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Bianchi's entropy resultwhat to ask, what to learn from it 
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#19
Apr2512, 08:35 PM

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#20
Apr2512, 08:39 PM

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#21
Apr2512, 08:46 PM

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#22
Apr2512, 08:49 PM

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[tex] \sum_f \frac{1}{j_f^p},[/tex] so presumably these converge and go to zero in the large [itex]j_f[/itex] limit. So maybe it's not so bad, but it would help to clarify the role of the limit. 


#23
Apr2512, 08:53 PM

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#24
Apr2512, 09:05 PM

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#25
Apr2512, 09:08 PM

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We still have to find out how L_{etc} is defined. I don't see it. There could be a discrepancy that needs to be fixed. Possibly just a typo. Or it could be all right and I'm just missing something. What we want is for L_f = j_f (rather than the sqrt of j(j+1).) Could things have actually been defined so it comes out that way? 


#26
Apr2612, 07:17 AM

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#27
Apr2612, 07:44 AM

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I reached out to bianchi for clarification about his area formula. In the interest of keeping his privacy, I will just summarize the main points of his brief reply that are apparently common knowledge.
In short, both [itex] \sqrt{j(j+1)} [/itex] and [itex] j [/itex] are acceptable area operators (they differ by an operator ordering ambiguity that vanishes as [itex] \hbar \rightarrow 0 [/itex] (which I guess here means something like [itex] j \rightarrow \infty [/itex] as fzero and others suggested). The two criteria for an area operator are apparently 1) that its eigenvalues go to j in the large j limit and 2) that its eigenvalue vanish for j=0. More systematically, bianchi is using a Schwinger oscillator type representation where we have two operators [itex] a_i [/itex] and the spins are [itex] \vec{J} = \frac{1}{2} a^+ \vec{\sigma} a [/itex]. The total spin of the representation can be read off from the total number [itex] N = a_1^+ a_1 + a_2^+ a_2 = 2j [/itex]. On the other hand, you can work out [itex] J^2 [/itex] for yourself to find [itex] J^2 = \frac{1}{4}( N^2 + 2N) [/itex] which one easily verifies gives [itex] J^2 = j(j+1) [/itex]. Thus by [itex] \vec{L} [/itex] bianchi appears to mean [itex] N/2 [/itex]. It is again interesting to see this kind of representation appearing in a useful way since it is quite important in condensed matter. 


#28
Apr2612, 08:04 AM

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One physical meaning one can attach to this expression is the following. If an observer of four velocity [itex] u [/itex] (with [itex] u^2 = 1[/itex]) measures the energy of a particle with four momentum [itex] p [/itex] then the energy measured is [itex] E = u\cdot p [/itex]. An observer hovering about the black hole horizon has [itex] u = (1/\sqrt{g_{tt}}) \partial_t [/itex]. On the other hand, the existence of the Killing vector [itex] \xi = \partial_t [/itex] means that the quantity [itex] \xi\cdot p [/itex] is conserved independent of r provided the particle of momentum p is following a geodesic. This quantity [itex] \xi \cdot p [/itex] is the energy of the particle measured at infinity [itex] E_\infty = \xi \cdot p [/itex]. Thus one sees that the energy measured by a hovering observer near the horizon is [itex] E = E_{\infty} /\sqrt{g_{tt}} [/itex] which diverges as the hovering observer approaches the horizon. Another perspective is that the hovering observer must fire her engines harder and harder to keep from falling as the horizon is approached. From the perspective of an inertial infalling observer the hovering observer is uniformly accelerated and hence experiences unruh radiation. The temperature of this radiation gets hotter and hotter as the hovering observer approaches the horizon (while the infalling observer sees nothing). Still another point of contact is Luttinger's old idea in condensed matter to model position dependent temperature by adding a spatially varying gravitational field e.g. for computing heat currents using Kubo formulae. Of course, this is most obvious in the Lorentz invariant context when one computes variations with respect to a background metric to evaluate correlators of the stress tensor i.e. [itex] \delta S = \int d^D x \sqrt{g} \delta g_{\mu \nu} T^{\mu \nu} [/itex]. 


#29
Apr2612, 08:08 AM

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Yes, I know this local temperature concept. However I think it has no relevance in defining entropy.



#30
Apr2612, 08:21 AM

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Well, I think it is a subtle notion, but it can be useful for studying entropy.
For example, consider Rindler space [itex] ds^2 =  \rho^2 d \eta^2 + d\rho^2 + dx_\perp^2 [/itex] where [itex] \eta [/itex] is like a time and [itex] \rho= 0[/itex] is the horizon. Imagine a CFT living in this space. Then one gets an entropy density of roughly [itex] s \sim T_\ell^d [/itex] (with d the space dimension). Integrating this over all space gives [tex] \int d\rho dx^{d1}_\perp \rho^{d} \sim \frac{A_\perp}{\rho^{d1}_c} .[/tex] [itex] A_\perp [/itex] is the cross sectional area and [itex] \rho_c [/itex] is some small distance cutoff. Putting something like the planck length in as a fundamental cutoff gives something that looks an awful lot like black hole entropy. Viewing the CFT in rindler space as an approximation to the near horizon of a black hole with matter, it looks like this calculation is giving a quantum correction due to the matter fields, the CFT, to the entropy of the black hole. 


#31
Apr2712, 06:19 AM

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#32
Apr2712, 04:37 PM

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#33
Apr2712, 06:23 PM

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Francesca, these are very instructive replies to Demy and to Atyy's question.
The researchers might be able to learn something by comparing Bianchi's thermodynamic entropy with the statistical statecounting done earlier. I imagine some are studying this comparison and it may prove fruitful. I was interested by what you said in reference to the last section of the paper "Partition Function and Spinfoams". That section goes beyond the main task of the paper and seems to point towards further work. Was that what you were referring to when you said "And finally there is the remarkable demonstration that the Spinfoam amplitude implies the right distribution, that yields Hawking entropy." For me, that section needs more elaborationit does not explain enough what is being donebut I suspect that would be more appropriately done in a second paper, not to overextend this one. 


#34
Apr2712, 07:56 PM

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The polymer microstate calculation had an explicit dependence on the Immirzi parameter. The only reason the present calculation does not have this dependence is because Bianchi uses a single pure state to do the calculation. Since the BH is not a pure state, the correct way to do the computation is to compute the energy from (9) in an ensemble. This will reintroduce the factors of [itex]\mu^*[/itex] and [itex]\gamma[/itex] that were found in the polymer paper.



#35
Apr2712, 08:43 PM

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#36
Apr2712, 08:48 PM

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In case anyone is new to the discussion the "polymer" paper just referred to is from a years and a half ago and is:
http://arxiv.org/abs/1011.5628 Black Hole Entropy, Loop Gravity, and Polymer Physics Eugenio Bianchi (Submitted on 25 Nov 2010) Loop Gravity provides a microscopic derivation of Black Hole entropy. In this paper, I show that the microstates counted admit a semiclassical description in terms of shapes of a tessellated horizon. The counting of microstates and the computation of the entropy can be done via a mapping to an equivalent statistical mechanical problem: the counting of conformations of a closed polymer chain. This correspondence suggests a number of intriguing relations between the thermodynamics of Black Holes and the physics of polymers. 13 pages, 2 figures The main paper we are discussing is the one Bianchi just posted this week. For convenience, since we just turned a page, I will give the link and abstract again: http://arxiv.org/abs/1204.5122 Entropy of NonExtremal Black Holes from Loop Gravity Eugenio Bianchi (Submitted on 23 Apr 2012) We compute the entropy of nonextremal black holes using the quantum dynamics of Loop Gravity. The horizon entropy is finite, scales linearly with the area A, and reproduces the BekensteinHawking expression S = A/4 with the onefourth coefficient for all values of the Immirzi parameter. The nearhorizon geometry of a nonextremal black hole  as seen by a stationary observer  is described by a Rindler horizon. We introduce the notion of a quantum Rindler horizon in the framework of Loop Gravity. The system is described by a quantum surface and the dynamics is generated by the boost Hamiltonion of Lorentzian Spinfoams. We show that the expectation value of the boost Hamiltonian reproduces the local horizon energy of Frodden, Ghosh and Perez. We study the coupling of the geometry of the quantum horizon to a twolevel system and show that it thermalizes to the local Unruh temperature. The derived values of the energy and the temperature allow one to compute the thermodynamic entropy of the quantum horizon. The relation with the Spinfoam partition function is discussed. 6 pages, 1 figure 


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