# Smolin:GR=EoS of SF, if he's right isn't that the ballgame? :-D

1. May 25, 2012

### marcus

Smolin makes the case that GR is the Equation of State of a given region's geometry considered as a thermodynamic system whose microscopic degrees of freedom are those of Spin Foam QG.

In short: GR=EoS of SF

The paper is here: http://arxiv.org/abs/1205.5529
General relativity as the equation of state of spin foam

He uses a family of accelerating observers to define the boundary of his region. Their worldlines describe a 3D surface S in his Figure 1. Time goes vertically in the Figure. Two dimensions are missing, necessarily, from the 2D picture.
You can see how the 4D region R is bounded on one side by S, on the other side by the Rindler horizons H which form behind any accelerated observer.

A rough analogy is the Gas Law PV=nkT viewed as the EoS of a bunch of little molecules whizzing and bouncing around in a box.
Here instead of molecules we have a bunch of little bits of geometric information (area, volume, angle) intersizzling and exchanging excitement inside this region R which Smolin gives the boundaries of. And now instead of the Gas Law, the coarse overall description is the GR equation.

Last edited: May 25, 2012
2. May 25, 2012

### marcus

This raises the prospect of a different approach to validating QG theories.

Suppose that in fact GR is the thermo EoS of some unspecified microgeometry degrees of freedom (like Jacobson 1995 says)

Then to validate a Quantum Geometry theory one does not take the "continuum limit" and get GR.
That is not the idea of molecular kinetics or stat mech or Gas Law thermo, and it should not be the paradigm.
What one has to show is that the spin foam micro degrees of freedom are the right discrete microscopic degrees of freedom that give rise to the correct Equation of State.
THAT THEY ARE THE RIGHT MOLECULES, so to speak.

There is a subtle difference, or maybe not so subtle. It seems to me that is the way Smolin is trying to go in this paper---an alternative to the naive straightforward "continuum limit" approach.
And it seems to me that this was the play that Ted Jacobson set up for. He did not specify the microscopic degrees of freedom but he broached the idea that whatever they were their equation of state coarsely describing their overall behavior would be the Einstein Field Equation of GR.

It is an intriguing approach to validating a QG theory, call it the Jacobson-Smolin gambit.

Last edited: May 25, 2012
3. May 25, 2012

### friend

How does this account for the fact that matter causes spacetime to seemingly curve?

4. May 25, 2012

### marcus

Well one way to find out is to notice that in many of the current papers (Smolin, Bianchi, FGP...) the authors keep referring to this 1995 paper by Jacobson. So something to do would be to check back and see what, if anything, it says that is relevant to your question:

http://arxiv.org/abs/gr-qc/9504004
Thermodynamics of Spacetime: The Einstein Equation of State
Ted Jacobson
(Submitted on 4 Apr 1995)
The Einstein equation is derived from the proportionality of entropy and horizon area together with the fundamental relation ∂Q=TdS connecting heat, entropy, and temperature. The key idea is to demand that this relation hold for all the local Rindler causal horizons through each spacetime point, with ∂Q and T interpreted as the energy flux and Unruh temperature seen by an accelerated observer just inside the horizon. This requires that gravitational lensing by matter energy distorts the causal structure of spacetime in just such a way that the Einstein equation holds. Viewed in this way, the Einstein equation is an equation of state. This perspective suggests that it may be no more appropriate to canonically quantize the Einstein equation than it would be to quantize the wave equation for sound in air.
8 pages, 1 figure Physical Review Letters 75:1260-1263 (1995)

There's more to say. I found some relevant stuff on pages 2-4 of the Jacobson paper. In J's scheme matter can apparently play a role here through its energy and entropy. He is describing things at the level of thermodynamics without having to specify microscopic detail.

Last edited: May 25, 2012
5. May 25, 2012

### marcus

==quote Jacobson==
Viewed in this way, the Einstein equation is an equation of state. This perspective suggests that it may be no more appropriate to canonically quantize the Einstein equation than it would be to quantize the wave equation for sound in air.
==endquote==

No one seems to have picked up on this yet. It tends to validate the direction that Loop gravity has moved, notably in the past 5 years but actually over more than a decade, away from an interest in canonical quantization of the Einstein equation.

To use an analogy, it would be silly to quantize the Gas Law PV=nkT. It is a classical EoS describing the collective behavior of a horde of quantum molecules. The molecules are quantum mechanical, for sure, and their individual behavior is quantum mechanical. One understands this is the underlying micro reality but one does not quantize the Equation of State!

What Smolin does is take the version of Spin Foam that has been developed in the past 5 years (specifically as formulated in arxiv 1102.3660 the Zakopane Lectures) and show that SF works as a theory of the underlying "horde of quantum molecules" from whose collective behavior the classical EoS can emerge.

Smolin's paper, especially if its findings are sustained by future research,would tend to justify the path taken by the many Loop researchers who for several years have neglected canonical quantization and the Hamiltonian constraint and have worked more vigorously on SF. Perhaps some had read Jacobson 1995 and taken the above suggestion to heart as a serious possibility, who knows?

Last edited: May 26, 2012
6. May 26, 2012

### fzero

I don't believe this is a fair assessment. The starting point of spin foams is the Holst action. It amounts to a quantization of the Einstein equation, just not the canonical one. Jacobsen's statement could apply equally well to any attempt to quantize the Einstein equation, since the canonical part is not crucial to the criticism.

Actually Smolin (and FGP) are not working entirely within spin foams. In order to show the Sciama property (40), they must use the projection of the Einstein field equations onto a onto a timelike normal vector, eq. (39). Smolin goes on to argue that this is equivalent to applying the Hamiltonian constraint. But the spin foam approach does not apply the Hamiltonian constraint! The only constraint is the linear simplicity constraint, which has nothing to do with the Hamiltonian constraint. In fact, Rovelli's Zakopane lectures lists making contact with the Hamiltonian constraint as open problem #14.

So it seems that Smolin does not actually have a proof, since he must already assume a component of the EFE. His argument is circular.

7. May 26, 2012

### marcus

I don't see anything unfair, so will reiterate with emphasis so as to make it clearer in case anyone is reading who hasn't followed Loop research.
===quote marcus===
==quote Jacobson==
Viewed in this way, the Einstein equation is an equation of state. This perspective suggests that it may be no more appropriate to canonically quantize the Einstein equation than it would be to quantize the wave equation for sound in air.
==endquote==

No one seems to have picked up on this yet. It tends to validate the direction that Loop gravity has moved, notably in the past 5 years but actually over more than a decade, away from an interest in canonical quantization of the Einstein equation.
==endquote==
Nothing unfair here. It's just history. After the trouble with Thiemann's Hamiltonian in the late 1990s the canonical (Hamiltonian) approach was essentially abandoned by almost everyone. The only continuing effort was by Thiemann and his students/co-workers. So the emphasis shifted to Spin Foam. Especially after 2005. (key work by Freidel et al and then later by Rovelli et al around that time)

The standard presentation of Loop Gravity does NOT start with the Holst Action. See the Zako lectures arxiv 1102.3660. The philosophy is to start with a quantum theory, define and develop it, and see if it recovers GR in the appropriate limit.
The Holst action is used as heuristic guide to guess the various forms of the Spin Foam vertex.
But you don't just quantize a classical theory (Holst or any other) and crank out Spin Foam QG.

Have a look at the first few pages of arxiv 1102.3660. This has been said explicitly and repeatedly, so that by now I think everyone who wants to has gotten the message. It's in line with keeping the options open: a decent Hamiltonian might be developed and the canonical approach might still succeed.

It's also perfectly in line with what Jacobson says. Use whatever classical action, like Holst, as a jumping off point and inspiration for defining Spin Foam amplitudes. The main thing then is to get a covariant 4D theory that provides 4D degrees of freedom so that GR has a chance to be the EoS.

Moreover, if you start with a canonical theory based on a 3D slice, with Hamiltonian constraint, you obviously are not very well set up to have that provide the "molecules in the box" for which the GR equation could be the thermodynamical EoS. The canonical formulation is not well-adapted for what Jacobson is talking about.

So there is a reason why he stresses especially that canonical quantization would not make sense, if what he conjectures is right.

The rest of my post may provide some additional clarification.
==quote==
To use an analogy, it would be silly to quantize the Gas Law PV=nkT. It is a classical EoS describing the collective behavior of a horde of quantum molecules. The molecules are quantum mechanical, for sure, and their individual behavior is quantum mechanical. One understands this is the underlying micro reality but one does not quantize the Equation of State!

What Smolin does is take the version of Spin Foam that has been developed in the past 5 years (specifically as formulated in arxiv 1102.3660 the Zakopane Lectures) and show that SF works as a theory of the underlying "horde of quantum molecules" from whose collective behavior the classical EoS can emerge.

Smolin's paper, especially if its findings are sustained by future research,would tend to justify the path taken by the many Loop researchers who for several years have neglected canonical quantization and the Hamiltonian constraint and have worked more vigorously on SF. Perhaps some had read Jacobson 1995 and taken the above suggestion to heart as a serious possibility, who knows?
==endquote==

Last edited: May 26, 2012
8. May 26, 2012

### fzero

The fact that Rovelli's lectures don't start with the Holst action is a matter of organization. Look at section V.A starting on page 24. There he explains how the spin foam variables are related to those of GR. He starts the paragraph after writing the Holst action (131) with the statement

"We are interested in the quantum states of this theory."

On page 27, while discussing polyhedra, he writes

"What is the relation with gravity? The central physical idea of general relativity is of course the identification of gravitational field and metric geometry."

(just above eq (143))

Rovelli is doing this because if you ever want to recover GR from a microscopic model, you had better have some idea of how the variables in your model connect with the variables of GR.

There are no states in the spin foam model that don't correspond to GR degrees of freedom, so it is some sort of quantization of GR. In fact, it is not just the degrees of freedom, but also the equations of motion of the BF theory that have a counterpart in the spin foam model. Rovelli explains below equation (136) how the requirement that the connection is flat, which follows from the EOM, appears in the prescription for the spin foam amplitude.

Jacobson is saying more than this. His line of reasoning is the same as what inspired his entropic gravity ideas. When he says that the EFE are an equation of state, he is saying that the only degrees of freedom are those of the quantum matter. No gravitational degrees of freedom are necessary for his argument and neither is any microscopic description of how the matter DOF interact. The EFE emerge from the macroscopic interactions of systems in thermal equilibrium.

As I've explained, Smolin is not using the usual spin foam theory, since he needs to posit the Hamiltonian constraint in his argument. In fact, he also has to assume that matter is consistently coupled to the spin foam model. These are serious gaps in his argument.

A further point of concern is that Jacobson's argument, as outlined above, does not require any microscopic theory of gravitational degrees of freedom. Where the microscopic degrees of freedom will matter is when we're away from local equilibrium.

To reiterate this point that Jacobson's argument hinges on local equilibrium rather than any microscopic details, you should note that in Smolin's argument, any details of how matter dof couple to the spin foam are completely irrelevant. The only thing that matters is that the stress tensor appears the appropriate way in the Hamiltonian constraint. But the Hamiltonian constraint already comes from the EFE (ignoring the fact that so far spin foam dynamics have been defined without it), so the proof is circular.

Last edited: May 27, 2012
9. May 26, 2012

### marcus

I don't see any place where he uses the LQG Hamiltonian constraint. He uses the time evolution Hamiltonian associated with the surface S, the locus of a family of observers. It's clear that this Hamiltonian is not zero on physical states (the way the LQG Hamiltonian is) and it's clear it has a nonzero expectation value. That type of Hamiltonian is all over the place in his paper.

But I have looked in vain for any appearance of the LQG Hamiltonian constraint :-(
so I think you must be mistaken.

Unless of course you can point out a spot where it explicitly appears...

10. May 26, 2012

### fzero

Read the discussion from section IV.B starting on page 7. Especially below (42) where he writes:

"But (Gab − 8πGTab )χa N b is proportional to a linear combination of the Hamiltonian and diffeomorphism constraints on Σ+."

11. May 26, 2012

### marcus

Mmmm, well that might be a place where the argument needs to be fixed up, because it isn't clear what the Hamiltonian is that he's referring to. One of Thiemann's proposed stable of Hamiltonians? If it is indeed such then you'd expect a reference to some paper.

There may also be some other (cleaner?) way to make that step in the argument.

Meanwhile, in case anyone has not seen Atyy's pointer to it, Ted Jacobson gave a great talk which is available as online video:
http://online.kitp.ucsb.edu/online/bitbranes_c12/jacobson/
and the first 18 minutes review just what we have been talking about. GR equation arising as EoS of some micro degrees of freedom.
Which Jacobson does not specify but which Smolin is arguing could well be those of Spin Foam QG.

He does not claim to have a PROOF of that yet, but he is making a plausible case for it being likely.

12. May 27, 2012

### fzero

It's fairly clear that Smolin is mixing concepts from canonical LQG and spin foams wherever he sees fit. For example, the comments surrounding eq (16) are straight from the canonical rulebook.

It really pays to look at the derivation in the FGP paper http://arxiv.org/abs/1110.4055. The relevant result is derived on page 3, eqs (15)-(19). They don't include a diagram, so it's probably helpful to use fig 1 or 2 from Smolin and translate the notation to keep things clear. Where the linearized EFE is introduced is in the form of the Raychaudhuri equation (17). If this were the straightforward Raychaudhuri identity, then the Ricci tensor $R_{ab}$ would have appeared, but the EFE has been used to write in terms of $T_{ab}$ instead.

The culmination is in the result (19) which relates a change in energy carried by matter degrees of freedom to a change in geometry. Either we need to use the EFE for that, or we must have some fundamental perspective of how matter interacts with geometry. I don't see any way around this. The spin foam approach at present doesn't offer either unfortunately, so we must conclude that Smolin's attempt at a proof fails for this reason.

Jacobson is actually very clear about what degrees of freedom the EFE is the equation of state of. From page 4 of http://arxiv.org/abs/gr-qc/9504004, above eq (1):

"We assume that all the heat flow across the horizon is (boost) energy carried by matter."

So the EFE is the EoS for the matter degrees of freedom.

I have to admit, I haven't listened to the KITP talk, but I did look through the Gravity Prize essay and didn't see any significant changes to the original picture I've been summarizing.

Now, as I said before, as long as the local equilibrium condition holds, the result is completely independent of the microscopic theory describing the matter and also completely independent of the microscopic gravitational degrees of freedom. We only require that the Rindler horizon satisfy the Bekenstein formula.

As I also mentioned earlier, entropic gravity is related to the present Jacobson argument. Namely, if only the matter dof are relevant to the EFE, then maybe we don't need a microscopic description of gravity at all. But I think that Jacobson correctly points out the flaw with that reasoning: the argument was made in the (near) equilibrium situation. It is clear that something more complicated is going on far away from equilibrium and that will probably require microscopic details.

13. May 27, 2012

### Chronos

Our inability to 'renormalize' gravity should be sufficient to suspect we are missing a key piece of the puzzle. My guess is GR is merely a better low energy approximation than Newtonian gravity.

14. May 27, 2012

### marcus

Another paper came out today which is connected with this small nexus of Loop Gravity papers. This time the author is Thanu Padmanabhan:

http://arxiv.org/abs/1205.5683
Equipartition energy, Noether energy and boundary term in gravitational action
(Submitted on 25 May 2012)
Padmanabhan indicates in his conclusions that his results are relevant to four recent Loop Gravity papers (references [10] and [11] by Frodden Ghosh Perez, by Bianchi, by Smolin, and by Bianchi Wieland:
==quote T.P. conclusions and references==
One motivation for writing this note stems from the recent interest in EN = TS in a few papers [10] which do not mention the connection between EN and the Noether charge, viz., that they are the same and EN is not a physical entity unrelated to previously known expressions! The relationship between EN and the boundary term of the gravitational action (which is essentially the relationship between the Noether charge and the boundary term of the action, a relationship that is probably of deeper significance) also seems to have gone unnoticed earlier. While this note was in the final stages of preparation, two papers appeared in the arXiv [11] which related EN to spinfoam based models and their boundary action, etc. However, as pointed out above, the relationship is actually very simple. It holds for the standard general relativistic action and its boundary term and is physically transparent once the connection between the Noether charge and EN is recognized.
...
...
[10] See for eg., E. Frodden, A. Ghosh, A. Perez, [arXiv:1110.4055]; E. Bianchi, [arXiv:1204.5122].
[11] L. Smolin, arXiv:1205.5529; E. Bianchi, W. Wieland, [arXiv:1205.5325].
==endquote==

15. May 28, 2012

### Paulibus

Smolin’s recent reconciliation of gravity with quantum theory seems a bit controversial. Judging from fzero’s posts it seems possible that some circular mathematical sophistry could be involved. Pardon me for barging in here.

The equation of state for a perfect gas, PV = RT, reconciles the behaviour of macroscopic, confined quantities of gas with the unobservable microscopic shenanigans ("intersizzling and exchanging excitement", as Marcus aptly says) of gas particles. This reconciliation is done within the context of a self-consistent (but limited) quantitative description of most of our contingent circumstances, called Classical Physics. It uses the concepts Pressure, Volume and Temperature from macroscopic Newtonian mechanics, school geometry and thermodynamics, all of which are also well understood microscopically in this same context. For example pressure is understood microscopically in terms of concepts like particle momentum conservation and mass. Pressure also features macroscopically in continuous fluid mechanics. The gas equation of state with its P,V,T concepts seems just an emergent and very convenient way of describing macroscopic behaviour for practical purposes, while ignoring detailed microscopic happenings.

Gravity in measurable macroscopic circumstances is accurately described as continuum geometry shaped by mass/energy, as Einstein’s field equations dictate. But Smolin doesn’t explain how the shaping (if any) by gravity of microscopic Loop Quantum geometry; "a bunch of little bits of geometric information (area, volume, angle)" as Marcus explains, can be described (curvature? changing of scale? closure failure when stepping around circuits? statistically described?) Maybe some kind soul can present a simple description of any microscopic changes (in geometry) to be expected in Loop Quantum Gravity from the proximity of mass/energy.

Or is an emergent equation of state sufficient to be expected from Smolin's kind of analysis?

Last edited: May 28, 2012
16. May 29, 2012

### Physics Monkey

It seems clear to me that, as fzero pointed out, Smolin is making a lot of additional assumptions in his argument. Of course, this is not to say that it doesn't teach us anything.

One idea, which I have repeatedly mentioned here, is that we could actually put some of Smolin's assumption on a better footing by considering asymptotically AdS spinfoams (whatever that means). We all know that AdS has a true hamiltonian because of its conformal boundary and hence has conventional time evolution. So let's start with some kind classical conformal boundary with a true Hamiltonian and "weld" onto this space a spinfoam in such a way that the time evolution is maintained. This should be an interesting hybrid of ads/cft and spin foams where Smolin's arguments should be better justified and maybe we could actually study the quantum geometry of asymptotically ads spaces.

17. May 29, 2012

### atyy

I wonder whether Donnelly is aiming to do something like that in http://arxiv.org/abs/1109.0036. His introduction says "We note also that the Hilbert space of edge states in SU(2) lattice gauge theory is closely related to the Hilbert space of the SU(2) Chern-Simons theory whose states are counted in the loop quantum gravity derivation of black hole entropy"

18. May 31, 2012

### Physics Monkey

Haha, I wish I knew.

19. Jun 1, 2012

### marcus

I was hoping to hear more along the lines of the PhysicsMonkey post that Atyy just quoted. Smolin's paper is frankly heuristic, it argues PLAUSIBILITY and breaks some new ground. Instead of expecting to canonically quantize GR and then recover GR in continuum limit, one uses the Holst action to GUESS the spinfoam degrees of freedom implicit in evolving geometry (the basic "molecules" of geometry. And sees if one recovers GR as their equation of state.
PhysicsMonkey's reaction was constructive: What can we learn from this? How might we make the argument stronger?
What will interest me will be to see if there is a followup to Smolin's paper along just those lines.
Since were on a new page, I'll recap and give the abstracts of the main papers being discussed.
A rough analogy is the Gas Law PV=nkT viewed as the EoS of a bunch of molecules whizzing and bouncing around in a box. Except that here instead of molecules we have a bunch of bits of geometric information (area, volume, angle) intersizzling and exchanging excitement inside this region R which Smolin gives the boundaries of. And now instead of the Gas Law, the coarse overall description is the GR equation.
This raises the prospect of a different approach to validating QG theories. Suppose that in fact GR is the thermo EoS of some unspecified microgeometry degrees of freedom (as per Jacobson 1995.)
Then to validate a Quantum Geometry theory one does not take the "continuum limit" and get GR.
What one has to show is that the spin foam micro degrees of freedom are the right discrete microscopic degrees of freedom that give rise to the correct Equation of State.
THAT THEY ARE THE RIGHT MOLECULES, so to speak.
Here are some relevant talks and papers:
http://arxiv.org/abs/1205.5529
General relativity as the equation of state of spin foam
Lee Smolin

http://arxiv.org/abs/gr-qc/9504004
Thermodynamics of Spacetime: The Einstein Equation of State
Ted Jacobson
(Submitted on 4 Apr 1995)
The Einstein equation is derived from the proportionality of entropy and horizon area together with the fundamental relation ∂Q=TdS connecting heat, entropy, and temperature. The key idea is to demand that this relation hold for all the local Rindler causal horizons through each spacetime point, with ∂Q and T interpreted as the energy flux and Unruh temperature seen by an accelerated observer just inside the horizon... This perspective suggests that it may be no more appropriate to canonically quantize the Einstein equation than it would be to quantize the wave equation for sound in air.
8 pages, 1 figure Physical Review Letters 75:1260-1263 (1995)

http://online.kitp.ucsb.edu/online/bitbranes_c12/jacobson/ [video and slides pdf]
Horizon Entropy, Higher Curvature, and Spacetime Equations of State
Ted Jacobson (Univ. Maryland)
24 May 2012

http://pirsa.org/12050053/ [video and slides pdf]
Black Hole Entropy from Loop Quantum Gravity
Eugenio Bianchi (PI Colloquium talk)
30 May 2012

Last edited: Jun 1, 2012