretations of the black hole entropy are known as the volume- and surface-interpretations respectively. You find a discussion of these both points of view in Ted Jacobson's paper "On the nature of black hole entropy" [gr-qc/9908031] and in the trialogue "Black hole entropy: inside or out?" [hep-th/0501103].

A recent contribution to this issue comes from Steve Hsu and David Reeb in their paper

Black hole entropy, curved space and monsters

Phys. Lett. B 658:244-248 (2008)

arXiv:0706.3239v2

...read the rest...

See related post here, with nice pictures.

## 22 comments:

Supposedly a high-IQ theoretical physicist, yet speaks positively about Klein's "The Shock Doctrine".

Anonymous: What I said de facto about Klein's book was a) I am reading it, though not finished b) I am skeptic about it because it presents a very single-sided point of view c) it is a useful source of data. Check for yourself what I wrote if you doubt it. Besides this, I am very sure you don't have any data on my IQ, supposedly or non-supposedly. Best.

B.

Hi Steve,

Thanks for the link and sorry about the cookie-monster picture, couldn't resist :-) Best,

B.

Re: cookie monster, neither could I!

Another example of the great utility of modern physics.

Or rather, the great fraud?

Naive question: if a monster exists, how, if at all, might its presence be inferred by present or future instruments?

If such objects exist it would be possible to fall into a black hole and discover a much richer variety of interiors than would be thought by "entropy counts the micro-states" camp.

But a (hypothetical) monster wouldn't have distinctive astronomical signatures?

From the outside a black hole resulting from the collapse of a monster looks just like any other black hole. There might be some implications for the endpoint of black hole evaporation, but that depends on some aspects of quantum gravity which we do not yet understand.

Hi Steve,

Why didn't you mention Bousso's bound, instead of just of Bekenstein's, in your article? Since you are very interested in intense gravitational fields, that one is the most general and successful formulation of a gravitational entropy bound that I am aware of. Unless, someone else find a bad fauld in his bound or something better.

I forgot whether we mention the covariant (Bousso) entropy bound in the paper, but at least when I give the talk I tend to mention that it is not violated. (IIRC, if you follow one of his lightsheets into the monster it only intercepts a finite amount of entropy before focusing to a point.)

But that is distinct from the question of whether all of the information in the interior monster state can re-emerge as Hawking radiation. Once we build a monster with S > A that becomes problematic even if Bousso's bound is satisfied.

Hi Steve,

Stupid question - Why is this solution called a monster?

I just made that up. From the perspective of the string theorists and AdS/CFT duality believers these configurations are monstrous -- they have to be excised from the Hilbert space or else they would contradict that camp's interpretation of black hole entropy and much more.

The interesting implication of monsters is that here you have a perfectly good semiclassical configuration in general relativity, but (according to one camp) it can't correspond to a state of the quantum theory of gravity (e.g., string theory) -- it ain't really in the Hilbert space.

In all other cases we are used to there are *many* distinct quantum states that correspond to a given macroscopic, semiclassical configuration. So quantum gravity, certain people would claim, is very very different and all monsters must die!

Hope that makes sense...

Interesting. Is the implication that it is not in Hilbert space or rather that there are other constraints that rule it out?

For example, not being in Hilbert space doesn't rule out weak solutions. The delta functional is not in L2 after all but it is a useful solution.

It can't be in the Hilbert space because the dimensionality of the Hilbert space (entropy) is bounded above for a compact object of some fixed mass M. In other words, if I give you an energy budget of M and a compact region of space then there are only supposed to be of order exp(M^2) possible distinct quantum states. This comes from so-called black hole entropy bounds: we imagine smashing your object into a black hole, which has entropy S = A = M^2, and since the entropy can only go up this is an upper bound on the entropy of your original object. The monsters have too many states -- their entropy can exceed M^2.

Sorry if that's confusing...

I think I understand. You've created a semiclassical object on a compact domain, whose entropy exceeds the maximum possible cardinality of a quantum state on that domain. Hence, it cannot exist if you believe the string theory interpretation of black hole entropy.

Perfectly summarized!

Probably incidental, but see this paper by Jonathan Oppenheim (2001):

Area scaling entropies for gravitating systems (gr-qc/0105101)

PS: Also see subsequent papers by Oppenheim (on arXiv.org) concerning thermodynamics in the presence of long-range interactions, all of which touch on the similarity of this situation to gravitating systems.

In Jonathan's construction he builds the object out of thin shells and assumes equilibrium. We do not assume equilibrium -- we simply want to count the number of distinct initial data with fixed ADM mass. This number can be larger than exp(A). Yet, each of the distinct configurations will collapse to a black hole with area A.

gs said...But a (hypothetical) monster wouldn't have distinctive astronomical signatures?

Steve Hsusaid...From the outside a black hole resulting from the collapse of a monster looks just like any other black hole. There might be some implications for the endpoint of black hole evaporation...In my untutored opinion, it would be noteworthy if a monster's anomalous concentration of states could never be inferred from the asymptotic outside. I had in mind the intermediate stage of a monster's life cycle. I was wondering if an external observer might detect the presence of a monster via its output or via the output from its interaction with the environment.

However I gather that not only are monsters unrealistic but, since you mention black hole evaporation, macroscopic monsters are

really reallyunrealistic...The construction has more to do with the internal logical consistency of models of quantum gravity than with any realistic phenomena.

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