Black hole entropy and the log of the golden mean

In summary, Rovelli finds the black hole entropy/area formula by a simple counting method. He uses the logarithm of the golden mean to find the BH entropy. This is similar to the first paper to be written on the subject, which was done in 1996.
  • #1
marcus
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there is a curious 1996 paper by Rovelli that gets the
black hole entropy/area formula by a simple counting method.

I say simple advisedly---a lot of combinatorics and counting is
not really simple at all but IMHO difficult---but in this amazing little
5 page paper the counting of partitions of a number, which is all it really is, really is simple

and like so much other combinatorics and elementary arithmetical jazz the golden mean shows up, so he is finding the BH entropy using the logarithm of the golden mean---got a chuckle out of that for some reason

well, here's the link if you want a look
http://arxiv.org/gr-qc/9603063 [Broken]

notice that he was just looking at a Schwarzschild hole, looking at the simplest thing he could, and oversimplifying indeed IMO not even paying attention like riding a bicycle with your eyes shut and the amazing thing is that his was one of the very first papers (either in string or loop) and he came within a factor the size of the, say, immirzi parameter (which he was ignoring). such things are governed by a kind of graceful luck I believe.

For comparison here is a 1996 string paper which is the first
instance of a string explanation.
http://arxiv.org/hep-th/9601029 [Broken]
It is by Strominger and Vafa and applies to 5-dimensional "extremal" holes by counting the degeneracy of "BPS soliton"
bound states.


Marcus
_______________

A foxhunt by British gentry has been described as "the unspeakable in pursuit of the inedible"
 
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  • #2
the counting

It turns out that all Rovelli has to do to find the entropy of the BH is to get a grip on something he calls N(M) which is

the number of sequences of positive integers p1, p2,...,pn of any finite length
which "add up" to a large number M, in a certain unusual way:

Σ sqrt(pi(pi + 2)) = M


This is almost like finding the number of PARTITIONS of the number M----the finite sequences of whole numbers that
add up to M in the usual sense
-------------------
Why is it that easy? Because for a given spin-network, or a given state of the geometry (a given giant polymer representing the exitation of the geometry that confers area and volume on things)
the area of the BH is

A = 8 pi hbar G Σ sqrt(pi(pi + 2))

that is just the renouned and revered quantum gravity area formula, where the p's are numbers 1,2,... called colors attached to segments of the polymer which happen to pass thru the surface----the edges of the graph confer area to surfaces they pass thru and the vertices confer volume to regions they lie within.
The graph is just a way of summarizing a lot of geometrical info like that in a way that is highly adapted to being quantized and makes for a good theory.

Because of background independence or so called covariance all that matters is the number of punctures and the colors on them---location is factored out as a "gauge" or physically irrelevant thing.
So all that matters is the p's and all he has to do is count them
------------------------------

And it's not too different from counting arithmetical partitions of integers.

To keep his backpack light, Rovelli defines

M = A/(8 pi hbar G)

So now counting the p's that give him A
A = 8 pi hbar G Σ sqrt(pi(pi + 2))
is the same as counting the p's that give him M
M = Σ sqrt(pi(pi + 2))
And N(M) ----the number of microstates----is the how many
p-sequences do that. As I mentioned at the top.
And the entropy is going to be the log of the number of (independent, i.e. really different) microstates that give the same
outward appearance namely the same area.

So he is going after the logarithm of N(M)

In fact Rovelli gets a grip on N(M) by clamping it between upper and lower bounds which do count ordinary-type partitions and which he denotes for convenience by N+(M) for the upper and by N-(M) for the lower



His N+(M) is just the number of finite sequences that satisfy a slightly less stringent condition namely

Σ sqrt(pi(pi)) = M

It is the same as before but with the 2 removed and it boils down to simply

Σ pi = M

The number of those things is 2M-1 so there is no trouble seeing

ln N(M) < M ln 2

To get his lower bound N-(M) he needs a slightly more stringent condition and he recalls the old Grade School thing
that A2 - B2 = (A + B)(A - B) and says

sqrt(pi(pi + 2)) = sqrt((pi + 1)2 - 1) which is about (pi + 1)

So his His N-(M) is just the number of finite sequences that satisfy a slightly more stringent condition namely

&Sigma; (pi + 1) = M

this means counting the partitions of M into chunks that are at least 2 in size and the Golden Mean comes in and he finds

ln N-(M) = M ln (1 + sqrt 5)/2 = M ln (G.M.)

Now he has some number D which is estimated to be between
ln (G.M.) and ln 2 and

ln N(M) = D M = D A/(8 pi hbar G) = (D/(8 pi hbar G)) A

the rest is mopping up
according to the official def of thermodynamical entropy
he is looking for Boltzmann's k times log of the number of states corresponding to a certain area. N(M) and N(A) are the same
state-count

S = k ln N(A) = M = [kD/(8 pi hbar G)] A

That means he has derived Beckenstein-Hawking except that the number in front is kD/8 pi instead of 1/4. But it's 1996 and that will be taken care of in time----what matters is the proportionality to the area (it could be some totally screwed up function of the area or no clear function of it at all but it isnt---it is some small number times the area). So it seems like a good place to stop.
 
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  • #3



Thank you for sharing this interesting information about black hole entropy and the log of the golden mean. It is fascinating to see how the golden mean, a mathematical concept often associated with beauty and harmony, can also play a role in understanding the mysteries of black holes. I find it amusing that the author of the paper found a way to use the golden mean in their calculations, and it goes to show that sometimes simplicity can lead to unexpected and elegant solutions.

It is also interesting to see the comparison with the string explanation for black hole entropy by Strominger and Vafa. It seems that different approaches and methods can lead to similar results, further highlighting the complexity and beauty of these phenomena.

On a lighter note, I couldn't help but chuckle at the foxhunt analogy. It is a humorous way to describe the pursuit of knowledge and understanding, and perhaps a reminder to not take ourselves too seriously in our quest for answers. Thank you again for sharing these insights.
 

What is black hole entropy?

Black hole entropy is a measure of the disorder or randomness inside a black hole. It is related to the number of microscopic quantum states that a black hole can have.

How is black hole entropy related to the log of the golden mean?

The log of the golden mean, also known as the logarithmic spiral, is a geometric pattern often found in nature. It has been proposed that the log of the golden mean may be related to black hole entropy through the holographic principle.

What is the holographic principle?

The holographic principle states that all the information contained within a region of space can be represented by the information on the boundary of that region. This theory has been applied to black holes, suggesting that the entropy of a black hole is proportional to its surface area rather than its volume.

What is the significance of the log of the golden mean in relation to black hole entropy?

The relationship between the log of the golden mean and black hole entropy is still a topic of ongoing research. Some scientists believe that it may provide a deeper understanding of the nature of black holes and the universe as a whole.

Can the log of the golden mean be used to solve the black hole information paradox?

The black hole information paradox is a long-standing problem in physics that arises from the conflicting theories of general relativity and quantum mechanics. While the log of the golden mean has been proposed as a potential solution to this paradox, it is still a subject of debate and further research is needed.

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