# The Entropy of Black Holes

2 views

### Aaron Bergman

Oct 20, 2003, 6:15:06 PM10/20/03
to

I hope I (or somebody else) hasn't already asked this before. A cursory
check of the google archives didn't turn up immediate results.

Anyways, every once in a while I dig up the papers on the black hole
entropy in loop quantum gravity and try to understand them. I admittedly
haven't spent too much time in this endeavor, so this is assuredly my
failing, not the authors. Nonetheless, I looked to Rovelli's book on his
webpage and examined his derivation. This crystallized some of my
half-remembered thoughts about the earlier works. The short question is:

How is the lqg derivation of black hole entropy not assuming that which
was meant to be proven?

As best I can tell, Rovelli's argument starts with the idea that the
degrees of freedom of a black hole should be found on its horizon and
then shows that the entropy is proportional (with an unknown
proportionality constant) to the area of the horizon. Is there a way
this calculation could have failed to produce this result?

ISTR that in older papers, spacetime was essentially divided into two
chunks, separated by the black hole horizon. Then, certain boundary
conditions were put on the horizon and a Lagrangian only describing the
portion of spacetime outside the horizon was quantized. Then certain
degrees of freedom are discarded as 'volume' degrees of freedom, leaving
only the 'surface' degrees of freedom behind. The log of the number of
these degrees of freedom again turns out to be proportional to the area.

Forgive me if I've mischaracterized the work here.

My problem is that there seems to be entirely too much a priori
knowledge used in these calculations. It's not terribly surprising that
if one assumes the degrees of freedom are localized on the horizon, you
get an entropy proportional to the area. The 'physical input' into the
problem seems to be effectively determining the answer. Can an ab initio
calculation be done? Am I just completely misunderstanding the work that
has been done?

Aaron

### Urs Schreiber

Oct 21, 2003, 6:13:21 PM10/21/03
to

Aaron Bergman wrote:

> I hope I (or somebody else) hasn't already asked this before. A cursory
> check of the google archives didn't turn up immediate results.

[...]

> How is the lqg derivation of black hole entropy not assuming that which
> was meant to be proven?

I have once asked this question in

where I wrote:

>"John Baez" <ba...@galaxy.ucr.edu> schrieb im Newsbeitrag
>news:apnhid\$q8b\$1...@glue.ucr.edu...
>
>[...]
>> since in that calculation we are counting
>> states of the horizon, not states of the bulk
>
>But why? Initially one wants to know the entropy of the Schwarzschild
>spacetime. From semiclassical calculations one knows that this is
>purely due to the horizon. But in LQG one does not want to use and
>rely on this semiclassical reasoning, but rather to justify it on
>deeper grounds. No?

John Baez' answer to that in
was

>In article <apo7bi\$h50\$1...@rs04.hrz.uni-essen.de>,
>Urs Schreiber <Urs.Sc...@uni-essen.de> wrote:
>
>>"John Baez" <ba...@galaxy.ucr.edu> schrieb im Newsbeitrag
>>news:apnhid\$q8b\$1...@glue.ucr.edu...
>>> since in that calculation we are counting
>>> states of the horizon, not states of the bulk
>>But why? Initially one wants to know the entropy of the Schwarzschild
>>spacetime.
>
>I guess it depends on who one is. :-)
>
>I want to know the entropy of a black hole, not the
>entire spacetime it sits in. For example, if we have
>a black hole in our actual universe, the entire spacetime
>is very far from the Schwarzschild geometry. Nonetheless,
>one can hope that the entropy of the black hole is well-defined
>and asymptotically proportional to the area of its event horizon
>(in the limit of black holes much larger than the Planck length).
>This raises the question of where the entropy actually "lives".
>Given that it's proportional the area of the horizon, a natural
>guess is: the entropy lives on the horizon.
>
>> From semiclassical calculations one knows that this is purely due to
>>the horizon. But in LQG one does not want to use and rely on this
>>semiclassical reasoning, but rather to justify it on deeper grounds. No?
>
>It would be nice to understand what happens to the singularity
>in loop quantum gravity, but we don't yet. So there are still
>mysteries about the precise nature of the interior of a black hole
>in loop quantum gravity. Right now, the best we can do is quantize
>gravity with boundary conditions asserting the existence of an event
>horizon - or more precisely, a marginally outer trapped surface.
[...]
>>Since one does not know yet how physical bulk states and boundary states are
>>related in LQG I suppose this still leaves the interesting possibility that
>>they really are in 1-1 correspondence, which would be holography.
>
>This would happen if there were really just *one* solution of the
>Hamiltonian constraint per state of the horizon.

My impression is that it boils down to the question of how
much the state on the horizon fixes the solution to the
Hamiltonian constraint in the bulk.

### Michael Petri

Oct 21, 2003, 6:27:51 PM10/21/03
to

"Aaron Bergman" <aber...@physics.utexas.edu> wrote
news:abergman-1F5537.09370220102003@localhost...

> ISTR that in older papers, spacetime was essentially divided into two
> chunks, separated by the black hole horizon. Then, certain boundary
> conditions were put on the horizon and a Lagrangian only describing the
> portion of spacetime outside the horizon was quantized. Then certain
> degrees of freedom are discarded as 'volume' degrees of freedom, leaving
> only the 'surface' degrees of freedom behind. The log of the number of
> these degrees of freedom again turns out to be proportional to the area.
>
> Forgive me if I've mischaracterized the work here.
>
> My problem is that there seems to be entirely too much a priori
> knowledge used in these calculations. It's not terribly surprising that
> if one assumes the degrees of freedom are localized on the horizon, you
> get an entropy proportional to the area. The 'physical input' into the
> problem seems to be effectively determining the answer. Can an ab initio
> calculation be done? Am I just completely misunderstanding the work that
> has been done?

I actually have the same problem. Now, I wouldn't be surprised, if the
problem just is that I don't understand the technicalities of the procedure,
that selects the horizon surface states and throws out all of the bulk
states.

Maybe it is helpful to formulate my problem somewhat differently: How can we
be sure, that the area of the spin-network state, whose entropy we determine
by counting all possible combination of punctures, *always* corresponds to
the area of the horizon of a black hole. I mean, when we usually measure an
area, it is *not* the horizon area of a black hole.

Best, Mike

### Mark

Oct 24, 2003, 2:03:02 PM10/24/03
to

Urs Schreiber <Urs.Sc...@uni-essen.de> writes:
>But why? Initially one wants to know the entropy of the Schwarzschild
>spacetime.

[and pertaining to the general question of why the "inside" isn't
counted in the total].

"John Baez" <ba...@galaxy.ucr.edu> schrieb im Newsbeitrag

>I guess it depends on who one is. :-)
>I want to know the entropy of a black hole, not the

>entire spacetime it sits in. [...] Nonetheless,

>one can hope that the entropy of the black hole is well-defined
>and asymptotically proportional to the area of its event horizon

[...]

>This raises the question of where the entropy actually "lives".
>Given that it's proportional the area of the horizon, a natural
>guess is: the entropy lives on the horizon.

Assuming it's localized at a point, however, is not a natural
assumption to make; since the total entropy is proportional to a
measure that is one dimension *less* than space.

That means it's more appropriate to think of it as being one dimension
*more* than a point: namely a thread that pierces THROUGH the event
horizon, rather than resides on it.

A bit is the ghost of a departed trajectory.

So the question of where it resides is misplaced. It's the entire
thread itself. Enclosing a region of spacetime by a surface,
you're actually counting the number of threads that pierce through
the boundary, not the number of "cells" that reside within the region.

### leo...@planck.com

Oct 26, 2003, 1:53:32 AM10/26/03
to
Aaron Bergman <aber...@physics.utexas.edu> wrote:

>The short question is:
>
>How is the lqg derivation of black hole entropy not assuming that which
>was meant to be proven?

If you haven't already done so, you might enjoy looking
at a 5 page 1996 paper by Rovelli called "Black Hole Entropy from Loop
Quantum Gravity"
http://arxiv.org/gr-qc/9603063

This gets a preliminary approximation by a simple elegant
combinatorial argument: counting of partitions, and makes the
mathematical content accessible without technical encumbrance.
However it is only an approximation. As you doubtless know the real
result first appeared in the 8 page 1997 paper of Ashtekar, Baez,
Corichi, Krasnov (known as ABCK) called "Quantum Geometry and Black
Hole Entropy" http://arxiv.org/gr-qc/9710007

In ABCK there is non-trivial mathematical content e.g. invoking Chern-Simons
theory but because the proof uses more in the way of machinery it is not
so clear what is happening and why the entropy turns out as it does.

The quantum state of the geometry is *not* confined a priori to the
horizon but can be visualized as a space-filling spin network. The
horizon area depends on this quantum state and is expressed a certain
sum of square-roots of colors or labelings of edges puncturing the
surface. This is simply the well-known LQG area formula---one of the
theory's basic results.

Rovelli uses elementary combinatorics to count the number of possible
relabellings of the spin network that would result in the same area.
He approximates the sum of square roots by a sum of integers, so that
he is now counting partitions (n-tuples adding up to the same thing.)
He gets that a certain ratio is bounded by the logarithm of the golden
mean and the logarithm of 2. The result remains approximate and whole
thing takes only a couple of pages.

>My problem is that there seems to be entirely too much a priori
>knowledge used in these calculations. It's not terribly surprising that
>if one assumes the degrees of freedom are localized on the horizon, you
>get an entropy proportional to the area. The 'physical input' into the
>problem seems to be effectively determining the answer. Can an ab initio
>calculation be done? Am I just completely misunderstanding the work that
>has been done?

I am not sure I can help you get over this feeling of being cheated.
The degrees of freedom determining the quantum state of the geometry
are not localized on the horizon. The fundamental hard theorem in the
picture is the LQG area formula telling you how the quantum state
determines the area of any given surface. The counting of microstates
with the same macrostate is very closely connected to counting states
that give the same *outward appearance* namely the same area.

### Michael Petri

Oct 28, 2003, 4:19:46 PM10/28/03
to
<leo...@planck.com> schrieb im Newsbeitrag
news:mailman.106669...@olympus.het.brown.edu...

> In ABCK there is non-trivial mathematical content e.g. invoking Chern-Simons
> theory but because the proof uses more in the way of machinery it is not
> so clear what is happening and why the entropy turns out as it does.

So how does Chern-Simons theory tell us, that the Quantum gravity area
formula refers to the event horizon of a black hole?

> The quantum state of the geometry is *not* confined a priori to the
> horizon but can be visualized as a space-filling spin network. The
> horizon area depends on this quantum state and is expressed a certain
> sum of square-roots of colors or labelings of edges puncturing the
> surface.

Here you have innocuously "slipped in" the assumption, that the quantum
gravity area formula refers to the area of the *event horizon* of a black
hole. What is the justification, other than that you already know, that the
result of the combinatorics, S propto A, is known to hold only for black
holes?

To make my criticism clear: LQG claims, that the area formula refers to
*any* area measurement of the physical state. So when I measure the area of
the computer screen in front of me, LQG allows me to determine the number of
"quantum distinguishable" possibilities (combinations of colors or labeling
of edges puncturing the screen) that amount to the macroscopic area of the
screen. But quite clearly the logarithm of this number, i.e. the entropy of
the screen, is much to high to be identified with the negligible entropy of
my computer screen. So what is the justification, that the LQG area formula
is only applied to black holes, but not to the "normal" measurements we
perform every day? It might be, that Cherns-Simons theory has a say in this,
but I neither see it, nor has anybody explained it in simple terms. Now, I
don't have to understand it myself, I would be quite content if somebody who
understands would just tell me with due authority, that Cherns-Simons theory
just does the trick. (Not that I would believe him/her, but I would be
motivated enough to check it myself ;-))

> I am not sure I can help you get over this feeling of being cheated.
> The degrees of freedom determining the quantum state of the geometry
> are not localized on the horizon.

But the punctures are all that "counts". And they are "localized" within the
area that you are measuring, which is (always) the horizon area of a black
hole, at least that's what you are saying.

> The fundamental hard theorem in the
> picture is the LQG area formula telling you how the quantum state
> determines the area of any given surface.

...the area of *any given* surface? Even if it's not the surface of a black
hole?

So you see the problem, now, do you ?! ;-)

Best, Mike

### John Baez

Oct 28, 2003, 4:56:54 PM10/28/03
to
In article <abergman-1F5537.09370220102003@localhost>,
Aaron Bergman <aber...@physics.utexas.edu> wrote:

>As best I can tell, Rovelli's argument starts with the idea that the
>degrees of freedom of a black hole should be found on its horizon and
>then shows that the entropy is proportional (with an unknown
>proportionality constant) to the area of the horizon. Is there a way
>this calculation could have failed to produce this result?

Sure - and in fact the ABCK (Ashtekar-Baez-Corichi-Krasnov)
calculation of black hole entropy does *not* get an entropy
proportional to the area of the horizon! The entropy is only
*asymptotically* proportional to the area of the horizon.
For small black holes there are quantum corrections.

>ISTR that in older papers, spacetime was essentially divided into two
>chunks, separated by the black hole horizon. Then, certain boundary
>conditions were put on the horizon and a Lagrangian only describing the
>portion of spacetime outside the horizon was quantized. Then certain
>degrees of freedom are discarded as 'volume' degrees of freedom, leaving
>only the 'surface' degrees of freedom behind. The log of the number of
>these degrees of freedom again turns out to be proportional to the area.
>
>Forgive me if I've mischaracterized the work here.

You've described it pretty well.

>My problem is that there seems to be entirely too much a priori
>knowledge used in these calculations. It's not terribly surprising that
>if one assumes the degrees of freedom are localized on the horizon, you
>get an entropy proportional to the area.

No, it's not terribly surprising - but we weren't aiming for shock
value. It's been pretty obvious ever since people discovered
the second law of black hole thermodynamics that that there must
be some degrees of freedom "living on the horizon" that give black
holes a quarter-bit of entropy per Planck length squared. Our goal
was not so much to rederive this known formula, as to give a precise
description of what those degrees of freedom might be.

So, to my mind the exciting aspect of this paper is that we:

1) found boundary conditions for general relativity describing
an "isolated horizon" - that is, one with no matter or
gravitational waves falling in.

2) saw that with these boundary conditions, general relativity
has a well-defined phase space in which the symplectic structure
is the sum of two parts: a "volume" part and a "surface" part.

3) saw that the "surface" part of the symplectic structure is
exactly the same as that for U(1) Chern-Simons theory on the
horizon - where the U(1) connection describes the intrinsic
geometry of the horizon, and the "level" of the Chern-Simons
theory is proportional to the horizon area.

4) saw that the curvature of the U(1) connection is zero except
where spin network edges poke through the horizon - this is
forced by the boundary conditions.

5) quantized the surface degrees of freedom (the U(1) connection)
and got a Hilbert space of states which have a clear geometrical
interpretation as "quantum geometries of the horizon".

6) computed the log of the dimension of this Hilbert space - the
entropy - and found that it was asymptotically proportional
to the area of the black hole.

>The 'physical input' into the problem seems to be effectively
>determining the answer. Can an ab initio calculation be done?

The physical input to a problem always determines the answer, of
course - it's all a question of how much suspense is involved.

If you find it insufficiently shocking that the entropy turned out
asymptotically proportional to the area, maybe you'd still agree
that items 1)-5) are somewhat interesting. To me, these are what
made the paper worthwhile: we got a nice relation between conditions
describing a black hole event horizon and Chern-Simons theory on the
horizon, which allowed us to understand black hole entropy in a
geometrical way: different microstates correspond to different
intrinsic geometries of the horizon.

But, you are right to be dissatisfied that we are *quantizing gravity
with boundary conditions that say an event horizon exists* instead
of *quantizing gravity in general, and finding certain states that
correspond to black holes*.

The second approach would be nicer, but it's too hard for us, at least
right now. Our calculation is a bit like the calculation of the Casimir
effect between conductive plates where you model the conductive plates
via a certain boundary condition on the electromagnetic field. It
gives the right answer, and it gives a certain amount of insight.
But, it's far from an "ab initio" calculation of the Casimir effect
where you use QED to describe copper atoms, then sheets of copper,
and finally the Casimir effect. The latter would be much harder,
and far from being exactly solvable in closed form.

### Lubos Motl

Oct 29, 2003, 2:56:17 PM10/29/03
to

On Tue, 28 Oct 2003, John Baez wrote:

> Sure - and in fact the ABCK (Ashtekar-Baez-Corichi-Krasnov)
> calculation of black hole entropy does *not* get an entropy
> proportional to the area of the horizon! The entropy is only
> *asymptotically* proportional to the area of the horizon.
> For small black holes there are quantum corrections.

That's right. Your calculation was constructed in such a way that it had
to lead to the entropy proportional to the area for *large* black holes -
in the limit where the areas behave properly etc. It's obvious that for
small black holes there might be *some* corrections (not necessarily the
correct physical ones). For example, the contribution from the spin J to
the area of regular surfaces in LQG goes like sqrt(J(J+1)) which is not
quite J+1/2, it is not quite J, but for large J these differences don't
matter and one can always use the classical intuition. According to the
same classical intuition, it was simply guaranteed that the entropy will
scale like the area in the limit where the whole calculation is local on
the horizon.

Don't get me wrong. The fact that the entropy is proportional to the area
*does* suggest that there should be a description that is local on the
horizon - and where this universal law is manifest. But the statement in
the previous sentence is not your idea - and it is unrelated to LQG. You
result had to agree with the rest of the paradigm. Nothing surprising was
going on.

> 1) found boundary conditions for general relativity describing
> an "isolated horizon" - that is, one with no matter or
> gravitational waves falling in.

The interior of the black hole is causally disconnected, so it is
guaranteed that there must exist a description appropriate for the
exterior observer where the interior does not matter: there had to exist
these boundary conditions. However, it is not quite the same statement as
"the interior does not exist". The interiors of stringy black holes are
not visible either (I apologize to Steve Shenker et al. for not including
the results of his recent papers), but we are not pretending that it is an
advantage of [our understanding of] the theory. On the contrary. It is
our ignorance that we don't quite know how the stringy laws should exactly
be applied to the interior.

> 2) saw that with these boundary conditions, general relativity
> has a well-defined phase space in which the symplectic structure
> is the sum of two parts: a "volume" part and a "surface" part.

That's a part of your description, but again I don't understand why you
think that it is a virtue. Such an intermediate result violates the
equivalence principle - the observer should feel nothing special when she
crosses the horizon, and your point of view has obviously no explanation
for this statement of general GR. On the contrary, it seems to be in a

The Chern-Simons counting is just a technical feature of your specific toy
model, and it is certainly not a general feature of all black holes, so
let me avoid these minor details. The leading result in your toy model can
be derived without any reference to Chern-Simons theory anyway. It is not
surprising that there is CS theory on the boundary - if you create a
3-dimensional domain wall in a 4-dimensional theory (or even non-theory)
whose local physics has a simple description, you must get a 3-dimensional
theory with a simple description, too. It does not matter whether it is
Chern-Simons theory. If you derived that the boundary theory were the
Clinton-Gore theory, you would expect us to be equally interested anyway.

You know, what is disappointing about your calculation is that there is
nothing surprising going on, no consistency check. The only nice surprise
could have been the numerical constant, but you obtain an incorrect one.
And you know that all proposed explanations of this incorrect value have
been proved wrong.

The proportionality itself (in the limit of large geometry) was included
as a starting point. A physicist who is used to check all statements and
their coherence in many complementary way must certainly feel that your
calculation does not say anything important about nature - it is just a
human construct that can't be pushed too far. Finally, the result does not
depend on any details of LQG.

This is a drastically different situation from string theory. In string
theory the whole result for the extremal and non-extremal black holes is
completely correct and completely nontrivial. The proportionality works,
but you can't see it even 30 seconds before you finish the calculation.
None is removing the interior by hand. The numerical factor works, and it
looks like another miracle. The result is invariant under all dualities
etc. I don't know whether you want to understand what we feel - but some
theories simply smell to be correct and others smell to be incorrect.
______________________________________________________________________________
E-mail: lu...@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
phone: work: +1-617/496-8199 home: +1-617/868-4487
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Superstring/M-theory is the language in which God wrote the world.

### Aaron Bergman

Oct 31, 2003, 6:30:43 PM10/31/03
to
In article <bnmon6\$pdf\$1...@glue.ucr.edu>, ba...@galaxy.ucr.edu (John Baez)
wrote:

> In article <abergman-1F5537.09370220102003@localhost>,
> Aaron Bergman <aber...@physics.utexas.edu> wrote:
>
> >As best I can tell, Rovelli's argument starts with the idea that the
> >degrees of freedom of a black hole should be found on its horizon and
> >then shows that the entropy is proportional (with an unknown
> >proportionality constant) to the area of the horizon. Is there a way
> >this calculation could have failed to produce this result?
>
> Sure - and in fact the ABCK (Ashtekar-Baez-Corichi-Krasnov)
> calculation of black hole entropy does *not* get an entropy
> proportional to the area of the horizon! The entropy is only
> *asymptotically* proportional to the area of the horizon.
> For small black holes there are quantum corrections.

That's to be expected for any quantum situation. It's clear that
something has been quantized here -- I'm just not sure what it is.

Aaron

### Urs Schreiber

Nov 1, 2003, 12:19:13 PM11/1/03
to
I am currently at the "Strings meet Loops" symposium at the
Albert-Einstein-Institute in Potsdam, Germany, and I have taken the
opportunity to interview A. Ashtekar after dinner on the entropy
caclulation of black holes in LQG and other issues. John Baez has also
Ashtekar said as I understood it.

Regarding the question of whether the proportionality (to first order)
of the entropy of a black hole to the area of its horizon is an
assumption or a result of LQG Ashtekar essentially made two points:

1) That the interior of the black hole does not enter the calculation
is a result of how the phase space of the system has to be
constructed.

I should have asked for further details on this point because I don't
understand it well. My impression was however that the interior of the
black hole can be removed due to purely kinematical considerations.
Apparently this step is trivial to those working in the filed, even
though to others it may contain the crucial information.

2) Even though it now follows that the entropy is entirely that of a CS
theory on the horizon, Ashtekar stressed that this does not imply at
all that we shoud automatically expect it to be proportional to the
area of the horizon. Rather, he said, the entropy of CS on a punctured
sphere depends crucially on the number of punctures, Since these
punctures are due to spin networks poking throught the horizon, the
nature of spin networks enters crucially into the calculation of the
area. Therefore it is nontrivial that the entropy calculated this way
for black holes really is propertional to the area.

We have talked about other things which I found highly interesting and
which I would like to discuss further on s.p.r., but right now I am
way too tired for that... :-)