Bianchi's entropy result-what to ask, what to learn from it

[marcus]

Bianchi's entropy result--what to ask, what to learn from it

I think most if not all here are familiar with the idea that entropy, by definition, is not an absolute but depends on the observer. (Padmanabhan loves to make that point. :-D) There may also be an explicit scale-dependence. And in the Loop context one expects the Immirzi parameter to run with scale.

Likewise black hole horizon temperature is highly dependent on how far away the observer is hovering. So there is this interesting and suggestive nexus of ideas that we need to pick apart and learn something from. Bianchi has just made a significant contribution to this.

http://arxiv.org/abs/1204.5122
Entropy of Non-Extremal Black Holes from Loop Gravity
Eugenio Bianchi
(Submitted on 23 Apr 2012)
We compute the entropy of non-extremal black holes using the quantum dynamics of Loop Gravity. The horizon entropy is finite, scales linearly with the area A, and reproduces the Bekenstein-Hawking expression S = A/4 with the one-fourth coefficient for all values of the Immirzi parameter. The near-horizon geometry of a non-extremal 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 two-level 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

 

[marcus]

Bianchi uses the classic Clausius definition of entropy ∂S = ∂E/T
and makes it very clear where the observer is hovering, at what distance from horizon.
So observer's measurement of E and T depends on that, but the effects cancel and to first order he gets S = A/4.

Earlier treatments of BH entropy did not use the Clausius relation. Instead, they employed state counting. One assumes the observer is low-resolution and can make only the coarsest distinctions. So the more states he confuses with each other, the more entropy. You take the log of the number of states and that's it. Or if it is a Hilbertspace of quantum states, you take the log of the dimension of the Hilbertspace.

Ted Jacobson made some critical comments about this in a 2007 paper, which is Bianchi's reference [20] at the end. I would be really interested to know Jacobson's reaction to Bianchi's paper.
http://arxiv.org/abs/0707.4026
Renormalization and black hole entropy in Loop Quantum Gravity
Ted Jacobson
7 pages
(Submitted on 26 Jul 2007)
"Microscopic state counting for a black hole in Loop Quantum Gravity yields a result proportional to horizon area, and inversely proportional to Newton's constant and the Immirzi parameter. It is argued here that before this result can be compared to the Bekenstein-Hawking entropy of a macroscopic black hole, the scale dependence of both Newton's constant and the area must be accounted for. The two entropies could then agree for any value of the Immirzi parameter, if a certain renormalization property holds."

Bianchi also introduces the concept of "quantum Rindler horizon" which I don't recall being used in earlier Loop BH entropy papers. If you know of an instance, please let me know--I could have simply missed it. Mathematically the idea of "γ-simple" unitary representations of SL(2,C) is intriguing and could turn out to be a fertile useful concept. It was already there, he just found a good terminology, I think, and occasionally in math that can be important.

I wonder if one might conclude that the bare value of the Immirzi is 0.2375. In many papers that study the long distance limit they let gamma go to zero----meaning that the region stays the same size but its geometry gets less fuzzy. Less "rumpled" like an unmade bed is rumpled. Then gamma=0.2375 would represent the maximal rumpling of nature. Just speculating

It's classy to use the Clausius definition of entropy.
In my humble opinion if you ever want a beard this is the kind to have:
http://en.wikipedia.org/wiki/Rudolf_Clausius (1822-1866)

Under no circumstances do you want one like this http://en.wikipedia.org/wiki/Ludwig_Boltzmann

 

[atyy]

If there's no state counting, isn't this just a semiclassical calculation, like Hawking's?

 

[Physics Monkey]

What intrigues me is the appearance of the boost generator. I would like to understand this operator better in the loop gravity context.

The connection between rindler horizons and the boost generator is long known, but I have a lot of interest in this topic because we have recently been able to put this connection to good use in condensed matter. Of course, all this spin network stuff reminds me of my old pal tensor networks, and I wonder if there is some grand synthesis (involving tensor networks, entanglement, holography, ...) possible here.

 

[atyy]

The connection between rindler horizons and the boost generator is long known, but I have a lot of interest in this topic because we have recently been able to put this connection to good use in condensed matter.

How?

 

[Paulibus]

...if you ever want a beard this is the kind to have:
http://en.wikipedia.org/wiki/Rudolf_Clausius (1822-1866)

Way to go. Got one like that.

 

[francesca]

If there's no state counting, isn't this just a semiclassical calculation, like Hawking's?

No, it's not semiclassical. In fact in the paper all the ingredients are derived from the full quantum theory. The relation energy-area was found by Frodden Gosh Perez using Einstein equations; here it is found using the boost generator given by Spinfoam Theory. The calculation in order to find Unruh temperature is done here again using the boost generator, it's completely new. And finally there is the remarkable demonstration that the Spinfoam amplitude implies the right distribution, that yields Hawking entropy.

 

[Physics Monkey]

Having read the paper a little more closely, I have some basic confusion about what is going on:

1. Although Bianchi claims that E and A don't commute, it looks like on the image under $Y_\gamma$ of the spatial spin network states they are essentially identical. This seems to be so because of the $\gamma$-simple constraint Eq. 6

2. Related to 1, in what sense can an eigenstate of energy and area possibly have an entropy?

3. Everything looks like a product state over facets, but I would expect entropy and thermalization to be associated with some interactions between facets.

4. Is there a $\rho$ for which $S = - \text{tr}(\rho \log{\rho})$?

5. What is the physical state space? Is it the finite spin network basis (given a set of punctures)? Surely the continuous space of SL(2,C) representations are not the physical states?

I should say that I haven't yet processed the temperature derivation section, although it looks like a standard unruh-type setup. Perhaps some of the answers can be found there, but many of these issues seem more basic as if they should be understood before tackling the issue of temperature.

 

[Physics Monkey]

How?

I am interested in the spectrum of the reduced density matrix of spatial regions inside bulk materials. This spectrum knows a lot about entanglement e.g. the entanglement entropy is computable from it. If $\rho_R = \exp{(-H_R)}$ is the reduced density matrix of region R in the ground state, then the spectrum of $H_R$ (defined by this equation) is the entanglement spectrum.

It is an old result in Lorentz invariant field theory that when R is the half space, say x>0, then $H_R = 2 \pi K$ with $K$ the boost generator mixing x and t. Thus for LI field theory we know the entanglement spectrum for a special subregion, the half space. The form of the operator $K$ is $K = \int_{x>0} dx\, dx^2 ... dx^d \left( x T^{tt} \right)$ (at t=0) and hence it looks like the physical Hamiltonian with an edge. We used this to show that in many cases the entanglement spectrum shares many universal features with the energy spectrum of a physical edge. In other words, the imaginary entanglement cut becomes a real physical cut in the system.

A simple example is provided by the fractional quantum Hall effect. In that case a physical edge always has a chiral edge mode circulating around the sample. Using the technology above we were able to show that the entanglement spectrum also has this chiral edge mode. So even on a system with no boundary you can, by looking at entanglement, detect the existence of protected chiral edge states.

 

[atyy]

No, it's not semiclassical. In fact in the paper all the ingredients are derived from the full quantum theory. The relation energy-area was found by Frodden Gosh Perez using Einstein equations; here it is found using the boost generator given by Spinfoam Theory. The calculation in order to find Unruh temperature is done here again using the boost generator, it's completely new. And finally there is the remarkable demonstration that the Spinfoam amplitude implies the right distribution, that yields Hawking entropy.

But isn't the action at the end a semiclassical one?

 

[marcus]

But isn't the action at the end a semiclassical one?

The main result(s) of the paper are proved in the first 4 pages up thru the section called
*Entropy of the Quantum Horizon*. You must be talking about some action that appears in pages 1-4, but I can't figure out which.

There is the section on page 5 which I see as kind of a postscript. It contains some interesting reflections and points to some future work (a paper which Wieland and Bianchi have in the works.) But that is not essential to the main work of the paper, it's more interpretive afterthought, and it does mention something that occurs in the "semiclassical limt of the Spinfoam path integral..." But that hardly means that the whole paper is proving things only at the semiclassical level (this is what some of your earlier comments seemed to be suggesting.)

 

[Physics Monkey]

The main result(s) of the paper are proved in the first 4 pages up thru the section called
*Entropy of the Quantum Horizon*. You must be talking about some action that appears in pages 1-4, but I can't figure out which.

There is the section on page 5 which I see as kind of a postscript. It contains some interesting reflections and points to some future work (a paper which Wieland and Bianchi have in the works.) But that is not essential to the main work of the paper, it's more interpretive afterthought, and it does mention something that occurs in the "semiclassical limt of the Spinfoam path integral..." But that hardly means that the whole paper is proving things only at the semiclassical level (this is what some of your earlier comments seemed to be suggesting.)

A simpler reason to worry about semiclassicality is found in the early pages, especially after Eq. 8 and Eq. 9. There Bianchi makes heavy use of the classical results to identify the right operator to call the "energy" of the horizon. One could worry in the usual way that this identification is semiclassical. For example, will the quantum hair proposal of Ghosh-Perez be captured by these identifications?

 

[fzero]

The Clausius relation is classical. The calculation done here is semiclassical because $\delta E$ corresponds to the addition of a single quantum of energy. In this regard, it's not that different from Hawking's approach. A fully quantum treatment must involve the counting of microstates.

In fact, the derivation of the energy of the black-hole is quite confusing from entropy considerations. Bianchi says that the Rindler surface is described by the state

$$|s\rangle = \otimes_f | j_f \rangle,$$

which results from a tesselation into the facets $f$. But this is a pure state and should have zero information-theoretic entropy. I'm not sure if it even makes sense to talk about other tesselations in this framework, but from the statistical point of view, one would want a mixed state obtained by summing over tesselations. The black hole should then turn out to be a maximal entropy configuration.

 

[marcus]

Having read the paper a little more closely, I have some basic confusion about what is going on:

1. Although Bianchi claims that E and A don't commute, it looks like on the image under $Y_\gamma$ of the spatial spin network states they are essentially identical. This seems to be so because of the $\gamma$-simple constraint Eq. 6

I've been appreciating your comments, since you know a lot about this. I'm glad you took an interest and read the paper. Part of the confusion could be due to problems with notation. I think of what we have now as a draft to which more explanation could be added.

I could be wrong but I don't think it says "E and A don't commute". The OPERATORS for energy and area are denoted H and A, are they not?
The letter E seems to denote a quantity. At one point he says E = <s|H|s>, so as a quantity it would commute with everything I suppose.

The energy operator H is defined by eqn (8) and seems to be composed of boost pieces. The area operator seems to be composed of rotation pieces. Correct me if I'm wrong these don't commute as operators, do they? Equation (6) just says they have the same matrix element form. Let me know if I'm saying something really stupid. So anyway I think on page 3, middle of first column, where he says "the energy does not commute with the area of the quantum horizon" what he means is "H and A don't commute."

Is this right? You are by far the expert in this context.

 

[fzero]

I've been appreciating your comments, since you know a lot about this. I'm glad you took an interest and read the paper. Part of the confusion could be due to problems with notation. I think of what we have now as a draft to which more explanation could be added.

I could be wrong but I don't think it says "E and A don't commute". The OPERATORS for energy and area are denoted H and A, are they not?
The letter E seems to denote a quantity. At one point he says E = <s|H|s>, so as a quantity it would commute with everything I suppose.

The energy operator H is defined by eqn (8) and seems to be composed of boost pieces. The area operator seems to be composed of rotation pieces. Correct me if I'm wrong these don't commute as operators, do they? Equation (6) just says they have the same matrix element form. Let me know if I'm saying something really stupid. So anyway I think on page 3, middle of first column, where he says "the energy does not commute with the area of the quantum horizon" what he means is "H and A don't commute."

Is this right? You are by far the expert in this context.

How is the operator $|\vec{L}_f|$ defined? I understand the subscript, but is it supposed to be the operator whose eigenvalue is the square root of that of $|\vec{L}_f|^2$? How do we find $j_f$ instead of $\sqrt{j_f(j_f+1)}$ (or has the limit of large $j_f$ been taken?)

In any case, the states $|E\rangle$ defined below (9) are simultaneous eigenstates of $|L|, L_z$ and $K_z$, so $H$ and $A$ commute on them as operators.

 

[marcus]

Just to be clear, what I said in post #14 was in reply to this of Physics Monkey:

...
1. Although Bianchi claims that E and A don't commute, it looks like on the image under $Y_\gamma$ of the spatial spin network states they are essentially identical. This seems to be so because of the $\gamma$-simple constraint Eq. 6
...

Physics Monkey also asked about the physical state space. It may help us to better understand the paper and even some of the notation if we read the first paragraph, where he refers to an earlier paper of his about black hole entropy. This is his reference [4]

==quote first paragraph==
Loop Gravity [2] has been shown to provide a geometric explanation of the finiteness of the entropy and of the proportionality to the area of the horizon [3]. The microstates are quantum geometries of the horizon [4]. What has been missing until recently is the identification of the near-horizon quantum dynamics and a derivation of the universal form of the Bekenstein-Hawking entropy with its 1/4 prefactor. This is achieved in this letter.
==endquote==
Here is [4]:
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
This was a year and a half ago and employed an entirely different method, namely (semiclassical) state-counting. But some of the notation and definitions undoubtably overlap, so this paper might be of use.

 

[fzero]

Just to be clear, what I said in post #14 was in reply to this of Physics Monkey:

Physics Monkey also asked about the physical state space. It may help us to better understand the paper and even some of the notation if we read the first paragraph, where he refers to an earlier paper of his about black hole entropy. This is his reference [4]

==quote first paragraph==
Loop Gravity [2] has been shown to provide a geometric explanation of the finiteness of the entropy and of the proportionality to the area of the horizon [3]. The microstates are quantum geometries of the horizon [4]. What has been missing until recently is the identification of the near-horizon quantum dynamics and a derivation of the universal form of the Bekenstein-Hawking entropy with its 1/4 prefactor. This is achieved in this letter.
==endquote==
Here is [4]:
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
This was a year and a half ago and employed an entirely different method, namely (semiclassical) state-counting. But some of the notation and definitions undoubtably overlap, so this paper might be of use.

In that paper, he uses $A_f \sim \sqrt{j_f(j_f+1)}$ and the entropy still has the coefficient of $\gamma$. The troublesome thing is that, in the new paper

$$E \sim \sum_f j_f$$

is not proportional to

$$A \sim \sum_f \sqrt{j_f(j_f+1)}.$$

Using the Clausius relation gives a correction to the area law. At first order, the correction is proportional to $N$, the number of facets. The 2010 paper, if it applies here, suggests in eq (19) that $N \sim A$, so this corrects the coefficient of the leading term (away from 1/4).

Edit: there is a mistake in the estimate above, see https://www.physicsforums.com/showpost.php?p=3884283&postcount=22

 

[atyy]

I am interested in the spectrum of the reduced density matrix of spatial regions inside bulk materials. This spectrum knows a lot about entanglement e.g. the entanglement entropy is computable from it. If $\rho_R = \exp{(-H_R)}$ is the reduced density matrix of region R in the ground state, then the spectrum of $H_R$ (defined by this equation) is the entanglement spectrum.

It is an old result in Lorentz invariant field theory that when R is the half space, say x>0, then $H_R = 2 \pi K$ with $K$ the boost generator mixing x and t. Thus for LI field theory we know the entanglement spectrum for a special subregion, the half space. The form of the operator $K$ is $K = \int_{x>0} dx\, dx^2 ... dx^d \left( x T^{tt} \right)$ (at t=0) and hence it looks like the physical Hamiltonian with an edge. We used this to show that in many cases the entanglement spectrum shares many universal features with the energy spectrum of a physical edge. In other words, the imaginary entanglement cut becomes a real physical cut in the system.

A simple example is provided by the fractional quantum Hall effect. In that case a physical edge always has a chiral edge mode circulating around the sample. Using the technology above we were able to show that the entanglement spectrum also has this chiral edge mode. So even on a system with no boundary you can, by looking at entanglement, detect the existence of protected chiral edge states.

That's fascinating. In the original case of a Lorentz invariant field theory on the half space, are there also "edge states"?

 

[Physics Monkey]

In that paper, he uses $A_f \sim \sqrt{j_f(j_f+1)}$ and the entropy still has the coefficient of $\gamma$. The troublesome thing is that, in the new paper

$$E \sim \sum_f j_f$$

is not proportional to

$$A \sim \sum_f \sqrt{j_f(j_f+1)}.$$

Using the Clausius relation gives a correction to the area law. At first order, the correction is proportional to $N$, the number of facets. The 2010 paper, if it applies here, suggests in eq (19) that $N \sim A$, so this corrects the coefficient of the leading term (away from 1/4).

I didn't even notice this at first, but it looks like bianchi is either doing the large j limit or made an important mistake?

 

[Physics Monkey]

That's fascinating. In the original case of a Lorentz invariant field theory on the half space, are there also "edge states"?

There certainly can be. Not all Lorentz invariant theories have protected physical edge states on a half space, but we showed that if they do then the half space entanglement spectrum (with no physical edge) also has the universal aspects of these physical edge states.

 

[Physics Monkey]

I've been appreciating your comments, since you know a lot about this. I'm glad you took an interest and read the paper. Part of the confusion could be due to problems with notation. I think of what we have now as a draft to which more explanation could be added.

I could be wrong but I don't think it says "E and A don't commute". The OPERATORS for energy and area are denoted H and A, are they not?
The letter E seems to denote a quantity. At one point he says E = <s|H|s>, so as a quantity it would commute with everything I suppose.

The energy operator H is defined by eqn (8) and seems to be composed of boost pieces. The area operator seems to be composed of rotation pieces. Correct me if I'm wrong these don't commute as operators, do they? Equation (6) just says they have the same matrix element form. Let me know if I'm saying something really stupid. So anyway I think on page 3, middle of first column, where he says "the energy does not commute with the area of the quantum horizon" what he means is "H and A don't commute."

Is this right? You are by far the expert in this context.

You're certainly right that what I mean is H and A. However, as we are discussing with fzero, it seems like Bianchi has used a strange expression for A (roughly just Lz) and Lz and Kz do commute (according to the lorentz algebra). Besides this issue I'm also confused about the state space, because if Eq. 6 holds for matrix elements between physical states, then it also follows that H and A commute. Now for some reason one seems to be using a vastly expanded set of sets where K and L are independent to analyze the physics, but I don't understand all these extra states and what they mean geometrically. I thought the physical Hilbert space was specified by the spin network states, that is the state of geometry at a fixed time. This also seems to be crucial for the detector analysis below where, under my naive reading, Bianchi uses both states of geometry and states of "energy" independently i.e. as a tensor product, which suggests K and L act on different spaces and hence commute.

 

[fzero]

I didn't even notice this at first, but it looks like bianchi is either doing the large j limit or made an important mistake?

Well I made a mistake. The corrections go like

$$\sum_f \frac{1}{j_f^p},$$

so presumably these converge and go to zero in the large $j_f$ limit. So maybe it's not so bad, but it would help to clarify the role of the limit.

 

[Physics Monkey]

Well I made a mistake. The corrections go like

$$\sum_f \frac{1}{j_f^p},$$

so presumably these converge and go to zero in the large $j_f$ limit. So maybe it's not so bad, but it would help to clarify the role of the limit.

I'm not sure what mistake you're referring to, but I think it could still be an issue. As I recall, in the very old days I heard that j=1/2 states played an important role for which obviously the different is substantial. Also in Fig. 1 Bianchi uses j=1 as an example, so I'm not convinced that Bianchi just meant for us to assume large j.

 

[fzero]

I'm not sure what mistake you're referring to, but I think it could still be an issue. As I recall, in the very old days I heard that j=1/2 states played an important role for which obviously the different is substantial. Also in Fig. 1 Bianchi uses j=1 as an example, so I'm not convinced that Bianchi just meant for us to assume large j.

Yes, I don't have an authoritative reference handy, but as I recall from earlier discussions, the large j limit (perhaps with j/N held fixed) is supposed to be a semiclassical limit. Bianchi does remark that his result does not include quantum corrections (below (3) and in the last paragraph of the conclusion).

 

[marcus]

... As I recall, in the very old days I heard that j=1/2 states played an important role for which obviously the different is substantial. Also in Fig. 1 Bianchi uses j=1 as an example, so I'm not convinced that Bianchi just meant for us to assume large j.

I agree (with Phy. Monk. post #23) As I recall j=1/2 predominates.
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?

 

[DrDu]

Likewise black hole horizon temperature is highly dependent on how far away the observer is hovering.

Why? Temperature is defined by two systems being in thermal equilibrium when they have the same temperature and by transitivity of the equilibrium. So if I put thermometer at large distance from the horizon and it is in thermal equilibrium with the black hole horizon, then any intermediate system which is in thermal equilibrium will be assigned the same temperature.

 

[Physics Monkey]

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 $\sqrt{j(j+1)}$ and $j$ are acceptable area operators (they differ by an operator ordering ambiguity that vanishes as $\hbar \rightarrow 0$ (which I guess here means something like $j \rightarrow \infty$ 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 $a_i$ and the spins are $\vec{J} = \frac{1}{2} a^+ \vec{\sigma} a$. The total spin of the representation can be read off from the total number $N = a_1^+ a_1 + a_2^+ a_2 = 2j$. On the other hand, you can work out $J^2$ for yourself to find $J^2 = \frac{1}{4}( N^2 + 2N)$ which one easily verifies gives $J^2 = j(j+1)$. Thus by $|\vec{L}|$ bianchi appears to mean $N/2$.

It is again interesting to see this kind of representation appearing in a useful way since it is quite important in condensed matter.

 

[Physics Monkey]

Why? Temperature is defined by two systems being in thermal equilibrium when they have the same temperature and by transitivity of the equilibrium. So if I put thermometer at large distance from the horizon and it is in thermal equilibrium with the black hole horizon, then any intermediate system which is in thermal equilibrium will be assigned the same temperature.

At the most basic level, this is gravitational red shift at work. There is a notion of local temperature given by $T_\ell(r) = T/\sqrt{-g_{tt}(r)}$ which in something like the scharzchild metric $g_{tt}(r) = -(1-r/r_S)$ gives a diverging temperature as the horizon is approached.

One physical meaning one can attach to this expression is the following. If an observer of four velocity $u$ (with $u^2 = -1$) measures the energy of a particle with four momentum $p$ then the energy measured is $E = u\cdot p$. An observer hovering about the black hole horizon has $u = (1/\sqrt{-g_{tt}}) \partial_t$. On the other hand, the existence of the Killing vector $\xi = \partial_t$ means that the quantity $\xi\cdot p$ is conserved independent of r provided the particle of momentum p is following a geodesic. This quantity $\xi \cdot p$ is the energy of the particle measured at infinity $E_\infty = \xi \cdot p$. Thus one sees that the energy measured by a hovering observer near the horizon is $E = E_{\infty} /\sqrt{-g_{tt}}$ 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. $\delta S = \int d^D x \sqrt{-g} \delta g_{\mu \nu} T^{\mu \nu}$.

 

[DrDu]

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

 

[Physics Monkey]

Well, I think it is a subtle notion, but it can be useful for studying entropy.

For example, consider Rindler space $ds^2 = - \rho^2 d \eta^2 + d\rho^2 + dx_\perp^2$ where $\eta$ is like a time and $\rho= 0$ is the horizon. Imagine a CFT living in this space. Then one gets an entropy density of roughly $s \sim T_\ell^d$ (with d the space dimension). Integrating this over all space gives $$\int d\rho dx^{d-1}_\perp \rho^{-d} \sim \frac{A_\perp}{\rho^{d-1}_c} .$$ $A_\perp$ is the cross sectional area and $\rho_c$ 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.

 

[Demystifier]

If there's no state counting, isn't this just a semiclassical calculation, like Hawking's?

I would put this way: If there's no state counting, then the entropy is merely a thermodynamic entropy (like Hawking's), not a statistical entropy.

 

[francesca]

I would put this way: If there's no state counting, then the entropy is merely a thermodynamic entropy (like Hawking's), not a statistical entropy.

That's right, but it would not say "merely": the microstate counting was already done in LQG, and Eugenio contributed to this with his nice paper "BH Entropy, Loop Gravity, and Polymer Physics", identying these microstates with the quantum geometry at the horizon. SO the "thermodynamical side" was the aspect somehow lacking... lacking in which sense? in the sense that Hawking calculation was done in the context of QFT in curved space, so spacetime was classical, while Eugenio's one is the thermodynamics of a quantum theory of gravity.

 

[marcus]

Francesca, these are very instructive replies to Demy and to Atyy's question.

If there's no state counting, isn't this just a semiclassical calculation, like Hawking's?

No, it's not semiclassical. In fact in the paper all the ingredients are derived from the full quantum theory. The relation energy-area was found by Frodden Gosh Perez using Einstein equations; here it is found using the boost generator given by Spinfoam Theory. The calculation in order to find Unruh temperature is done here again using the boost generator, it's completely new. And finally there is the remarkable demonstration that the Spinfoam amplitude implies the right distribution, that yields Hawking entropy.

I would put this way: If there's no state counting, then the entropy is merely a thermodynamic entropy (like Hawking's), not a statistical entropy.

That's right, but it would not say "merely": the microstate counting was already done in LQG, and Eugenio contributed to this with his nice paper "BH Entropy, Loop Gravity, and Polymer Physics", identying these microstates with the quantum geometry at the horizon. SO the "thermodynamical side" was the aspect somehow lacking... lacking in which sense? in the sense that Hawking calculation was done in the context of QFT in curved space, so spacetime was classical, while Eugenio's one is the thermodynamics of a quantum theory of gravity.

Indeed we are talking about Black Hole thermodynamics and the Bekenstein Hawking entropy S_BekHaw was thermodynamic. So likewise should Bianchi black hole entropy S_Bianchi be thermodynamic according to Clausius. This was one of the aims and achievements of the paper. And he evaluates it in a purely quantum way. Not in the context of QFT on a fixed curved space. So definitely not semiclassical . And as you said earlier, it is completely new!
The researchers might be able to learn something by comparing Bianchi's thermodynamic entropy with the statistical state-counting 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 elaboration--it does not explain enough what is being done--but I suspect that would be more appropriately done in a second paper, not to overextend this one.

 

[fzero]

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 $\mu^*$ and $\gamma$ that were found in the polymer paper.

 

[marcus]

... 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 $\mu^*$ and $\gamma$ that were found in the polymer paper.

Why so? I see no reason that combining states of the form (9) to make a mixed state would need to introduce a $\gamma$. Please explain.