Penrose's argument that q.g. can't remove the Big Bang singularity

Haelfix

Negative entropy does not make sense. It is defined (for a microcanonical ensemble) as the logarithm of the number of microstates. You cannot, by definition, have a negative value.

Now, whether entropy is or is not defined in the quantum gravity regime is one question. However if you believe in unitary physics, you do run into a contradiction at some stage from the global point of view. So it is true that there is a problem of principle.

If you take a state in the far past pre bounce (where slices are nice, well behaved and semiclassical), and a state in the far future post bounce (likewise), and derive that the former has higher entropy than the latter, that does violate the second law (and unitarity) regardless of what tricks you want to pull in the middle. Amongst other catastrophes, it implies that you do not have reversible physics.

Now, as I said, these types of stat mech arguments are essentially a problem with all proposals really (eg an infinite finetuning in a boundary conditions or alternatively a discontinuity in the laws of physics).

Interestingly there might be a way out if you believe in observer complementarity in which case inflation might potentially resolve some of the finetuning (b/c crucially the all important volume factor enters (and dissappears) from the picture). See recent papers by Banks et al

marcus

These are interesting questions. In a covariant theory one does not a priori have time or time slices. But one can still have entropy defined.
Rovelli is currently working on this and has proposed a definition of entropy in the LQG context. http://arxiv.org/abs/1209.0065

Have a look at Appendix Section D on pages 7 and 8,
and again at section F, on page 4.

tom.stoer

There are several problems
- w/o QG you can't define and therefore you can't count microstates
- w/o thermodynamics you can't define Q, T and dS = δQ / T, therefore you can't identify a macrostate
- w/o a Hamiltonian H (or with H ~ 0) you cannot define E etc.
- you can't define the density operator ρ b/c you neither know the states nor the probabilities for the states

Demystifier

Marcus, even though the precise definition of entropy may be problematic in LQG, one should never expect entropy decrease. As Sir Arthur Stanley Eddington famously said:
"If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equations—then so much the worse for Maxwell's equations. If it is found to be contradicted by observation—well these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation."

Naty1

If Steve Carlip is on to something about THE SMALL SCALE STRUCTURE OF SPACETIME being two dimensional, [Post # 14] one has to wonder if hidden in the details of quantum spacetime foam are other restrictions on degrees of freedom.

Marcus: "He was charming and had great slides but the argument was handwaving and not convincing.

In the talk by Penrose I attended he did not address this at all, just waved his hands. So he actually did not make logical contact with LQG. But it was otherwise a delightful and stimulating talk about his new (Conformal Cyclic) Cosmology idea."

That sounds exactly like one of his talks online linked to in another thread. It is very worthwhile for a broad overview of some interesing issues in cosmology and I thought Penrose readily admitted there were a lot of unanswered questions remaining.

marcus

This summarizes the challenges very well! As you can see, all these are being directly confronted in 1209.0065, and in addition there is one more: TIME. A gen. covariant theory is timeless. So to give the 2nd law meaning he has to define "thermal time" (an idea of time as emergent from the statistical state) and restrict to Gibbs states.

That's a memorable and hilarious quote. Penrose used it in several of his CCC talks I heard in 2005 and 2006 (Cambridge, Princeton, Berkeley,...) including the one at Berkeley that I attended. Even though the quote is extremely well-known, this did not prevent Penrose from using it with great verve and relish at the start of his presentation each time he gave the talk.

tom.stoer

Thanks

Yes; that's closely related to H ~ 0

marcus

Indeed, you had already identified the time problem by implication, in your post. It did not really need to be mentioned again, by me.

I have to say (again) I find this set of problems (thermodynamics without time, statistical mechanics without time, or with time observer-dependent/emerging from the state) truly exciting.

There are many concepts of entropy, various definitions. As I am coming to see it, what seems most interesting and fundamental IMHO is vonNeumann entropy---that which is zero on pure quantum states and which is defined on trace-class operators rho, representing mixed quantum states. It's really neat, and it reminds me of the Shannon information-theory definition.

marcus

Bianchi's November paper uses the vonNeumann entropy and the concept of Gibbs state in its key step: equation (14).
It also has a reference to a paper by Don Marolf which caught my attention--I'm going to check it out now: hep-th/0310022 "Notes on space-time thermodynamics and the observer-dependence of entropy."

http://arxiv.org/abs/1211.0522
http://arxiv.org/abs/hep-th/0310022

marcus

At first sight Don Marolf's 2003 paper (that Eugenio pointed us to) is quite interesting:

http://arxiv.org/abs/hep-th/0310022
Notes on Spacetime Thermodynamics and the Observer-dependence of Entropy
Donald Marolf, Djordje Minic, Simon Ross
(Submitted on 2 Oct 2003)
Due to the Unruh effect, accelerated and inertial observers differ in their description of a given quantum state. The implications of this effect are explored for the entropy assigned by such observers to localized objects that may cross the associated Rindler horizon. It is shown that the assigned entropies differ radically in the limit where the number of internal states n becomes large. In particular, the entropy assigned by the accelerated observer is a bounded function of n. General arguments are given along with explicit calculations for free fields. The implications for discussions of the generalized second law and proposed entropy bounds are also discussed.
14 pages. Phys.Rev. D69 (2004) 064006

==quote Marolf Minic Ross==
We will show that the entropy associated with a simple localized matter system in flat and otherwise empty space is not an invariant quantity defined by the system alone, but rather depends on which observer we ask to measure it. An inertial observer will assign the usual, naïve entropy given by the logarithm of the number of internal states. However, an accelerated observer (who sees the object immersed in a bath of thermal radiation) will find the object to carry a different amount of entropy. Note that in the context we will consider both observers are able to describe the object with the same degree of precision; the issue is not that our object is partially hidden behind the Rindler horizon.
It is of course well known that the inertial and Rindler observers already ascribe a different entropy to the Minkowski vacuum, as this is a thermal state with divergent entropy [11] from the Rindler point of view. Considering both this fact and the background structures necessary for standard discussions of thermodynamics, Wald has argued for some time [12] (see also the last part of [13]) that entropy is an extremely subtle concept in general relativity – even for ordinary matter systems – and that we still lack the proper framework for a general discussion. Our results are in complete agreement with this philosophy and may be considered a next small step in pursuit of this goal. ...
==endquote==

Entropy is meaningless without the specification of an observer. Mathematically speaking, one cannot apply the 2nd law without specifying an observer.
In the context of the LQG bounce it is not clear to me that one can define an observer who passes thru the extreme density regime when gravity is violently repellent. In what sense can one have an observer in the expanding phase coming out of the bounce who is the SAME as the observer going in? It will be interesting to see how these issues are resolved.

I see this 2003 paper of Marolf Minic Ross as the STARTING POINT for Rovelli's September 2012 paper 1209.0065. Basically CR is taking the first steps toward defining a truly General Relativistic thermodynamics and stat mech. Something that is not trivial and has the potential to dig up a new way to conceptualize the microstates of geometry (on which matterfields live).

tom.stoer

fine - but that does not address gravitational entropy

marcus

Indeed it doesn't! The 2003 Marolf et al paper only addresses a highly simplified picture: flat space, a material object, two observers. More recent papers (e.g. Padmanabhan as I recall) have emphasized the observer-dependence of entropy repeatedly and in more general terms. I think you are as or more aware of this than I am, so I won't go link-hunting.

What I would like to see worked out soon is the LQC bounce thermodynamics in the terms introduced in 1209.0065.
That would be fascinating and I suspect that Bianchi is moving in that direction. He has been doing basic innovative research on the Loop BH thermodynamics and now one would want to see that carried over to LQC bounce thermodynamics. I have to deal with something offline now but will try to be back here soon. Interesting bunch of ideas!

Finbar

I think we are missing the essential point here. The idea of LQC with a bounce is that the universe at some point collapsed under it's own gravity bounced and formed a big bang. Is this right? On the other hand when matter collapses from some generic initial conditions we expect it will form a black hole and ultimately matter will be compressed to Planckian densities. Even if at this point QG kicks in and the singularities are removed it won't lead to a state anything close to the unique state needed form a big bang.


So one does not have to worry about how entropy is defined. After all entropy is just a useful concept to introduce when think about statistical ensembles of states. Instead the problem is a fine tuning one.


If you think about it though if you accept an infinite universe either temporally and/or spatially all states will be realised at some point. So perhaps the big bang was just a fluke in an otherwise orderless universe.

tom.stoer

I think we don't miss the point.

Looking at our expanding universe it seems to be obvious that it evolves from a low-entropy initial state to a high-entropy final state. But looking at a collapsing and bouncing universe it is unclear how the low-entropy initial state can be formed based on a collaps to a high-entropy final state which becomes the initial state of following expansion

marcus

Well F. it sounds like your word against the equations and your word against the computer.

The LQC bounce has been both reduced to equations and simulated numerically many times with lots of variations---with anisotropy with perturbations with and without inflation. The upshot is that the "big crunch" collapse of a spatially finite classical universe typically DOES lead big bang conditions and an expanding classical universe. The result is remarkably robust---the people who do the modeling do not find there is a need for fine-tuning.

This is not to say that Nature IS this way. What it says is that in this theoretical context with this version of quantum cosmology a big crunch tends to rebound in a big bang fairly robustly.

Black hole collapse has also been studied in the LQG context--that is very different. In a BH collapse, there is some MATTER that collapses, but the surrounding space does not. In a LQC cosmological collapse the whole of space collapses and rebounds. I'm sure you are well aware of the difference.

Something I would like to see would be a LQC numerical simulation of a bounce starting with a universe containing one or more black holes. I do not know of that being done, perhaps the Loop BH model is not as well developed as the cosmological model. Or it simply is not feasible numerically, too messy, for the time being.

tom.stoer

... and I think there is another issue: the LQC models always have finitely many gravity and matter d.o.f. so they are always in a pure state and have entropy zero

marcus

something suspiciously like a mixed state arises here:
http://arxiv.org/abs/1211.1354
An Extension of the Quantum Theory of Cosmological Perturbations to the Planck Era
Ivan Agullo, Abhay Ashtekar, William Nelson

earlier analysis used Liouville measure on space of solutions to calculate probabilities of specific outcomes of the bounce. (Ashtekar Sloan March 2011)


In part simply as a reminder to myself, I post a handy checklist of five research fronts where LQG may be developing or changing--short abbreviated names to make the list easy to remember and review. General Relativistic thermodynamics and related is a major one:

GR Thermo (incl. GR stat mech http://arxiv.org/abs/1209.0065 and horiz. entang. entrpy http://arxiv.org/abs/1211.0522)
TGFT (tensorial group field theory, see Carrozza's ILQGS talk and http://arxiv.org/abs/1207.6734)
HSF (holonomy spinfoam models, see Hellmann's ILQGS talk and http://arxiv.org/abs/1208.3388)
twistorLQG (see Speziale's 13 November ILQGS talk and http://arxiv.org/abs/1207.6348)
dust (ways to get a real Hamiltonian incl. field of obs./clocks, see Wise's ILQGS talk and http://arxiv.org/abs/1210.0019)