Loop Quantum Gravity: Explained for Physics Laymen

[marcus]

Originally posted by marcus
... So then the ES area formula is

A_S = (4 ln 3)\frac{2}{3} \Sigma (j_n + 1/2)

the fraction 2/3 is what still needs discussion

It is getting clearer that in the case of a black hole event horizon the area could gain and lose in steps of (4 ln 3) because that is already appearing in the formula.

Gour and Suneeta give a thermodynamical argument that the maximum entropy, for a hole of a given area, is achieved by having all or virtually all of the
edges passing thru the surface be labeled j = 1.

In that case you can easily see that the terms you add up in the sum are (1 + 1/2) which is 3/2

that cancels the 2/3

So the basic picture of a black hole that these two scholars give us is that it is like a pincushion punctured by jillions of little network edges all labeled 1 and that the area is just
(4 ln 3) times the number of punctures!

So in the course of random fluctuations or vibrations or whatever, the kind of jangling jitter always happening in the world, the hole is always gaining or losing area in quantum steps of size (4 ln 3)

The mass-energy of a black hole, and also properties like temperature and entropy, are related to its surface area. Having a quantum handle on the area helps get a grip on other things as well.

So, these upstart "equidistant spectrum" people say, the number 4 times the logarithm of 3 is a good number to remember in connection with LQG theory of black holes.

 

[marcus]

Some people might be interested in working thru some elementary arithmetic around Gour/Suneeta equation (8), having to do
with "degeneracy" or with counting states.

Suppose we think of the BH surface as punctured by edges all of the same j, either j = 1/2, 1, or 3/2,

then to make a given area there have to be this many punctures
in the three cases
N_{1/2} = Q
N_1 = (2/3)Q
N_{3/2} = (1/2)Q.

For larger j, you get to have fewer punctures because each puncture contributs more, according to that area formula. The number Q is just an alias for N-sub-1/2 to show the relation between the numbers in a clean way. You could replace it simply by
N_{1/2
and not even use the symbol Q.

The dimension of the state space associated with that many punctures all having that particular spin is
g(j, N_j) = (2j + 1)^{N_j}
This is their equation (7),
officially it's called the degeneracy for a particular spin j it's the dimension of a tensorproduct of a lot of hilbertspaces.


OK now we evaluate their equation (7) using what we already know about N's.

g(1/2, N_{1/2}) = (1 + 1)^{N_{1/2}}= 2^Q
g(1, N_1) = (2 + 1)^{N_1} = 3^{\frac{2}{3}Q}
g(3/2, N_{3/2}) = (3 + 1)^{N_{3/2}} = 4^{Q/2} = 2^Q

So the degeneracy for j = 1/2 and j = 3/2 is the same in both cases and LESS than that for j = 1. So they say there is a pile-up with virtually all the punctures being j = 1 and achieving the maximum entropy with is the logarithm of this state-counting degeneracy thing.

that is their equation (8) which seems like the key step in the paper and it is kind of elementary so I copied it in
Gour and Suneeta seem smart and Polychronakos too. It has the air of being pretty reasonable, just different from the way LQG originally came down with Rovelli and Smolin. I wonder how this will sort out.

 

[marcus]

Oh yeah the Bekenstein-Hawking or whatever formula for the entropy of a BH should come out of this without much trouble.

I should only need to take the logarithm of the degeneracy
and say what Q is and that should give it.

Let us take logarithm of both sides of this equation

g(1, N_1) = (2 + 1)^{N_1} = 3^{\frac{2}{3}Q}

entropy = (\frac{2}{3}Q)ln 3

Q = \frac{A}{(2/3)4 ln 3} (see footnote*)

entropy = \frac {A}{4}

So it comes out really easily.


* this expression for Q comes from the area formula
I wrote a couple of posts back. the one cleaned up so
it didnt have the eyesore gamma sticking out like sore thumb

So hey here is the famous entropy formula: S = A/4

 

[marcus]

Originally posted by marcus
...the ES area formula is

A_S = (4 ln 3)\frac{2}{3} \Sigma (j_n + 1/2)

...

this is the area formula I was referring to
if all the punctures have j=1 then to get the sum right
the number of punctures has to be the area divided by
(4 ln 3)\frac{2}{3}
this is what I was using in the previous post


This ES area formula has a nice quantum correction to the
Hawking radiation spectrum
for very long wavelengths comparable in size to the black hole itself

It looks like a black body spectrum for short wavelengths but
as Polychronakos says,
"the high-frequency exponential part of the spectrum is accurately reproduced, the discreteness there being inconsequential. This is the energy range in which photons (and other emitted particles) behave essentially like classical particles...For frequencies close to the thermal frequency
[that is the kT freqency where T is the temp of the BH]
however, the wavelength of the photons becomes comparable to the size
of the black hole and they sense global properties of its geometry. Back reaction due to geometry change at emission and absorption of such photons is expected to be important, the energy of these photons being of the same order as the energy spacing of the black hole. A deviation from ideal black-body spectrum, which assumes a fixed metric and ignores back-reaction, would seem reasonable..."

good old Polychronakos!
http://arxiv.org/hep-th/0304135
page 9

I think these ES people make a reasonable case for the idea.
have to give it some more thought, hope others too

 

[8LPF16]

Marcus,


Would you look at my chart in Theory Development "Resonating Vibrational Potentials", and tell me what you think?


LPF

 

[meteor]

marcus, if you want to discuss about the equally-spaced area spectrum here, then I will post this recent paper:
http://arxiv.org/abs/hep-th/0401187
"Area spectrum of Kerr and extremal Kerr black holes from quasinormal modes"
Authors: Mohammad R. Setare, Elias C. Vagenas
Abstract:
Motivated by the recent interest in quantization of black hole area spectrum, we consider the area spectrum of Kerr and extremal Kerr black holes. Based on the proposal by Bekenstein and others that the black hole area spectrum is discrete and equally spaced, we implement Kunstatter's method to derive the area spectrum for the Kerr and extremal Kerr black holes. The real part of the quasinormal frequencies of Kerr black hole used for this computation is of the form $m\Omega$ where $\Omega$ is the angular velocity of the black hole horizon. The resulting spectrum is discrete but not as expected uniformly spaced. Thus, we infer that the function describing the real part of quasinormal frequencies of Kerr black hole is not the correct one. This conclusion is in agreement with the numerical results for the highly damped quasinormal modes of Kerr black hole recently presented by Berti, Cardoso and Yoshida. On the contrary, extremal Kerr black hole is shown to have a discrete area spectrum which in addition is evenly spaced. The area spacing derived in our analysis for the extremal Kerr black hole area spectrum is not proportional to $\ln 3$. Therefore, it does not give support to Hod's statement that the area spectrum $A_{n}=(4l^{2}_{p}ln 3)n$ should be valid for a generic Kerr-Newman black hole.


The area spectrum of normal Kerr black holes is not evenly spaced: Rovelli will be happy. The area spectrum of extremal Kerr black holes is evenly spaced: Bekenstein will be happy

 

[marcus]

Originally posted by meteor
...
http://arxiv.org/abs/hep-th/0401187
"Area spectrum of Kerr and extremal Kerr black holes from quasinormal modes"
Authors: Mohammad R. Setare, Elias C. Vagenas
...
...
The area spectrum of normal Kerr black holes is not evenly spaced: Rovelli will be happy. The area spectrum of extremal Kerr black holes is evenly spaced: Bekenstein will be happy

Meteor, this is great to have! Already something contesting, at least in part, the equal spacing idea and the logarithm of 3. I will post part of the abstract again, with emphasis, and then give a supporting link.

--------part of abstract----
...The resulting spectrum is discrete but not as expected uniformly spaced. Thus, we infer that the function describing the real part of quasinormal frequencies of Kerr black hole is not the correct one. This conclusion is in agreement with the numerical results for the highly damped quasinormal modes of Kerr black hole recently presented by Berti, Cardoso and Yoshida. On the contrary, extremal Kerr black hole is shown to have a discrete area spectrum which in addition is evenly spaced. The area spacing derived in our analysis for the extremal Kerr black hole area spectrum is not proportional to ln 3. Therefore, it does not give support to Hod's statement that the area spectrum

A_{n}=4l^{2}_{p}ln 3

should be valid for a generic Kerr-Newman black hole.
-----end quote from abstract---

Here is the recent paper they refer to, posted January 13.

http://arxiv.org/gr-qc/0401052


"Highly Damped Quasinormal Modes of Kerr Black Holes: A Complete Numerical Investigation"

Emanuele Berti, Vitor Cardoso, Shijun Yoshida
Comments: 5 pages, 3 figures

----from their abstact----
We compute for the first time very highly damped quasinormal modes of the (rotating) Kerr black hole. Our numerical technique is based on a decoupling of the radial and angular equations, performed using a large-frequency expansion for the angular separation constant...
---end quote---

----exerpt from Berti/Cardoso/Yoshida text---

Black holes (BHs), as many other objects, have characteristic
vibration modes, called quasinormal modes
(QNMs). The associated complex quasinormal frequencies
(QN frequencies) depend only on the BH fundamental
parameters: mass, charge and angular momentum.
QNMs are excited by BH perturbations (as induced, for
example, by infalling matter). The early evolution of a
generic perturbation can be described as a superposition
of QNMs, and the characteristics of gravitational radiation
emitted by BHs are intimately connected to their
QNM spectrum. One may in fact infer the BH parameters
by observing the gravitational wave signal impinging
on the detectors [1]: this makes QNMs highly relevant in
the newly born gravitational wave astronomy [2, 3].
Besides this “classical” context, QNMs may find a very
important place in the realm of a quantum theory of gravity.

General semi-classical arguments suggest [4] that on
quantizing the BH area one gets an
evenly spaced spectrum of the form
A_n = 4 log (k) (l_P)^2 n; n = 0, 1, ... (1)
where l_P is the Planck length, and k is an integer to be
determined.

Hod [5] proposed to fix the value of k, and
therefore the area spectrum, by promoting QN frequencies
with a very large imaginary part to a special position:
they should bridge the gap between classical and quantum
transitions. Hod obtained, for the Schwarzschild
BH, k = 3.

Following his proposal, further enhanced by
the prospect of using similar reasoning in Loop Quantum
Gravity to fix the Barbero-Immirzi parameter [6], the interest
in highly damped BH QNMs has grown considerably
[7].

There is now reason to believe that the connection
between QN frequencies and the BH area quantum
is not as straightforward as initially suggested. However
a relation between classical and quantum BH properties
does seem to exist, even in non-asymptotically flat
spacetimes [8].

A prerequisite to study this connection is
to compute QN frequencies having very large imaginary
part. So far this problem has been solved only for a few
special geometries: Schwarzschild BHs [9, 10, 11, 12, 13],
Reissner-Nordstr¨om (RN) BHs [11, 12, 13], the Ba˜nados-
Teitelboim-Zanelli BH [14], and the four-dimensional
Schwarzschild-anti-de Sitter BH [15].
-----end quote---

 

[meteor]

Just found the homepage of Elias Vagenas
http://ns.ecm.ub.es/~evagenas/
And there's a nice animation of chairs falling in black holes, but more important, he lives in my city! Perhaps I will ask him some questions. If you want some questions for him, just post it here

 

[marcus]

Originally posted by meteor
Just found the homepage of Elias Vagenas
http://ns.ecm.ub.es/~evagenas/
And there's a nice animation of chairs falling in black holes, but more important, he lives in my city! Perhaps I will ask him some questions. If you want some questions for him, just post it here

Que viva Barcelona!

 

[meteor]

Thanks! marcus, do you speak spanish?

 

[marcus]

more from Elias V. paper

unfortunately not, although I can read a little.
Here is from the conclusions paragraph.
I tried to put in equivalent symbols, some of which didnt copy.

----exerpt----
In this paper we have evaluated analytically the area spectrums of Kerr and extremal Kerr black holes by implementing Kunstatter’s approach. The area spectrum of Kerr black hole was derived by using as real part of its quasinormal frequencies a function of the form m*Omega.

It was shown that the area spectrum is discrete but not evenly spaced.

Furthermore, an unexpected feature of the area spectrum is that it depends explicitly on the Kerr black hole parameters, i.e. the mass and the angular momentum. It is clear that since the novel numerical results show that the real part of the quasinormal frequencies of Kerr black hole is not just a simple polynomial function of its Hawking temperature and its angular velocity (or their inverses), further theoretical study is needed.

We have also shown that the area spectrum of extremal Kerr black hole is discrete and equidistant. The corresponding lower bound is universal, i.e. independent of the black hole parameters, but it is not proportional to ln 3. Therefore, it does not provide any support to Hod’s statement that the area spectrum of the form

A_n = (4*planck area*ln 3)n

should be valid for a generic Kerr-Newman black hole.

Finally, it is now known that the asymptotic quasinormal frequencies of Reissner-Nordstr¨om black hole are given by a QNM condition involving exponentials of its temperature.
It seems likely that the asymptotic quasinormal frequencies of Kerr black hole will also be described by such an analytic formula. We hope to return to this issue in a future work.
---end quote---

I am forgetting to go to sleep. will sign off now. and resume
in morning.

 

[Ivan Seeking]

Boy, I have some reading to do. Great job Marcus!

 

[marcus]

Ivan, thanks for the encouragement! these last posts are my first reading of those papers, trying to get at the basics, and as such are a bit disorganized. could be edited down to less than half the length. Meteor brought in the first paper and we just started reading papers from the "evenly spaced" BH spectrum people. This is high risk in the sense that the ES people could be wrong. they are "revisionists" who want to modify the majority's tenets. I find it fascinating but would not urge my interest on others.

I just dug up a bunch of earlier papers on the BH spectrum and
"QNM". Protect your free-time for serious things like sunday drives in the country, picnics, and folkdancing! This BH business is going to get more confusingly worse before it gets better.

There are basically at least two types of spectrum of interest here.

*there is the LQG area operator that measures the area of some physical surface defined by some material thing. And it has (discrete) eigenvalues which are the possible outcomes of measuring area.

*then there is spectrum of the Hawking radiation from a BH, which Hawking decided was a perfect blackbody curve for a certain temperature, T_BH. But other people apparently think that the hawking radiation might not be perfect blackbody and might
deviate from perfection at energies less than T_BH
(imagine you see Boltzmann k in front of that temp, making it energy, but Boltzmann k = 1) or at wavelengths comparable to the BH radius, i.e. longer wavelengths. this is a bit unsettling to contemplate because the blackbody curve is so beautiful and it has long been accepted as gospel (by me at least) that hawking radiation has that perfect continuous spectrum

like the Cosmic Microwave Background, right? the CMB has this perfect blackbody spectrum. but then it turned out not to, and it is often the deviations you find out about later that are interesting (a philosophical reflection that applies to other things as well)

*then to make it worse there is the spectrum of vibrations of the BH itself. presumably these are related to hawking radiation and the processes by which the BH radiates away energy and gains energy as stuff falls in, and these "ringing modes" can be calculated more or less classically as one would calculate vibration modes of some other object. and since BH area depends on mass, as the hole gains or loses energy it will be gaining or losing eventhorizon area. Lots of interconnected things (entropy, area, mass, gravity, temperature, vibration modes)

I had better list the QNM ("quasinormal mode") papers. I wish the whole business were not so controversial

 

[marcus]

Shahar Hod
Bohr's Correspondence Principle and the Area Spectrum of Quantum Black Holes
http://arxiv.org/gr-qc/9812002

Hod's equation (8) says "the area spectrum for the quantum Schwarzschild black hole is given by

A_n = 4*planckarea*ln 3*n
for n = 1,2,...

which you could say means that in the simplest (Schw.) BH case
the area comes in steps of (4 ln3) Planckarea units.
and he says this on the basis of the "Bohr correspondence principle"
that "transition frequencies at large quantum numbers should equal classical oscillation frequencies" (Hod page 5)

plus some classical oscillation frequencies calculated by Nollert in 1993 (H-P Nollert Phys Rev D 47 5253)
------------------------------
Olaf Dreyer
Quasinormal Modes, the Area Spectrum, and Black Hole Entropy
http://arxiv.org/gr-qc/0211076

Dreyer's paper is only 4 pages and contains useful exposition.
The impact of his paper was complicated by the fact that it contains a proposal which did not really catch on with LQG people. He suggested changing a key symmetry group from SU(2) to SO(3) and his elders-and-betters took him up on it. (Ashtekar, Rovelli, etc kept on using SU(2) and refused to take the bait)
But if you just ignore the suggested change of group, the paper itself is a nice brief exposition that lays out the situation and draws the basic connections.

the QNM of a (Schw.) black hole with mass M is a complex number omega
whose real part tells the frequency and whose imaginary part indicates damping. Dreyer's equation (5) says

\omega = \frac{ln{3}}{8\pi M}+ \frac{i}{4M}(n + 1/2)

For a Schw. BH area and mass are related by A = 16M^2
so simple calculus (dee by dee M) says change in A is
32MΔM

but the Bohr correspondence says ΔM (equivalently the size of an energy step) should correspond to the resonance
\frac{ln{3}}{8\pi M}
and if you multiply that by 32M, you get
4ln{3}

so energy changing by the amount that Hod and Bohr say
translates into area changing by this 4 times natural log of 3.

Now the puzzle becomes explaining 4ln3 in the context of Loop gravity and Dreyer presents the ES suggestion in his equation (18) as one solution. But in the next paragraph he rejects it!
"The problem with this approach is that it does not give the Bekenstein-Hawking entropy if one follows the same procedure as above..."

Note that what Gour and Suneeta did (the authors of that paper Meteor introduced us to) was to GET the Bekenstein-Hawking entropy by NOT following "the same procedure as above" but by following a different procedure. According to ancient wisdom there is more than one way to skin a cat.
---------------------
hard to do even rudimentary justice to all the turmoil and ferment.
Lubos Motl and his friend Andy Neitzke enter the picture here (Andy also sometimes comes to PF and signs himself neitzke IIRC but mostly doesn't post and just reads) and beyond them quite a considerable crowd

 

[meteor]

Another paper by Shahar Hod:
http://arxiv.org/abs/gr-qc/0307060
"Asymptotic quasinormal mode spectrum of rotating black holes"
Abstract:
"Motivated by novel results in the theory of black-hole quantization, we study {\it analytically} the quasinormal modes (QNM) of ({\it rotating}) Kerr black holes. The black-hole oscillation frequencies tend to the asymptotic value $\omega_n=m\Omega+i2\pi T_{BH}n$ in the $n \to \infty$ limit. This simple formula is in agreement with Bohr's correspondence principle. Possible implications of this result to the area spectrum of quantum black holes are discussed."

In this paper, Hod insists into apply Bohr's correspondence principle in order to determine the value of the fundamental area in a theory of quantum gravity

 

[meteor]

I've talked with Mr. Vagenas,and he doesn't speak spanish at all (he is greek), so the interview was in english.He says that calculating the QNM of Kerr Black holes is very difficult, because they don't have the analytic formula, have to apply numerical methods. The QNM of Schwarzschild BH are better understood that the QNM of Kerr BH because Schwarzschild BH only depend on mass, while the Kerr BH depend also on angular momentum. In any case, in a Kerr black hole, normal modes are better understood that quasinormal modes
I've asked him about his preferences between string physics and LQG, and he says that he is not an expert and cannot say
He also says that he is going to visit the forum
I feel myself important now, after talking with a high level scientist

 

[marcus]

Originally posted by meteor
I've talked with Mr. Vagenas,...He also says that he is going to visit the forum...

That is good news! I will, like you, express my satisfaction

that Mr. Vagenas has been invited and may in future vist the forum.

I got the same impression, that so far everyone has been defeated by rotation.
As long as the hole has no angular momentum, then it is either Schwarzschild (plain vanilla) or ReissnerNordstrom (electrically charged), and they seem to be able to find the vibration modes.
But if it rotates there is always some trouble with the calculation.

Hod has proposed two formulas for the rotating case and each time
people have tried them out and found they appear not to work (dont agree with the computer calculations). Maybe third time lucky.

I visited Elias Vagenas homepage as you suggested and admired the cascade of chairs falling into the black hole.

 

[marcus]

Loop Gravity area spectrum and Schw. hole vibration modes

I should sum up.
Maybe the first big result in Loop Gravity was the 1994 finding of Rovelli and Smolin that the quantum operator that measures area has a discrete set of possible values----a discrete spectrum----consisting of a sequence of multiples of the Planck unit of area.

The Rovelli/Smolin spectrum is not an "evenly spaced" spectrum, consisting of whole-number multiples of some area quantum.

However in January of this year Gilad Gour and V. Suneeta (both at the University of Alberta in Canada) argued that the LQG area spectrum should be revised. This solves several problems and may raise others.

The revision is foreshadowed by a 1992 paper of Edward Witten who used a similar quantum correction in casimir elements. And Gour/Suneeta are not the first to propose doing this. But their paper makes the case very clearly so I will take it as one representative of a recent line of research.

Gilad Gour, V. Suneeta
"Comparison of area spectra in loop quantum gravity"
http://arxiv.org/abs/gr-qc/0401110


-----------------------

The 1992 paper by Edward Witten (Journal of Geometrical Physics 9 (1992) 303-368) gives an example of "regularizing" a casimir element by changing it from
j(j+1)
to
(j + 1/2)^2

You can see the two things are not really very different. They differ only by a quarter, one is
j^2 + j
and the other is
j^2 + j + 1/4

Witten is not the only person to make this kind of change, which has been used by others as well---and now we are going to see the revisionists, including Gour and Suneeta, apply it to the Loop Gravity area spectrum.
------------------------------


In 1994 Rovelli/Smolin calculated the area spectrum in LQG to consist of "squareroot casimir terms"
\sqrt{j(j+1)}


Now in 2004 the revsionists are proposing to put in
(j + 1/2)^2
which, when it goes under the squareroot sign just comes out
a very simple
(j + 1/2)

There is a coefficient out front that everything gets multiplied by
and when that is done the spectrum turns out to be whole-number multiples of a "quantum of area" which (expressed in Planck terms) is:

4 log {3}

This is the "equidistant spectrum" version LQG area, also called the "evenly spaced" spectrum.
Either way the abbreviation ES would do as a tag.

There is a certain so-far undetermined paramter in the theory which may still be adjusted and would, according to the ES view, turn out to be the natural logarithm of 3, divided by 3 pi:

\frac{log {3}}{3\pi}


===========
Alexios Polychronakos
"Area spectrum and quasinormal modes of black holes"
http://arxiv.org/hep-th/0304135

Alekseev, Polychronakos, Smedbaeck
"On the area and entropy of a black hole"
http://arxiv.org/hep-th/0004036

 

[marcus]

In LQG a quantum state of the geometry, or the gravitational field, is represented by a spin network consisting of nodes and edges. The nodes contribute volume to regions containing them and the edges contribute area to surfaces they pass thru. Smolin's SciAm article does a good job of presenting this.

In the Rovelli/Smolin (not ES) version, the area of a surface S
defined by some material object is this sum

A_S = 8\pi l_P^2 \gamma \sum \sqrt{j_n(j_n+1)}



Here the surface is intersected by N edges, indexed n = 1,...,N, and edge #n is labeled by spin
j_n
----------------------------------

In the evenly spaced (ES) version the area gets changed to

A_S = 8\pi l_P^2 \gamma \sum (j_n + 1/2)



l_P^2 is the Planck unit of area, the square of the Planck length, and gamma is the Immirzi parameter, which still has to be determined.
-----------------------------------

As I have written it here the ES area formula is slightly messier than it needs to be. If one puts in the value they propose for gamma then it simplifies to:

A_S = \frac{8 log 3}{3} \sum (j_n + 1/2)

or, if you prefer

A_S = (4 log 3)\frac{2}{3} \sum (j_n + 1/2)

The fraction 2/3 is going to get eaten up later so this
reveals the important thing: that quantum of area
4 log 3.
---------------------------

I'm writing "log", instead of "ln" for the natural logarithm
because it's a little easier to read. It is base-e logarithms,
not base-10, that we are using.

-----------------------------

Gour and Suneeta show that this formula reproduces the Bekenstein-Hawking result that

entropy = A/4

For details see their paper. Earlier in this thread I went thru their argument. It looked to me like several recent papers were saying
much the same thing. I don't know how much of this is original with Gour and Suneeta. But their article is clear and recent and complete.

 

[marcus]

So in the ES version the area of a surface has this rather simple formula---expressing the area in Planck area units:

A_S = \frac{8 log 3}{3} \sum (j_n + 1/2)

=======
It's possible this version of the area will replace the original 1994 Rovelli/Smolin version. Now let's see what happens when we apply this version to a Schwarzschild black hole. We want the area of the hole's event horizon (EH). Classically this is

A_{EH} = 16\pi M^2

But we can also use the ES formula, slightly rewritten:

A_{EH} = (4 log 3)\frac{2}{3} \sum (j_n + 1/2)

Gour and Suneeta consider all the quantum states (microstates) that correspond to a given area A and calculate the degeneracy. They find that what dominates numerically are states where almost all the spins are one. The ES formula reduces therefore to:

A_{EH} = (4 log 3)\frac{2}{3} \sum (1 + 1/2)

the 2/3 and 3/2 cancel and we have

A_{EH} = (4 log 3)N

where N is the number of spin network edges intersecting the event horizon. N is the number of "area quanta", in effect, each quantum of area consisting of 4 log 3 Planck units.

This result is interesting because it matches what Shahar Hod and others have found about the vibration modes of the Schw. black hole.

And on the other hand it matches the Bekenstein-Hawking entropy formula.
entropy = A/4

So there is this aspect of the ES version producing a comfortable fit. On the other hand there have been objections to it. What I have seen in the way of counterarguments have been answered by Polychronakos. Would anyone like to present objections to the evenly spaced spectrum and have us consider what Polychronakos says about them?


For details see the Gour/Suneeta paper.
http://arxiv.org/gr-qc/0401110
Earlier in this thread I went thru their argument. It looked to me like several recent papers were saying much the same thing. I don't know how much of this is original with Gour and Suneeta. But their article is recent and relatively complete.

the Loop Gravity "surrogate sticky" has other links
https://www.physicsforums.com/showthread.php?s=&postid=140731#post140731
these are to other recent papers about the area spectrum and
to research on BH quasinormal vibration modes, including
links to a couple of papers by Lubos Motl (an occasional PF poster)

 

[marcus]

the quasinormal mode business

the business about vibration frequencies of black holes is interesting. Sometimes they are called black hole "ringing frequencies"

the whole (hole) structure including the event horizon has a kind of rigidity and can vibrate like a giant bell

(or like a little bell, in the case of smaller BH's)

I calculate a black hole with the same mass as the sun would ring
at a frequency that you could play on the piano----two octaves above middle D
Such a hole would have about a 4 mile diameter (or 6 km)

this is just approximate, to give an idea.

A more massive, larger, hole would have a deeper ringing tone.
If a star 4 times the mass of the sun were to collapse and form a black hole with 4 solar masses, it would ring 2 octaves lower pitch---
so around middle D on the piano.
----------------------

maybe it would be a good idea to learn how to calculate the vibration frequency of a Schw. BH. from its mass, I mean.

the symbol often used for frequency is omega
the frequency that goes with the mass M is
\omega_M = \frac{log3}{8\pi M}

This is in natural units, the usual Planck units. In Planck terms the mass of the sun is 10³⁸ and the frequency of middle D on the piano is 10^-40

so if you want omega to equal the middle D frequency, you can just solve for M
M = \frac{log3}{8\pi M}10^{40} = 4.3 x 10^{38}

It comes to roughly 4 times the mass of the sun.
Middle D on the piano is a pitch I can sing, and also people with high voices can (it's high for me and low for them), so I use it as a reference pitch some
especially since it is 10^-40 Planck.

this way I know that I or any of us could sing the ringing pitch of a BH with 4 times solar mass.

 

[marcus]

connection to LQG, by Bohr's correspondence principle

Bohr's correspondence principle is a not a law but more a strategy for finding things out
It says that if you do a classical calculation on a system and get a frequency then you can multiply by hbar to get get an energy step and you can expect to find that energy transition in the quantum version.

So people like Shahar Hod and all those who came after him have made classical calcuations of the ringing frequency of Schw. BHs and
found this formula in the previous post

And you can multiply by hbar (which is one in natural units) and get a quantum of energy----or mass (it is the same number in our units).

So because of the Bohr principle, and because it rings at
log 3/8piM
the hole must be gaining and losing energy in little steps of
log 3/8piM

And that means its surface area is gaining and losing area in steps of
4 log 3!
This is pretty nice. It is the quantized area spectrum of LQG discussed in preceding posts.

I will go thru the steps to show that

\Delta M = \frac{log 3}{8 \pi M}

corresponds to

\Delta A = 4 log 3

Well it is freshman calculus, there are no steps to go thru
you just note the relation of area to mass for Schw. black holes

A = 16\pi M^2

and differentiate it

\Delta A = 32\pi M \Delta M

and plug in what you know from Shahar Hod about Delta M

and solve for Delta A

 

[8LPF16]

Marcus,

Are you using the values of 294 for middle D freq.(and 73.5 for two octaves lower)?

Is there a maximum # of solar masses?

(at 33 it should be at freq. of a photon @color "blue", and would make a good limit - "c")


quote"This is in natural units, the usual Planck units. In Planck terms the mass of the sun is 10^38 and the frequency of middle D on the piano is 10^-40"unquote


Do you mean 10^-40 down from 10^38?

LPF

 

[marcus]

Originally posted by 8LPF16

Is there a maximum # of solar masses?

there certainly is. In my previous post I was just doing rough estimates, not exact calculation. So in that spirit, the maximum size of a star is roughly 100 solar masses. Sources differ---I have seen an estimate of 60 solar masses. Some people might say more than 100. But let's just say 100.
Chroot and others (Phobos, Labguy, Nereid,..) would know more exactly.

the point is that a young star of 100 solar masses would burn so brightly it would blow itself apart with its own light
(the more massive the star, the hotter and denser the core and the more rapidly it consumes its fuel, light exerts pressure, at a certain point the light would be so intense as to prevent more material from condensing...light pressure fights the gravity that is trying to collect the mass and build the star)
we should make a new thread in Astro forum, like in Astrophysics,
"How big can a star be?"

Originally posted by 8LPF16

Are you using the values of 294 for middle D freq.(and 73.5 for two octaves lower)?

Again I was doing rough estimates. Middle D on the piano is
about 10^-40 of Planck frequency.

I believe you are familiar with Planck units so you know there is unit of energy E_P
and the units are based on hbar, so its convenient to use hbar and say

E = hbar ω

the Planck frequency is the angular frequency that corresponds to Planck energy----one radian of phase per Planck time unit---best to stick with angular format consistently when using hbar.

So A is 440 cycles per second----same as 880pi radians per second.
And every halfstep is the twelfth root of 2.
The musical interval [D EF G A] represents
seven halfsteps. So 880pi divided by 2^7/12
But you and I know that is a major fifth interval and pythagoras would have divided by 1.5
however 2^7/12 is 1.498
well not to quibble---just divide 880pi by one or the other
it comes to about 1845 radians per second.

the frequency is the same whether you express it in radians per second or cycles per second----the note sounds like the note.
cyclic format and angular format are just two different formats
for describing a single reality

Now Planck frequency, if you slow it down by a factor of 10⁴⁰, is 1855 radians per second.
And if your piano has standard tuning the middle D is 1845.
It is a small percentage difference---dont know if one could hear it.
(a halfstep is 6 percent and this is about half a percent)

so imagine your piano is tuned so Middle D is 1855---"planck tuning"
are you comfortable speaking of frequencies in angular format.
it is what physicists are doing when they use omega as the symbol for frequency and write
E = hbar \omega

would you like a thread about this? which forum?

 

[marcus]

LPF,

We were talking about the ringing frequency of a black hole.
I recalculated and got that a hole with FIVE solar masses would
ring at around middle D.
I think earlier I said four. Sloppy back-of-envelope arithmetic!

the more massive the lower the pitch.
less massive raises the pitch
So divide the mass by two and the pitch will go up an octave,
For a rough back of envelope calculation, dividing by five (to get the sun's mass) is like going up two octaves
so a one solar mass hole rings at about 2 octaves above middle D.
but that is not exact. Would you like to know more precisely
for any reason?

black holes as gongs :)

POSTSCRIPT EDITED IN AFTER
LPF, as you suggested I did start a thread (in PF's "Stellar Astrophysics" forum) about the resonant pitch of a stellar-mass
black hole.

Alejandro Rivero, your questions about LQG area and volume
spectrum are too deep for me to reply to right away. I hope
someone else may respond (while I take a little time to think).

 

[8LPF16]

Marcus,


Yes, yes, yes. Please start another thread, as I have many questions, and they will be abstract to LQG. (at first)

I will look for thread later today - thanks!


LPF

 

[arivero]

This thread seems to be most interesting that usual, and I am really sorry I can not contribute at the technical level.

About all these area and volume spectra, there was a couple of things amazing, to me:

-one of them is that both volume and area are quantised. In quantum mechanics, while the area in phase space is quantised, the operators limiting these area, namely position and momentum, are not.

I suposse that the fact that area operators do not conmute for intersecting areas is the technical trick letting us to quantise the volume within (as well, surely, as preserve ordering and position of space chunks).

-related to this, I wondered if the quantisation of 3D volume is too strong a requisite. Naively I have expected just quantization of the dinamically generated 4D volume.

Time ago, the founding fathers discussed a lot about the question of mapping a lattice into a finer one. One of them, Zeta, sustained that it was not possible to build the lattice if one of the lattice coordinates was time. Another one, Delta, followed upon him and concluded that it was possible to do the map if all the coordinates were spatial, but he agreed (surely) that something pesky happens if time is involved. This ZD-principle is in some sense our guiding rule to quantum mechanics. But LQG goes an step further and tell us that even in the static case, without considering time, you can no iterate the mapping below Plank length. It is fascinating, but I wonder if it is a necessary condition or, perhaps, an excesive one. Have spin-networks in (3+1)D space been built? Do they induce quantised 3D volumes and 2D areas?[edited postscript]Just after sending this, I find that gr-qc/0212077 shows continuous spectrum in space-like lines!

Ah, by the way, a third founding father, Alpha, thought that the method of Delta was "non-rigourous".

 

[nonunitary]

Hi there,

It is interesting to see the interest that the so-called ES-area spectrum has generated. See for instance,

Gilad Gour, V. Suneeta
"Comparison of area spectra in loop quantum gravity"
http://arxiv.org/abs/gr-qc/0401110

I have strong reasons to believe that such operator does not make sense within the LQG framework.
If I can, I will write a longer post later. For now, just two comments:

1.- The standard Non-ES spectrum of Rovelli-Smolin has been obtained by different regularization procedures and the "standard" Casimir is selected. The corrected version seems somewhat ad-hoc, but that would not be a strong argument if it not were by the fact that:

2.- The fact that the ES operator counts zero-j spin networks and assigns area to them is what makes it non-sense. Let me explain. In LQG a good operator should be well defined on the space of states that is constructed (via a GNS construction) using the C-* holonomy algebra.
The zero-j spin networks correspond to an element of the algebra corresponding to the zero-loop, or in other words the identity element. This means that we can add ar remove closed loops with zero-j for free to a state and get the "same physical state".
The ES-area operator yields different areas each time one adds or removes the zero-j loop.

Therefore the operator is not even well defined on the Hilbert space of the theory.

 

[marcus]

Originally posted by nonunitary


...I have strong reasons to believe that such operator does not make sense within the LQG framework.
If I can, I will write a longer post later. ...

I hope you have time later and can expand on this.

In fact I have been reading the Gour/Suneeta paper which you mention, and I have been wondering about this problem of a j = 0
edge contributing area.

This does not appear to be addressed by Gour/Suneeta or by the other ES papers I have looked at.

For example, I didnt find any mention of it in a 2003 paper by Polychronakos
http://arxiv.org/hep-th/0304135
although this paper does reply to one or two other possible objections.

BTW can you pass on to us any news of the recent conference in Mexico City that you told us was planned for this past weekend?

 

[nonunitary]

Marcus,

I hope I will have the time to write this soon, either here in for another forum. I am not surprised that Suneeta, Gour, and Polychronakos do not mention this (probably unaware of this problem), since they are not LQG "experts". These are the kind of things that the people who have seen the transition from the old Loop representation to the C*-algebra stuff to finally spin networks and foams, would know. I think it is important to clear this point since it distracts from more fundamental problems like:
what are the QNM are really telling us?
Can we live with the new value of the Immirzi parameter, SU(2) and some exclusion principle (as by Corichi and Swain)? Is supersymmetry relevant (as Ling and others suggest)?
If ln(3) is relevant for uncharged non-rotating solutions, what can we make of the fact that this number does not show up in more general cases?
Should we ask LQG to answer this from the first place?
...

As for the LQG meeting in Mexico I have heard that it was a big success. Lot's of progress in agreeing on several conceptual points regarding spin foams, the hamiltonian constraint, semiclassical issues and phenomenoly. Also, lots af ideas of where to go and what to look at came out of the discussion. I think that after this first "NAFTA" meeting on LQG there will be more on a rotating basis.