View Full Version : Loop Quantum Gravity
I read about the Loop Quantum Gravity theory in some science magazine during my break at work. I wasn't reading attentively enough to make anything out of it. All I know is it is another theory to unite General Relativity and Quantum Mechanics, I think. Anyone care to share their knowledge on the subject for a physics laymen? Thanks.
If you're wondering whether LQG is something worth spending your time on, the answer is that it really isn't. Hardly anyone in the physics community has ever taken it seriously and have always been and continue to be confident that the whole program is fundamentally flawed (the basic reason is that the assumption in LQG that GR should be valid as a basis of quantization all the way up to planck energies is wildly implausible given what we know about the other interactions and the likely behaviour and ultimate fate of quantum field theories at high energies).
Pathetically, some PF members for their own questionable reasons continue attempts to convince other members (they sure wouldn't be able to do so with physicists) that there's real parity between LQG and strings, but really, all of the good ideas are being generated by string research and none by LQG. The twistor program recently rehabilitated within string theory is just the most recent example of how all roads appear to lead to strings and that the best hope for LQG is that it will ultimately share the same fate.
Only a tiny, tiny fraction of papers on the subject of QG deal with LQG, and when compared to the thousands upon thousands of stringy papers submitted each year, are really of no consequence. All serious research in QG/TOE (not to mention a great deal of mathematical research) really is being driven by string theory and this will continue to be the case for a very, very long time to come.
It's often said that string theory makes no predictions, but as is well-known, this is untrue. It turns out that the only self-consistent string theories necessarily contain GR and in this sense, string theory predicts spacetime! In fact for reasons related to this, string theory is the only bonafide quantum gravity theory we've ever had. LQG appears not to contain GR and is therefore not a QGT or even a physics theory but rather just some pretty but in the end vacant mathematics. This is not just my personal view, it's the view of the overwhelming majority of researchers in this field, though they don't argue about it since they choose not to deal with the misguided hobbiests in forums like these.
One development likely to finally kill LQG off for good will be the generation in future accelerators of particles associated with the long predicted supersymmetries which have motivated so much important research in all aspects of physics (along with many aspects of mathematics as well) and required by strings for it's self-consistency.
So, if you enjoy spending lots and lots of energy trying to understand a theory which few outside forums like this one take seriously, then go straight ahead andworry about LQG. If on the other hand you want to know about whats actually going on in the physics community, keep an eye on string theory.
jeff wrote: One development likely to finally kill LQG off for good will be the generation in future accelerators of particles associated with the long predicted supersymmetries which have motivated so much important research in all aspects of physics (along with many aspects of mathematics as well) and required by strings for it's self-consistency. ... and/or the discovery of such particles in astronomical research (e.g. cosmic ray detectors, GLAST).
There are also a number of other astronomy projects that may provide solid support for - or refutations of - SST (SMT?) and/or LQG fairly soon, esp the neutrino telescopes and gravitational wave detectors.
jeff -> is string theory at the point where it can account for neutrino oscillations yet? how about predicting the mass of any neutrino mixtures (with non-zero mass)?jeff wrote: It's often said that string theory makes no predictions, but as is well-known, this is untrue. It turns out that the only self-consistent string theories necessarily contain GR and in this sense, string theory predicts spacetime! In fact for reasons related to this, string theory is the only bonafide quantum gravity theory we've ever had. That's true, but isn't it a post-diction? What about predictions, in the sense of something specific and quantitative about things as yet undiscovered, such as the mass of the Higgs? How do SST/SMT and LQG compare in terms of their respective abilities to make specific, quantitative predictions (explicitly excluding post-dictions)?
The observation of supersymmetry, or the failure to find it (should that happen) is not a test of Loop Gravity since the theory can take it or leave it. Rovelli and Smolin have both made that point IIRC---anybody want a reference?
It is sometimes said that string "requires" SUSY for its "consistency". So if LHC does not find SUSY that is certainly bad news for string (but it does not effect loop either way.)
And if, on the other hand, SUSY is found, this also does not effect loop either way. The spin networks just have to carry more fields, more quantum numbers.
Whether or not SUSY is found, string can still be wrong---supersymmetry is NOT in itself a final test of string theory.
Either one or the other or both of the two theories, Loop and String, can be wrong---gotta keep an open mind, the proof is in the empirical pudding[:)]
I very much appreciate that you responded marcus. Thankyou.
Originally posted by Nibles
I read about the Loop Quantum Gravity theory in some science magazine during my break at work. I wasn't reading attentively enough to make anything out of it. All I know is it is another theory to unite General Relativity and Quantum Mechanics, I think. Anyone care to share their knowledge on the subject for a physics laymen? Thanks.
hello Nibles, yeah I guess we might refrain from squabbles and get around to answering your question which basically was "what is Loop Gravity?" The way the term is used it includes several recently developed research lines---still small in terms of the number of people working them, but definitely experiencing an upswing, and getting results and gaining visibility.
The first hardcopy book on the subject will be published this year
or next---Rovelli's "Quantum Gravity", about 350 pages, Cambridge University Press.
Also hardcopy textbook by Thiemann "Lecture Notes on Quantum Gravity" or some such title will be published by Springer Verlag, Berlin.
Before this everything was online, or in scholarly journals. You did not have Loop books. So the field is getting established.
this year there are already more Loop conferences scheduled than last.
I listed some in the "Intuitive Loop---Rovelli's Program" thread
in the LQG String Brane Forum here at PF. I try to keep that
thread up to date on study resources and current events. So the
upcoming conference links should be near the end of the thread.
Also there was that Scientific American article by Lee Smolin, and the hardcopy magazine Physics World had its November issue devoted to Quantum Gravity and gave equal billing to Rovelli about Loop and Susskind about String. Loop has far fewer people than String but is nevertheless gaining parity in certain respects.
The main names are Rovelli, Smolin, Bojowald, Thiemann, Freidel, Livine, Ashtekar, Corichi, Perez, Roche, Noui... I am leaving too many out. The main centers are Marseille, Berlin, Mexico City, Penn State, Waterloo, again it is hard to be complete.
Experimental tests of Loop Gravity are expected soon---especially with GLAST starting 2006----and "Quantum Gravity Phenomenology" has become a hot topic with collaboration going on between LQG people and leading phenomenologists like Kowalski-Glikman and Amelino-Camelia.
Phenomenology basically just means figuring out how to check theories against reality--how to get testable predictions from theory. Ted Jacobson, one of the founders of LQG with Smolin and Rovelli, is working primarily in phenomenology now---using astronomical data to narrow down and guide theory.
I have some discussion of developments in that area in recent posts in the "Intuitive LQG--Rovelli's program" thread.
So that is a birds-eye view of the research activity called Loop Gravity. There are links to plenty to read at that other thread.
Now I'll try to give a sketch of the various lines of theoretical development that come under this heading----quantizing General Relativity, spin networks, spin foams (the 4D version of spin networks), the hardcore principles that characterize all these approaches.
Actually Smolin's January 2004 SciAm article did that so well it is a shame to repeat it---I take it you read Smolin's article, which was the "cover story" of that SciAm issue.
Yes, I believe it was Smolin's article that I read. Anyways, thanks for the replies, I suppose if LQG is a rediculous theory, as most seem to think, then there is no need to bother with it. It's not something I need to know, just something I was curious about since it was in the magazine [g)].
Originally posted by Nereid
is string theory at the point where it can account for neutrino oscillations yet? how about predicting the mass of any neutrino mixtures (with non-zero mass)? What about predictions, in the sense of something specific and quantitative about things as yet undiscovered, such as the mass of the Higgs?
Since strings contain QFT, it can accomodate all of these ideas.
Originally posted by Nereid
That's true, but isn't it a post-diction?
The point is that strings must contain GR.
Originally posted by Nereid
How do SST/SMT and LQG compare in terms of their respective abilities to make specific, quantitative predictions (explicitly excluding post-dictions)?
The only way to get LQG to make any kind of conventional QFT prediction is to somehow put the requisite structure in by hand. On the other hand, string theory is unavoidably a theory of everything and so automatically contains all the requisite machinery. The problem for string theory is then how to break the symmetries needed to produce low energy phenomenology consistent with observation.
Originally posted by marcus
The observation of supersymmetry, or the failure to find it (should that happen) is not a test of Loop Gravity since the theory can take it or leave it. It is sometimes said that string "requires" SUSY for its "consistency". So if LHC does not find SUSY that is certainly bad news for string (but it does not effect loop either way.) And if, on the other hand, SUSY is found, this also does not effect loop either way. The spin networks just have to carry more fields, more quantum numbers. Whether or not SUSY is found, string can still be wrong---supersymmetry is NOT in itself a final test of string theory.
Suppose that while deciding between becoming a christian (string theory proponent) or a jew (LQG proponent) christ (supersymmetry) showed up. How persuasive would you find claims advanced by jews that this doesn't matter since they've no need of him and besides, it's possible that his 2nd coming could be incorporated into judiasm if necessary? My guess is not very.
Originally posted by marcus
Either one or the other or both of the two theories, Loop and String, can be wrong---gotta keep an open mind, the proof is in the empirical pudding[:)]
My basic point is that although this may be true, it's important to be honest and clear with PF members that only a handful of researchers have ever taken LQG seriously and that really strings dominate. It's not fair to take advantage of the lack of sophistication and knowledge of layman to coax them into spending enormous amounts of valuable energy learning ideas that have little bearing on what's currently driving high energy research.
Originally posted by marcus
... [LQG is] definitely...gaining visibility.
But not within the physics community.
I dont think this thread has any further purpose now that Nibles is satisfied that he doesnt have to know any more about Loop Gravity.
After all, it began with him asking about LQG---we never got around to saying anything very real about the subject but regardless of whatever curiosity he had is finished.
But I'll take the opportunity to give some of my own reasons (which others may share or not as they choose) why I am interested in Loop Gravity. I dont normally do this---I dont think anyone here should have to justify their physics interests. And I dont think I need to explain or justify liking to discuss LQG and report on recent papers etc.
Loop Gravity is interesting for me because it says something about the fundamental texture of space and time---even about their very existence---and because some strands of the theory are surprisingly close to being testable (contrary to what was expected a few years back)----and because as a research field it is still small, each person's contribution counts----and because it is just now "coming of age" with its first hardcopy books (having been mostly online till now)---and because the researchers themselves are obviously excited.
And because of a couple of notions that make Loop special.
----here's a footnote----
I take for granted you know that classical 1915 Relativity itself is BI and DI----no fixed background geometry, no absolute space, time and space coordinates have no physical meaning. No absolute time or any notion of an ideal clock: since relativistic mechanics is about relations between observables (not about evolution in time, though one of the possible observables can be a reading from some particular real-world material clock).
BI, background independence, says you have no prearranged background geometry---the shape of space is freely variable and dynamic.
DI, diffeo-invariance, says any smooth deformation of a solution is still a solution. If the shape of space and the distribution of energy in it are related according to the theory, and if you then re-map the shape and the energy, they stay in the right relation. Solutions--like silly-putty--can be stretched and squeezed and bent around at will.
BI and DI are the core principles of classical 1915 Relativity and they are what makes Loop Gravity different from stringy theories for example.
-------end of footnote---------
One suspicion I have about these things is that if GR is a large-scale limit of some more fundamental theory or class of theories, then those theories must also be BI AND DI.
A major stumbling block for string theory has been that by not being Background Independent it fails to really reproduce GR---only a kind of fake version that is not BI (sorry if the abbreviation bothers you because also used in the classified sex ads but it is needed here too).
A corollary of this is that string theory has not produced a candidate model of gravity that has GR as a large-scale limit. Since GR is BI and DI, you need some candidate which is itself also BI and DI, and because of string's basic limitations, such is not forthcoming.
Incidentally this central failure of string theory (Tom Banks mentioned it when he said the long hoped-for goal of Background Independence in string theory was a "chimera"---I think he meant an elusive will o' the wisp) is why evidence of supersymmetry wouldn't make me more interested in string theory. SUSY is a liability for string, not an asset----predicted in an earlier theoretical context but something the theory cannot do without.
------now to get to the heart of why I'm interested----
BTW remember YOU don't have to be interested in LQG. I dont care if you are or arent. I like to report on it and discuss it with others who share my interest. Some people may feel threatened by an open discussion of LQG but that's their problem.
What I see happening in recent physics is Relativity taking initiative and driving progress. A large fraction of current theoretical and observational developments is about extremely curved spacetime situations or those with dynamic geometry----quasars, gammaray bursts, neutron star mergers, black holes, inflation, big bang, singularity removal, ripples in the CMB, the integrated Sachs-Wolfe effect, dark energy, accelerated expansion, cosmological constant. All these things are born out of GR. The interesting things in the news (physics-wise) are increasingly turning out to be General Relativity-type things.
HEP physics is based on static geometry, it is built on fixed backgrounds which are NOT highly curved as a rule and not dynamic.
Minor perturbations of geometry can be considered, but the underlying space is not freely-evolving. A fixed space either can't even expand or at best does so in a rigid artificial fashion.
Fixed space theories can't cope gracefully what is increasingly of central interest. So HEP is out and GR is in---its that simple. What represents the leading edge of new understanding has started to be mainly in the precincts of Relativity (and I dont mean 1905 "special", which is not BI and DI and shouldnt even count as relativity).
What I am saying here is an observation about the history of science and in particular physics. The HEP (high energy physics) paradigm with its fixed absolute space was in charge for a while and did great things. And now the GR paradigm has taken the initiative and is calling the important shots.
People with a heavy investment in HEP-style thinking may sometimes feel envy or resentment---their sense of priority or privilege is somehow threatened---or they're in denial. Whatever. It does not concern me personally and it doesn't really have much to do with Loop Gravity.
Loop Gravity is just the most direct way to quantize classic 1915 BI and DI General Relativity. In the process of doing that it may come up with an explanation of the cosmological constant---there are hints of that in the Girelli/Livine paper (link in the other thread).
And it may come up with some non-commutative geometry or some quantum groups---like in the Noui/Roche paper (link in the same place).
Or with high-energy, planck scale modifications of GR, of which classic GR is the large-scale limit. Or a modification of Lorentz symmetry that makes both the speed of light the same for all observers and other planck quantities the same as well (like in DSR
and work by Kowalski-Glikman, Smolin and several others)
Links to all those papers are posted here
http://www.physicsforums.com/showthread.php?s=&postid=124320#post124320
in the "Rovelli's program" thread
In other words there are possibilities for interesting development stemming from insisting on really quantizing GR---preserving the core ideas of Background Indep and Diffeo-Invariance---and not giving up and falling back on some Absolute-Space-with-Gravitons substitute for GR.
So there are some reasons I'm interested.
Also I am not looking for something to believe in.
Physical theories arent religions. The idea of setting out to choose one to believe in---to decide whether to be a this-proponent or a that-proponent---seems ridiculous. I can happily contemplate the possibility that all theories in sight are wrong.
The question for me is where are the new concepts of space and time
coming from that will allow quantum mechanics to be put together
with general relativity. Rovelli discusses this in his book. From their birth early last century, QM and GR have had an incompatibility at the foundations level--they've been like oil and water.
Uniting them will cause a basic re-thinking of space and time. I guess I'd like to be on hand as more of nature gets to be understood and able to watch the deeper ideas as they emerge.
Marcus,
I really enjoy reading your posts. Thanks.
Originally posted by Tsunami
Marcus,
I really enjoy reading your posts. Thanks.
I appreciate the compliment because I can return it,
I enjoy yours----largely because of the gift for one-liners,
the elements of humor and surprise.
the Joan Crawford's daughter inset caused some discomfort
but that may merely have been an idiosyncratic reaction
following the development of Loop Gravity means sharing
in a common conjecture that the planck scale (which is different
from the string scale) is fundamental and intrinsic to nature.
That means there is a fundamental length and a fundamental frequency in the universe.
The length is almost exactly (within half a percent) of
E-38 of a mile
the frequency is almost exactly (within a tenth of a percent or so) of E40 times middle D on a conventionally tuned piano.
So if you know what a mile is, imagine 10-38 of it and that is Planck length.
And if you know what middle D sounds like, imagine a frequency which is imagine 1040 times higher and that is Planck frequency, the reciprocal of planck time, one event per planck time unit, one radian of phase or turn per planck unit, whatever.
The conjecture which is a semiconscious assumption in the minds of people involved with this kind of quantum gravity is that this frequency is "universe frequency" and this length also a universal---a threshold scale for new physics. Or a limit of some kind, the way the speed of light is a speed limit.
It is not yet clear HOW these new scales are built into nature but the suspicion is that they are built into her as deep proportions.
We think we understand fairly well how the speed of light is built in--thru all those special relativity formulas and thru the (lorentz) group of symmetries. But how, in what symmetries or formulas are these other scales present.
An early hint was when Rovelli and Smolin calculated the Loop Gravity area and volume operator's spectrum and found that all the possible eigenvalues (in a quantum theory, the outcomes of measuring some observable thing like a volume or area) were order-one multiples of the planck area and the planck volume.
Maybe they were wrong! The wheel is still turning.
But they may in fact be right and, in any case, even if the result eventually needs to be revised it seems likely that area and volume will still turn out to be a discrete set of order-one multiples of the planck units. That is these units of area and volume are basic to nature and to nature's space. It is a space that can only have areas and volumes chosen from that discrete set.
And then last year Girelli and Livine (part of a generation 20 or more years younger than Rovelli and Smolin) posted this exquisite 4-page paper that SPEED was quantized in little speed-steps too, in a way that incorporates the cosmological constant or vacuum energy density or "dark energy" density---something that can be incorporated as a small positive quantity rather easily in the theory, that "fits" in Loop Gravity without much trouble. Great that speed is quantized, but even better that determining the steps involves the density of dark energy.
So that is the suspicion, the conjecture----that the basic stuffs of geometry are quantized----that nature is constructed that way: dyed-in-the-wool so it wont wash out.
You can find the Girelli/Livine article link, if you want it,
in this post of useful Loop links:
http://www.physicsforums.com/showthread.php?s=&postid=124320#post124320
For links to the National Institute of Standards and Technology page that gives values for the planck length, timescale, temperature scale and such:
http://www.physicsforums.com/showthread.php?s=&postid=121318#post121318
Originally posted by marcus
I appreciate the compliment because I can return it,
I enjoy yours----largely because of the gift for one-liners,
the elements of humor and surprise.
Why, thank you very much!! [;)]
the Joan Crawford's daughter inset caused some discomfort
but that may merely have been an idiosyncratic reaction
Yes, I agree. That's why I posted it. [:D] I hate being alone in my discomfort zone. [g)]
Originally posted by marcus
following the development of Loop Gravity means sharing
in a common conjecture that the planck scale (which is different
from the string scale) is fundamental and intrinsic to nature.
That means there is a fundamental length and a fundamental frequency in the universe...
Totally fascinating stuff, but, being the non-nerd that I am, it certainly gives me the worst twisty-face!!![6)] But keep it comin'![:D]
Originally posted by Tsunami
...keep it comin'![:D]
a distinctive thing about quantum gravity is that the planck scale of area and volume come out as a result---not something put in by hand.
starting from very general principles like background independence and diffeomorphism invariance (Einstein called it "general covariance") rovelli and smolin were able to derive the area and volume spectra and they turned out to be a discret set of multiples of the planck area and volume units---the natural units (at least up to order-one factors) of area and volume.
how to say this? It had been suspected on general theoretical grounds for more than half a century that a new picture of spacetime would emerge at planck scale. But when LQG came along it pointed like a compass-needle at that scale, without having been told by its inventors to do that. (smolin, rovelli 1995)
so if someone wants to follow the ongoing development of quantum gravity they would do well to get a comfortable familiarity with the planck units, hence this informal introduction.
down at the end of this post I will just directly define them in the immediate hard-*** algebraic way, as is usually done, without a serious attempt at motivation. So if you want to avoid the gradual step-by-step introduction, just scroll down and there will be the usual formulas.
-------gradualist treatment of planck units------
one way to get a handle on them is to take a fresh look at one of the most basic equations in science, the 1915 Einstein equation, the central equation in our model of how gravity works: General Relativity. this equation relates the density of energy in a region ("joules per cubic meter", "footpounds per cubic foot") to the curvature in that region. It says the two are proportional!
Except for a factor of 8\pi which is no big deal, and the fact that they use special symbols for curvature and energy density, this equation is customarily laid out this way:
curvature = \frac{G}{c^4}energy density
Curvature in this context is measured in units of reciprocal area ("per square meter", "per square foot").
Now just suppose that we deviate a tiny bit from the customary layout and write the Einstein equation this way:
\frac{c^4}{G}curvature = energy density
This quantity c^4/G is the planck unit of force, that is, the force unit which belongs to a system of units that Max Plack discovered in 1899 and presented to his contemporaries as a natural (rather than artificial or man-made) system of units.
So in 1915 Einstein discovered that nature knows about the force
c^4/G. It is the force that connects the amount of curvature in a region to the energy density there.
For dimensional reasons a force is the only type of quantity that can do that. Multiplying a curvature ("per sq. foot") by a force ("pounds") gives a pressure ("pounds per square foot") and that is dimensionally the same type of quantity as an energy density
("footpounds per cubic foot"). Or say the same thing substituting metric newtons for pounds and meters for feet. Newtons per sq. meter is the same as joules per cubic meter. The longandshort is that if you multiply a curvature by a force you get an energy density and the only thing you CAN multiply a curvature by to get an energy density is a force. Einstein found that the force that works in this context
is the unit force c^4/G in planck's 1899 system of units
----------------
we can get all the planck units from this force, and stuff you already know like the speed of light
People COULD have realized that planck units were basic as early as 1915, but they did not and it was still a bit surprising in 1995 when the area and volume spectra were found to be multiples of planck area and volume units. Our species is not quick to catch on, sometimes.
-----------
To summarize
around 1900 Planck discovered that nature knows about a certain ratio of energy to frequency called hbar. (when using hbar you need to express frequency in angular format, radians per unit time, but that is a technicality involving a factor of 2pi and I wont belabor it)
in 1905 Einstein reminded everybody that nature knows about c the speed of light, in fact you could almost say nature is obsessed with the speed of light. that was the year he expounded the universal speed limit and E = mc^2 and a bunch of other things involving c.
in 1915 Einstein showed that nature knows about a certain force
c^4/G which is the force unit belonging to Planck's system of natural units. It turns up as the central constant in General Relativity: the thing that relates the lefthand side to the righthand side in the main GR equation. If you have a book where you can look up the metric values of G and c, or if you just know them, then you can easily calculate what the planck force unit is---just follow the formula c^4/G. You will get the answer in terms of the metric force unit, the so-called "newton" which is about a tenth the weight of a kilogram in normal gravity.
----------------
all the other planck quantities come from these three that were already immanent and obvious in 1900, 1905, 1915.
in any system of units the unit power is always equal to the unit force multiplied by the unit speed (in our case c^4/G multiplied by c)
If you work out c^5/G with a calculator you get planck power unit is
3.6E52 watts. Lots of watts.
In the planck system, hbar is the ratio of unit energy to unit frequency. So unit power divided by hbar gives the square of unit frequency, namely c^5/hbarG
unit frequency = \omega = \sqrt{\frac{c^5}{Ghbar}}
this is a frequency expressed in angular format, so the convention is to use the symbol omega for it, instead of the letter f.
Anybody who wants can immediately find out what the planck unit energy is at this point because it is hbar\omega
-----------
in a consistent system of units the unit length is equal to the unit speed divided by the unit frequency
so in our case
unit length = \frac{c}{\omega} = \sqrt{\frac{Ghbar}{c^3}}
-------
that's about it for the definitions, now we have unit force, power, frequency, time (the reciprocal of frequency), and length----the rest derive in familiar ways from these. the mass unit, for instance, is equal to the energy unit divided by the square of the unit speed (the square of the speed of light) and so on like that.
-----------
now the question is: what sizes are these units. Of course now you have the formulas for many of them so you could calculate them out in metric terms. But to save you the bother, the best way I know is just look them up at the NIST website.
BTW the NIST "fundamental constants" website has the planck temperature unit too---which is the planck energy unit divided by boltzmann's constant.
Beyond just always looking them up, there are some facts about them that are not too hard to remember. Like 2E-30 planck temp is a reasonably good reference temperature to remember---it's about 10 Celsius or 50 Fahrenheit. And E38 planck length is a mile.
-----direct no-nonsense definitions----
unit time = t_P = \sqrt{\frac{Ghbar}{c^5}}
unit length = l_P = \sqrt{\frac{Ghbar}{c^3}}
unit energy = E_P = \sqrt{\frac{c^5hbar}{G}}
unit temperature = T_P = \frac{\sqrt{\frac{c^5hbar}{G}}}{k}
-------direct no-nonsense explanation---------
the only way that it is possible to cook up a quantity with the dimension of time using the quantities G, hbar, c is this definition written here and simple constant multiples of it, but why bother to scale the thing by an extra number?
since c is going to be unit speed in the system, it must be unit length divided by unit time
so to get unit length simply multiply tP by c (unit time by unit speed)
since hbar is the product of energy with time, and since it is going to be a unit quantity in the system, it must be equal to the unit energy multiplied by the unit time
so to get the unit energy simply divide hbar by the unit time
the boltzmann k is a ratio of energy to temperature and it is a unitary ratio (like c and hbar) in the system
so to get the unit temperature, divide the unit energy by k
I'm trying to remember how to spell the original titles of the two basic references. Planck's 1899 paper is little-known but lays out the system and gives essentially the same values for the basic natural units that we use today
Planck (1899). "Ueber irreversible Strahlungsvorgaenge. Fuenfte Mitteilung." Koeniglich Preussische Akademie der Wissenschaften (Berlin). Sitzungsberichte: 440-480.
Einstein (1916) "Grundlage der allgemeinen Relativitaetstheorie"
Ivan Seeking
Jan19-04, 03:24 PM
After reading the last post, Tsunami looked at me and said that her face is now permanently stuck in the twisted position. Thanks a lot Marcus!
[:D]
Originally posted by Ivan Seeking
After reading the last post, Tsunami looked at me and said that her face is now permanently stuck in the twisted position. Thanks a lot Marcus!
[:D]
LOL!
well Ivan you know I never have a sure notion of who, if anyone,
reads these posts. Or for whom, and at what level, to write
Did you ever read from the 3 article series "Scaling Mount Planck"
in Physics Today, by Frank Wilczek.
Wilczek is one of todays great (senior) theoretical physicists, at least in my view, and he gave one of best explanations Ive seen for why one ought to try to understand Planck scale and think about things from Planck scale perspective.
It is hard tho because the natural units are so extreme, many of them.
But they seem more and more to be a part of nature.
Tell Tsunami to smile enigmatically (as if she were nature) instead of looking puzzled
the calm enigmatic smile is the other side of the twistyface and much easier on the eyes
Ivan Seeking
Jan19-04, 05:08 PM
Originally posted by marcus
[B]Did you ever read from the 3 article series "Scaling Mount Planck"
in Physics Today, by Frank Wilczek.
If not I will...I don't recall reading the series.
Tell Tsunami to smile enigmatically (as if she were nature) instead of looking puzzled.
the calm enigmatic smile is the other side of the twistyface and much easier on the eyes
Really don't worry. She has looked twisted and pain ridden since the day we got married. [8)]
I am goint to say diffeomorphism invariance a lot for awhile just to annoy her. By the way, she wanted to know if this - diffeomorphism invariance - has anything to do with Michael Jackson?
Originally posted by Ivan Seeking
If not I will...I don't recall reading the series.
In case you want to look at them online
Earlier articles (June, Novemember 2001, August 2002)
http://www.physicstoday.org/pt/vol-54/iss-6/p12.html
http://www.physicstoday.org/pt/vol-54/iss-11/p12.html
http://www.if.ufrgs.br/~jgallas/wilczek.html
Much in them is too technical for me or general readers but
I believe there is an understandable perspective that comes thru
despite this. See what you think.
Wilczek gives a way to understand why the planck time is so brief and the planck length so tiny
he starts by considering the question of why gravity is so weak
compared to other forces, and defines a number N as a measure or indicator of its weakiness, and about halfway thru the first paper he says:
"...Thus hbar and c appear directly as primary units of measurement in the basic laws of these two great theories. Finally, in general relativity theory, spacetime curvature is proportional to the density of energy--and G (actually c^4/G) is the conversion factor.
If we accept that G is a primary quantity, together with hbar and c, then the enigma of N's smallness looks quite different. We see that the question it poses is not, "Why is gravity so feeble?" but rather, "Why is the proton's mass so small?" For in natural (Planck) units, the strength of gravity simply is what it is, a primary quantity, while the proton's mass is the tiny number sqrt N.
That's a provocative and fruitful way to invert the question, because..."
The tiny number sqrt N that he is talking about is one over the rather large number 13.01 quintillion. It is one of a small bunch of mysterious numbers that are really fundamental constants, like 137 or 1/137. It is easiest to remember it as 13E18 or 13 quintillion.
What Wilczek says is that our theory is being challenged to explain this number 13 quintillion, and (surprisingly enough) according to him it is making some progress towards explaining it!
Remember Feynmann saying something like this about the number 137, that in his opinion every physicist worth his salt should have that number pasted up on the wall to remind him to try to figure out why it was that----why it was 137 instead of something else.
(I dont always distinguish between numbers and their reciprocals as you see [;)])
So why is planck length so tiny, or the universe's time unit so brief. It has the naive ring of a child's question "why is the sky blue why are the clouds white etc." as historical questions can have, sometimes I think.
well its the same as asking why Planck energy is so large
because the time unit is only as small as the frequency unit is high, and hbar times that frequency is the unit energy
so brief time corresponds to high energy (as we know so well!)
energy being more or less the same scale as mass, the question comes down to
"why is the planck mass 13 quintillion times the proton mass?"
So he boils the problem of understanding planck scale (and the weakness of gravity) by reinterpreting it this way
"what causes the proton mass to be 1/(13E18) of the natural mass unit"
now the problem is with the proton
the natural unit of mass is just the natural unit and we challenge our theory to come up with the small number which is the reciprocal
of 13 quintillion.
It seems so simple, I can even feel comfortable with it. Maybe you and/or Tsunami can too! All we need to do is trust that the people we pay to be theoreticalphysicists have all stuck the number 13 quintillion up on the study wall or over the shaving mirror and that
eventually some one of them will come up with a reasonable explanation of why it is what it is.
I guess I trust them to that extent. so I can feel comfortable about the sizes of the Planck units.
and this is the scale that appears to be emerging in (for instance the area and volume spectra of) loop gravity.
erm......gonna need a little absorption time, here.....
Originally posted by Tsunami
erm......gonna need a little absorption time, here.....
Hello Tsunami, I am embarrassed to say that
while you have been friendly and encouraging I have not
written any posts specifically for you---too busy talking to myself
and possibly to Ivan and others.
So I have not reciprocated. Which, come to think of it,
is rude. So I would like to make amends for this, if
possible. Maybe by responding to some question or idea of yours
in a comprehensible way.
Have you met Nereid and selfAdjoint here, by the way. They
tend to be nice and answer people's questions in a way so that
the person understands and is left feeling good about
having asked. By contrast, I often fly off on tangents.
It was probably a mistake, for example, for me to get started
talking about Planck units----though they truly have something
to do with quantum gravity, quite a lot actually.
Maybe we can get this thread back on track and make it have
more genenral interest. Any ideas or suggestions?
Originally posted by marcus
Hello Tsunami, I am embarrassed to say that
while you have been friendly and encouraging I have not
written any posts specifically for you---too busy talking to myself
and possibly to Ivan and others.
And I'M embarrassed to say that here I thought that ALL OF THESE POSTS IN THIS WHOLE FORUM were written JUST FOR ME!! Imagine my surprise to find out that this is not so! [:D] I'm actually very relieved to know I won't be held responsible to know the contents of your last post! [;)]
So I have not reciprocated. Which, come to think of it,
is rude. So I would like to make amends for this, if
possible. Maybe by responding to some question or idea of yours
in a comprehensible way.
Not to worry. I'm thinking there's probably not a rude bone in your body. But I do have an idea forming...
Have you met Nereid and selfAdjoint here, by the way. They
tend to be nice and answer people's questions in a way so that
the person understands and is left feeling good about
having asked. By contrast, I often fly off on tangents.
It was probably a mistake, for example, for me to get started
talking about Planck units----though they truly have something
to do with quantum gravity, quite a lot actually.
Yes, I've 'met' Nereid and selfAdjoint. I like both of thier posts very much. Wolram is another whose posts I enjoy. Don't worry about your tangents...I'm pretty 'tangential' myself. [;)] Planck units are good, especially with all these special relationships they seem to have with everything else.
Maybe we can get this thread back on track and make it have
more genenral interest. Any ideas or suggestions?
Here's my idea. Let's let this thread continue on in it's own little nerdy fashion[:D], and I'll start a thread or two of my own later today. I need to get more than one explanation to all my questions so I can pull it all into a better understanding of the more important aspects and theories in today's physics. Hope to see you in MY threads![;)][:D]
I'm happy with that.
This thread has a lot that is just my personal views of Loop Gravity and continuing (with that reminder) here is the gist of LQG or its main interest for me.
It reinforces a historical change in the view of space and time that was already in General Relativity. GR already has freedom from background geometry and it already has what A.E. called "general covariance" which deprives the spacetime continuum of physical existence and leaves the fields of energy and spatial relationship the only things with invariant meaning. Contemporaries call these two feature of General Relativity by the names BI and DI (Background Independence and Diffeomorphism Invariance) and Smolin did a good job explaining them for wide audience in his SciAm article.
But these are not specifically LQG things. They are aleady in 1915 GR and LQG merely confirms them by confirming GR and making it harder to ignore. A lot of people are in denial about the 1915 message because it does away with absolute Minkowski space (beloved of Particle guys) and absolute Newtonian space which is the mothers milk imbibed the Freshman year by all physicsmajors. So the 1915 message is disturbing and there has always been this resentful dream floating around of throwing out what we learned from GR and replacing it with some imagined "more fundamental" theory that wouldnt have BI and DI and would still have the beloved spacetime continuum.
What LQG does that is new---the essential thing I believe---is lend credence to a burgeoning thicket of new theories called Double Special Relativity which say that there is something else besides the speed of light that is the same to all observers.
There is currently a kind of uncontrolled growth of DSR theories and they all take seriously the possibility that there is something real about the planck scale. This would mean Minkowski space is the wrong space (even tho everything in Particle physics is built on it).
The reason that LQG lends credibility to DSR theories (which are growing wild and need to be pruned back by experiment) is that
when Rovelli and Smolin (1995) calculated the spectra of the area and volume operators----essentially the possible results of microscopic measurements of area and volume, beyond current technical capability---they found they were a bunch of fractional and rootly multiples of the planck unit area and planck unit volume.
Another thing LQG does is show you in concrete detail how to throw away the old Riemannian manifold and still keep the fields.
You can chuck out the spacetime continuum (which does not have physical existence in GR, being a mere "gauge" or mathematical convenience) and still describe quantum states of the gravitational field using networks or knots.
Or maybe I should say "abstract graphs". The kind of thing that is just a set of points with a list of which points are connected to which others. It does not have to live in a conventional Euclidean space or anywhere in particular because it is an abstract "combinatorial" being-----you write down a list of nodes and a list of links----or you draw a networklike thing on a piece of paper---but the thing itself does not have to be embedded in some oldfashioned absolute space in order to exist. It exists.
And all kinds of fields can be defined on this graph, or net, or knot, just by coloring its nodes and its links.
And a quantum superposition of enough such graphs will look and behave sufficiently like familiar space that it will fool most people.
the graph is pure relation, abstracted out of any spatial or temporatl continuum. the continuum can be thrown away and all the fields defined relationally, on the graph
one of the Loop people likes to call the quantum states of the gravitational field "polymers" because to him they seem like huge ball and stick molecular models---we are talking jillions of nodes.
the nodes supply volume to whatever physical region contains them and the links supply area to whatver surface they pass thru
What this is doing is REIFYING or thing-ifying a basic insight that Einstein already had and made fundamental to General Relativity. The continuum does not physically exist, what counts are the interrelationships, the intersections and meetings of worldlines. These have a physical meaning that does not go away when you morph the space.
If the paths of two particles cross, then they still cross after you morph all space and time. So that coincidence or intersection has meaning, although the coordinates on the map do not.
That was already there in 1915, but the 1990 loop business with its "spin networks" just gives a concrete representation of a purely relational quantum state. Something they could calculate with---and when they calculated area and volume spectrums they encountered the natural units. It think that's basically the story (in its briefest outline)
Originally posted by marcus
...And a quantum superposition of enough such graphs will look and behave sufficiently like familiar space that it will fool most people.
the graph is pure relation, abstracted out of any spatial or temporal continuum. the continuum can be thrown away and all the fields defined relationally, on the graph...
so the main thing the Loop approach does is finish (what Einstein began in 1915 namely) getting rid of the classical continuum.
by offering a blurred superposition, a quantum dogpile, of all the possible relationship graphs
graphs that describe proximity or contact between things, saying what neighbors on what
and a blurred heap of enough such graphs will act in a way that resembles the old familiar continuum (but not exactly, UV infinities go away in the quantum fields of particles, a cosmological singularity disappears, microscopic quantization of geometric features area and volume occur, new problems arise)
now string theory has always been done in the old-fashioned continuum---indeed mostly in the simplest and flattest space of all, Minkowski space---and in that context it would not work unless the space had an unrealistic number of dimensions like 26 or 10 depending on what particular variant of string theory one was considering.
But what would happen if you set up a string to vibrate in the new picture of space, this quantum heap of networks?
Thiemann says that some friends at the Albert Einstein Institute (AEI) at Berlin (like Hermann Nicolai, a string theorist) kept urging him to try this. He tried it and some problems that had bothered string theory went away.
the analysis no longer required 10 or 26 dimensions, you could do it in any number of dimensions including the number we see (the obvious 3+1)
no more ghosts, no more negative normed Hilbert spaces
no more tachyons (particles faster than light)
no dependence on supersymmetry, can adopt or not adopt, depending on
whether experiment shows the extra particles exist or not
-------------------------
It looks to me as if Berlin is going to be pretty strong in gravity.
Bojowald is already there
Thiemann is currently in Canada but will probably go back
in this new paper he is using results of Hanno Sahlmann and ones he got with Sahlmann. Sahlmann will probably go back.
The AEI is part of the Max Planck Institute for Gravitation Physics there---they held the October 2003 conference called "Loops meets Strings" and Hermann Nicolai was the co-host with Abhay Astekar.
The top people in DSR (doubly special relativity) are in Europe. Kowalski-Glikman is right down the road from Berlin, in Wrocz.
Carlo Rovelli has already moved to Marseille.
It looks like the center of gravity in 21st Century theoretical physics is shifting or has already shifted to Europe
---------------------
Thiemann's new paper just scratches the surface. Everything has to be checked and extended to more cases (he only does a boson string, he or somebody needs to do a fermion one)
there is a huge amount of open research territory in this direction
as Thiemann indicates (but you can gauge for yourself without that)
-----------------------
what was Edward Witten's last paper about? anybody remember?
this is a high-risk thread, I guess, so I'll open this here can of worms
a minority of comparative newcomers in Loop Gravity are proposing the possibility that the area spectrum is not what Rovelli/Smolin calculated in 1994 but instead (if expressed in units of the planck area, the natural unit of area)
consists of multiples of
4 ln 3
here are three papers explaining that possibility, and why it might be true
Gilad Gour, V. Suneeta
"Comparison of area spectra in loop quantum gravity"
http://arxiv.org/abs/gr-qc/0401110
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
Interestingly enough, the last of these papers cites a 1992 paper of Edward Witten which gives an example of when "regularizing" a casimir element changes it from
j(j+1)
to
(j + 1/2)^2
For the most part that is all this little fracas is really about.
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
And there are times when "quantum corrections" require changing from the first to the second. The Edward Witten paper apparently affords one example of this:
Journal of Geometrical Physics 9 (1992) 303-368
and they give other instances as well.
Anyway in 1994 Rovelli/Smolin calculated the area spectrum in LQG to consist of "squareroot casimir terms"
\sqrt{j(j+1)}
And now these other people like Polychronakos say they should have considered making a quantum correction and putting in
(j + 1/2)^2
which, when it goes under the squareroot sign just comes out
a very simple
(j + 1/2)
So for them the area of a physical surface, like a tabletop or a beachball or a blackholeeventhorizon will consist of a sum of
terms like that
This is the "equidistant spectrum" version LQG area.
also called the "evenly spaced" spectrum.
Either way the abbreviation ES would do as a tag.
and as a consequence, in the ES version a certain so-far undetermined paramter in the theory which may still be adjusted would, according to these minority voices, equal the natural logarithm of 3, divided by 3 pi:
\frac{ln 3}{3\pi}
this is the Immirzi parameter and its a wild-card in the Loop Gravity deck, which disturbs some people and not others. Theories under construction just have undetermined parameters. It is a blessing LQG has so few, I think, and I dont worry about this parameter and its eventual fate. Some Loop/Foam theories look like they may get rid of the parameter altogether but that hasnt happened conclusively. So if there is this parameter, well, maybe it will turn out to be the log of 3 divided by 3 pi (and maybe it wont) and if it does turn out to be that someone will doubtless find an ingenious explanation for why.
But that is something to look forward to, and not worry about at the moment.
ranyart
Jan31-04, 05:35 PM
Originally posted by marcus
Theories under construction just have undetermined parameters. It is a blessing LQG has so few, I think, and I dont worry about this parameter and its eventual fate. Some Loop/Foam theories look like they may get rid of the parameter altogether but that hasnt happened conclusively. So if there is this parameter, well, maybe it will turn out to be the log of 3 divided by 3 pi (and maybe it wont) and if it does turn out to be that someone will doubtless find an ingenious explanation for why.
But that is something to look forward to, and not worry about at the moment.
My money is on a little exercise in Feynman 'Negative Probability', functions.
There is a root to the problem, and at the same time there is a definate answer, inner probability against outer probabilities! just a teaser to a great time to be alive!
Hi ranyart! well I am not placing any bets for now.
I just want to see what it is these "equidistant spectrum" (ES)
people are saying
To a passerby it might seem a bit picky or obsessive-compulsive for people to be concerned with whether an area formula is this or that, when the numbers are so close. But that attention to little numerical nits comes with the territory and so lets be picky
We are talking about a physical surface, S, defined by some material object, and a quantum state of geometry called a spin network state.
Geometry is the same as the gravitational field. So the a spin network is also a concise description of the gravitational field.
Particles and/or other fields can be located on the network. Information about area and volume are contained in the network. Each node contributes a bit of volume to whatever region contains that node. Each link or edge of the network contribute a bit of area to whatever the edge passes thru.
Each edge is labeled with a number j (called a "spin" by Roger Penrose who first used these networks). If a particular surface is intersected by N edges, indexed n = 1,....,N, and edge #n is labeled by spin
j_n
then in the evenly spaced (ES) version the area is
A_S = 8\pi l_P^2 \gamma \Sigma (j_n + 1/2)
Here gamma is the famous Immirzi number and
l_P^2 is the planck unit of area, the square of the planck length
Now basically all these people seem to be saying, it seems to me, is that this formula is only a little bit different from the Rovelli/Smolin (let's abbreviate it RS) formula and that it works out better. They get that in the case of a Schwarzschild black hole the
area will change by increments of 4 ln 3. (In natural units) and
that seems to be nice and compatible with other findings, including the big one of Shahar Hod.
for comparison, I'll write down the RS version of the area. You will see it is not much different. Same spin network quantum state of the gravitational field (determining distances angles areas volumes etc) and same surface, intersecting the same N edges with the same N spin labels: In the not-evenly spaced (RS) version the area is
A_S = 8\pi l_P^2 \gamma \Sigma \sqrt{j(j+1)}
Originally posted by marcus
...Then in the evenly spaced (ES) version the area is
A_S = 8\pi l_P^2 \gamma \Sigma (j_n + 1/2)
...
l_P^2 is the planck unit of area, the square of the planck length
and in the evenly spaced (ES) version the area is
A_S = 8\pi l_P^2 \gamma \Sigma \sqrt{j(j+1)}
...
In physical models parameters accumulate like barnacles on the bottom of a ship, depending on history and who discovered what when.
So what happens if we just put in the value for gamma that they want.
namely ln3/3pi. If Tsunami were to look at this and see the gamma she would not be pleased. Neatness and simplicity matter. So let us get rid of it. Make the formulas as clean as we can.
And if we take for granted the area is going to be measured in Planck units of area, then we dont have to include that in the formula either. 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.
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.
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.
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
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
Marcus,
Would you look at my chart in Theory Development "Resonating Vibrational Potentials", and tell me what you think?
LPF
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
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---
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
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!
Thanks!! marcus, do you speak spanish?
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
Feb1-04, 05:07 AM
Boy, I have some reading to do. Great job 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, TBH. But other people apparently think that the hawking radiation might not be perfect blackbody and might
deviate from perfection at energies less than TBH
(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
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 doesnt post and just reads) and beyond them quite a considerable crowd
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 in to apply Bohr's correspondence principle in order to determine the value of the fundamental area in a theory of quantum gravity
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[:))] [:))] [:))]
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.
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
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 dont know how much of this is original with Gour and Suneeta. But their article is clear and recent and complete.
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 dont 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
http://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)
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.
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
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
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 27/12
But you and I know that is a major fifth interval and pythagoras would have divided by 1.5
however 27/12 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?
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).
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
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 continous spectrum in space-like lines!
Ah, by the way, a third founding father, Alpha, thought that the method of Delta was "non-rigourous".
nonunitary
Feb5-04, 08:24 PM
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.
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
Feb6-04, 08:14 AM
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.
nonunitary, thanks much, both for your reflections on the area
spectrum (the ES, non-ES issue) and for the report on the conference. I am glad to hear it turned out well and is likely to be repeated!
Originally posted by nonunitary
...
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)?
...
in case anyone is interested in following up on the references
John Swain's original paper was posted May 2003
http://arxiv.org/gr-qc/0305073
"The Pauli Exclusion Principle and SU(2) vs SO(3) in Loop Quantum Gravity"
He expanded it for publication and reposted last month
http://arxiv.org/gr-qc/0401122
the article by Ling and Zhang is
http://www.arxiv.org/abs/gr-qc/0309018
"Do Quasinormal Modes Prefer Supersymmetry"
(this is not a recommendation of Swain's or Ling's ideas, but just in case a reader wants to see what nonunitary was referring to, the investigation of BH quasinormal modes has caused a ferment of ideas among which ES is only one contender, it has also brought new people into LQG: for example Swain is an experimental particle physicist who was drawn into LQG by this. As nonunitary indicates some (if not all) of the ES people are newcomers too)
Originally posted by nonunitary
Marcus,
...
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 phenomenology. 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.
Please let the rest of us know if there are any talks from the conference posted or any further reports are available.
About what you said on the subject of ES area spectrum:
may I translate the main objection you have
from loops to spin network states?
Is the objection then that
there can be edges of the spin network with zero spin (?)
and that one wishes they would not contribute area but they do contribute area because of the term (j + 1/2).
Right now I am not clear about the problem. Maybe one
defines spin networks so that the miniumum j is 1/2?
EDIT: Alejandro Corichi recently posted a clarifying 3page paper on this question:
http://arxiv.org./gr-qc/0402064
"Comments on area spectrum in Loop Quantum Gravity"
this paper may be said to settle the ES hash or
cook the ES goose
nonunitary
Feb7-04, 08:27 AM
About what you said on the subject of ES area spectrum:
may I translate the main objection you have
from loops to spin network states?
Is the objection then that
there can be edges of the spin network with zero spin (?)
and that one wishes they would not contribute area but they do contribute area because of the term (j + 1/2).
Right now I am not clear about the problem. Maybe one
defines spin networks so that the miniumum j is 1/2?
Marcus,
You are right. Maybe I was not clear enough. A closed loop is a particular case of a closed graph, and we can define a spin network there by assigning reps. of SU(2) to it, labelled by j. If one choses j=0, one has the trivial function (explained in my previous post). One then defines spin networks starting with j=1/2 and do not include the $j=0$ case. One could include it, but then everytime one has a statement, one would have to add something about the j=0 case. It is not convenient and might lead to confusion.
Now, if one had three-zillion edges with j=0, it is not that one wished that they do not contribute. The statement is much stronger:
if there is an operator that "sees" the j=0 edges, then it is not a well defined operator of the theory. This is a result that comes of the very precise and rigurous ways in which the Hilbert space of the theory is built. My favorite way of seeing this problem is by the C*-algebra argument I used, but I am sure that one can come up with more explanations.
There is an analogy in ordinary QM, where the Hilbert space is L^2 "functions". Actually they are equivalence class of functions where two functions define the same state is they are the same "almost everywhere". Suppose we want to define the operator that has the action "evaluate the wave funtion at the origin".
Clearly, the action of the operator on two functions of the same class that hapen to take different values at the origin will be different. The operator does not respect the equivalence classes and is therefore not well defined.
early on, around page 2 I think, Tsunami urged that this thread
"continue on in nerdy fashion"
and it has done so, becoming a thread for introducing and discussing interesting aspects of LQG
on page 3 we opened the can of worms of the LQG Area Spectrum
which a minority (nonunitary points out that they are newcomers also) wants to revise so that it is evenly-spaced (ES)
IIRC the first link was one Meteor posted on the "surrogate sticky" thread, and he also supplied one other ES link here.
Nonunitary showed where the roadblocks are to adopting the proposed ES spectrum.
Pages 3-5 of this thread contain discussion of the area spectrum and some Black Holery. Thanks to all who participated---it was interesting and we may get back to it.
-----------------------------------
Now a new topic. 2+1 quantum gravity.
Ordinarily one thinks of studying 3+1 dimensional gravity.
Can anything useful be learned from studying gravity in 2 spatial
and one temporal dimension----in 3D instead of 4D?
Some famous people have found it interesting enough to explore and write about
S. Deser, R. Jackiw, G. 't Hooft "Three-dimensional Einstein Gravity" (1984)
then a few years later (in 1988) Edward Witten "(2+1)-Dimensional Gravity As An Exactly Soluble System"
S. Carlip has a Cambridge monograph on it "Quantum Gravity In 2+1 Dimensions".
---------------------------
Apparently despite the encouraging title of Edward Witten's paper there are still good many problems to solve about even this dumbed-down or toy version of gravity. Maybe it should not be called toy. And there is a widely shared suspicion that successfully quantizing gravity in 3D will give lots of hints as to how to do it 4D.
Some of the people in LQG who are currently working on 3D(or have irons in the fire) in random order:
Laurent Freidel
Karim Noui
Alejandro Perez
David Louapre
Etera Livine
I think what we might to do is just check out a little of what they are doing to keep track of what is happening in the 3D quantum gravity
department.
Does anyone have other suggestions----they are welcome too.
Here are some links
http://www.physicsforums.com/showthread.php?s=&postid=128813#post128813
to work by the LQG people just mentioned
in the 2+1 D direction
of particular interest I think is a series of 4 papers that is in the works, the first one is out
and David Louapre says to expect the second in a few weeks
1. L.Freidel and D. Louapre,"Ponzano-Regge model revisited I: Gauge fixing, observables and interacting spinning particles"
http://arxiv.org./hep-th/0401076
2. L.Freidel and D. Louapre, “Ponzano-Regge model revisited II: Mathematical aspects; relation with Chern-Simons theory, DSU(2) quantum group and link invariant". To appear.
3. L.Freidel, E. Livine and D. Louapre, “Ponzano-Regge model revisited III: The Field Theory limit”. To appear.
4. L.Freidel and D. Louapre, “Ponzano-Regge model revisited IV: Lorentzian 3D Quantum Geometry”. To appear.
----------------------
Karim Noui and Alejandro Perez have one in the works called
"Three dimensional loop gravity coupled to point particles".
Ambitwistor referred to a talk by Perez at Penn State last fall about this. Louapre mentioned a more recent talk by Karim Noui.
Freidel/Louapre and the other two are probably getting results along some similar lines.
I want to read some in the Freidel/Louapre paper number 1. because it gives an overview of what they intend to accomplish in this series of papers.
BTW in case you looked at Freidel and Louapre's paper
and didn't recognize that unusual symbol on page ten
it is the Hebrew letter "daleth"
(I asked David to be sure)
Freidel/Louapre hep-th/0401076:
"In our paper, we consider the spin foam quantization of three dimensional gravity coupled to quantum interacting spinning particles. We revisit the original Ponzano-Regge model in the light of recent developments and we propose the first key steps toward a full understanding of 3d quantum gravity in this context, especially concerning the issue of symmetries and the inclusion of interacting spinning particles.
The first motivation is to propose a quantization scheme and develop techniques that could be exported to the quantization of higher dimensional gravity. As we will see, the inclusion of spinless particles is remarkably simple and natural in this context and allows us to compute quantum scattering amplitudes. This approach goes far beyond what was previously done in this context by allowing us to deal with the interaction of particles.
The inclusion of spinning particles is also achieved. The structure is more complicated but the operators needed to introduce spinning particles show a clear and beautiful link with the theory of Feynman diagrams [26]."
Marcus,
Are spinless particles one's with spin zero, or particles that are not expressed in this value?
Can you give short definition of quantum scattering amplitudes?
Thanks!
LPF
my original post was mistaken, thanks to sA for the correction
there is an entry-level discription of "amplitudes"
in Feynmann's book "QED The Strange Theory of Light and Matter"
around page 78. Do you have a library where you could
borrow this book. It is very thin (150 pages) but costly
to buy.
selfAdjoint
Feb14-04, 08:58 AM
They really mean that the math of spin does not apply to these particles. They are also called scalar particles, and they are frequent subjects in developing new ideas because they are easy to work with. Actually there are no scalar particles in today's physical theories except the hypothetical Higgs partical, and you bet these scal particles are not intended to model the Higgs.
eigenguy
Feb14-04, 11:51 AM
Originally posted by selfAdjoint
---the math of spin does not apply to---scalar particles.
The math of spin does apply: spin-0 means lorentz scalar, which is where "scalar" comes from.
Originally posted by selfAdjoint
there are no scalar particles in today's physical theories except the hypothetical Higgs partical
There is the dilaton of ST.
Hello all, I'm quoting some exerpts to give an idea of context and what the article's about. This first quote was from page 3.
Freidel/Louapre hep-th/0401076
"In our paper, we consider the spin foam quantization of three dimensional gravity coupled to quantum interacting spinning particles. We revisit the original Ponzano-Regge model in the light of recent developments and we propose the first key steps toward a full understanding of 3d quantum gravity in this context, especially concerning the issue of symmetries and the inclusion of interacting spinning particles.
The first motivation is to propose a quantization scheme and develop techniques that could be exported to the quantization of higher dimensional gravity. As we will see, the inclusion of spinless particles is remarkably simple and natural in this context and allows us to compute quantum scattering amplitudes. This approach goes far beyond what was previously done in this context by allowing us to deal with the interaction of particles.
The inclusion of spinning particles is also achieved. The structure is more complicated but the operators needed to introduce spinning particles show a clear and beautiful link with the theory of Feynman diagrams [26]."
The next exerpt is from page 20:
Freidel/Louapre hep-th/0401076
"...
It is well known that, at the classical level, three dimensional gravity can be expressed as a Chern-Simons theory where the gauge group is the Poincare group. The Chern-Simons connection A can be written in terms of the spin connection ω and the frame field e,
A = \omega_iJ^i + e_iP^i
where Ji are rotation generators and Pi translations.
Moreover, since the work of Witten [44], it is also well known that quantum group evaluation of colored link gives a computation of expectation value of Wilson loops in Chern-Simons theory. Our result
therefore gives an exact relation, at the quantum level, between expectation value in the Ponzano-Regge version of three dimensional gravity and the Chern-Simons formulation...
[footnote] κ = 1/4G is the Planck mass in three dimensional gravity"
So why is the Planck mass different in 3 dimensions?
got kind of a glimmering
but has anybody thought thru this one and
got an explanation ready?
In 4D---the world we have----the planck quantities are such and such.
But, according to this paper in 1+2 dimensions----their 3D world---the planck mass is
\frac{1}{4G}
thats what the footnote on page 20 says, that i just quoted
anyone want to comment or differ with this or explain?
answered my own question
in 1+2D
newtons constant has dimensions of
inverse mass
http://arxiv.org/hep-th/0205021
s'what I thought cause of force falling off
as reciprocal of distance instead
of sq. recip
the mass unit in 3D is basically 1/G
order one coefficients like 4 or 8pi are
mostly a matter of convention (how you
write the einstein equation, the 8pi business)
lets put c and hbar in explicitly and see what the planck units
actually are in (1+2)D
thing to notice is that in 4D we have
GM^2 = \hbar c
because GM^2 has to equal the unit force x area (inverse sq. law)
and that equation defines the pl. mass in 4D
but in 3D GM^2 will equal the unit force x distance!
and that is the unit energy in the system: Mc^2, so we have instead
GM^2 = M c^2 which solves to
M = \frac{c^2}{G}
After that, easy, unit energy is
E = \frac{c^4}{G}
and unit freq is
\omega = \frac{c^4}{G\hbar}
That makes unit time
T = \frac{G\hbar}{c^4}
and unit distance
L = \frac{G\hbar}{c^3}
nonunitary
Feb16-04, 11:37 AM
I was meaning to elaborate on the reasons why the ES-are espectrum of Alekseev and colaborators is not well defined.
Too late, this guy must have read my posts and took part of them, added a new argument with graphs and posted:
http://arxiv.org/abs/gr-qc/0402064
I think I have to agree with him. What he didn't say though is that one might be abre to define a new operator that somehow "ignores" a j=0 edge, but there is some work involved in showing that it is possible.
Anyway, farewell to the ES-area operator of APS.
Originally posted by nonunitary
Too late, this guy must have read my posts and took part of them, added a new argument with graphs and posted:
http://arxiv.org/abs/gr-qc/0402064
What a coincidence! I posted that very same link yesterday evening
my comment was it "cooks the ES goose"
http://www.physicsforums.com/showthread.php?s=&postid=142694#post142694
he does seem to have some of the same arguments
so your posts have been in a certain sense prophetic
Originally posted by nonunitary
... this guy must have read my posts and took part of them, added a new argument with graphs and posted:
http://arxiv.org/abs/gr-qc/0402064
I think I have to agree with him. What he didn't say though is that one might be abre to define a new operator that somehow "ignores" a j=0 edge, but there is some work involved in showing that it is possible...
again you were prophetic, the guy has added a paragraph to his
conclusions and updated the preprint
(it is now a little longer and is dated 17 February instead of 13 February)
and the addition includes the case where the operator is
ad hoc made to ignore any j=0 edge
so the spectrum is ES except for a double-size space at zero.
the author does not like this case but he includes it (with a warning) presumably for the sake of completeness
I checked the Gour/Suneeta paper (gr-qc/0401110) and it did not
seem to disturb their calculation of BH entropy
I could not see any reason to accept or reject, it appeared (at least for now) to be just an arbitrary ad hoc fix.
I am looking for previous discussion of BH entropy, BH area, in LQG context.
Sauron recently posed some questions about entropy and LQG in another thread and hopefully there is something relevant to that here.
------here's some of Sauron's post-------
I have a few generic questions/reflections about some of the themes LQG is addressing.
Let´s begin by the question of entropy. My deal is whether the concept of entropy makes sense in GR at all. At least in the same sense as in ordinary statistical mechanics.
I know about two main results. The one, of wich i have a reasonable understanding , about the black hole area behaving like entropy. I also have notice about (but no understanding at all) results of Penrose relating the Weyl tensor to entropy, at least in cosmological scenarios.
The question is that in the microcanonical device the entropy is related to the number of micro-states compatible with an energy. But in GR there is no a good (and less local) definition of the energy of the gravitational field....
--end of exerpt--
http://www.physicsforums.com/showthread.php?p=195126#post195126
I am trying to connect Sauron's post with earlier discussion we've had about LQG and entropy.
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