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

[bcrowell]

I came across this argument in the book The Nature of Space and Time, which is based on a series of lectures given by Hawking and Penrose. Although it relates to Penrose's Weyl curvature hypothesis (WCH), it does not depend on it, and that, to me, makes it a lot more interesting, since I wouldn't bet a six-pack on the validity of the WCH.

As a preliminary, Penrose observes that (in my possibly inaccurate paraphrase):

(1) The Big Bang was not a generic state. A generic Big Bang state would have had a large Weyl curvature, but the universe we see looks nothing like the one that would have resulted from such an initial state. Our Big Bang appears to have had a small or even vanishing Weyl curvature.

(2) The evolution of our universe has led to a state with nonvanishing Weyl curvature. (At black hole singularities, we even have diverging Weyl curvature.)

At the end of his first lecture, someone in the audience asks whether he thinks quantum gravity removes singularities. He says:

I don't think it can be quite like that. If it were like that, the big bang would have resulted from a previously collapsing phase. We must ask how that previous phase could have had such a low entropy. This picture would sacrifice the best chance we have of explaining the second law. Moreover, the singularities of the collapsing and expanding universes would have to be somehow joined together, but they seem to have very different geometries. A true theory of quantum gravity should replace our present concept of spacetime at a singularity. It should give a clear-cut way of talking about what we call a singularity in classical theory. It shouldn't be simply a nonsingular spacetime, but something drastically different.

What do folks here think of this? It seems pretty compelling to me, and yet the practitioners of loop quantum cosmology seem to be very convinced at this point that they're on the right track with models in which the big bang singularity is removed.

Presumably he has his cosmic cyclic cosmology (CCC) model in mind here (this was in 1996). Although CCC no longer looks viable, that doesn't resolve the issue he raises, which seems pretty model-independent.

The possibility that occurs to me is that the big bang singularity is removed by quantum effects, the entropy of the universe was minimized at the big bang, and there is time-reversal symmetry, so that the thermodynamic arrow of time was reversed in the universe before the big bang. Thermodynamically, the big bang would then look like an extremely unlikely thermal fluctuation, but presumably whoever set the boundary conditions of the universe got to choose to make it that way.

 

[Finbar]

You seem to suggest the existence of a creator?

Thermodynamics isn't time symmetric so you can't have the universe as a thermal state the decreases in entropy before the big bang started. The ccc models I guess are some kind of loop hole in the 2nd law like mapping states of high entropy to states of low entropy. I'm not sure how much sense that makes though.

 

[bcrowell]

Thermodynamics isn't time symmetric so you can't have the universe as a thermal state the decreases in entropy before the big bang started.

I think this depends on where you think the thermodynamic arrow of time comes from. If you think it comes from the fact that the big bang was a low-entropy state, then in a crunch-bang scenario, I think it's perfectly natural to imagine that the thermodynamic arrow of time flipped at the big bang.

But if that isn't the option one picks, then what counterargument is there to Penrose's?

 

[Finbar]

The second law of thermodynamics has an explanation from statistical physics. I can't understand how you could explain this flip? To create some state before the big bang that created the special low entropy state at the big bang would require some fine tuned pre-big bang state. It can't begin from a generic 'crunch'.

I don't think there is a couter argument to Penrose's argument. Something drastic has to happen to space-time at the big bang.

 

[bcrowell]

The second law of thermodynamics has an explanation from statistical physics.

Well, not really. It has an explanation from (1) statistical physics plus (2) the assumption that the universe used to be in a low-entropy state. You can't do it without ingredient #2.

I can't understand how you could explain this flip? To create some state before the big bang that created the special low entropy state at the big bang would require some fine tuned pre-big bang state. It can't begin from a generic 'crunch'.

Fine-tuning is required no matter what. We observe that the big bang had low entropy compared to a maximum-entropy big bang. This is a ridiculous amount of fine-tuning, and it's simply an observed fact.

I don't think there is a couter argument to Penrose's argument. Something drastic has to happen to space-time at the big bang.

...in which case loop quantum cosmology is trivially wrong and not worth pursuing? Seems unlikely that its practitioners would never have considered this issue.

 

[friend]

Well, not really. It has an explanation from (1) statistical physics plus (2) the assumption that the universe used to be in a low-entropy state. You can't do it without ingredient #2.

How can you tell the difference between the early universe being in a low entropy state or being in complete thermal equilibrium? I mean, if everything then was the same everywhere, isn't that the definition of thermal equilibrium?

Maybe complete thermal equilibrium of the entire universe is acutally equivalent to everything being in one state that can degenerate, and that's how you can go from one cycle of the universe to the next.

 

[bcrowell]

How can you tell the difference between the early universe being in a low entropy state or being in complete thermal equilibrium? I mean, if everything then was the same everywhere, isn't that the definition of thermal equilibrium?

We have a FAQ about this: https://www.physicsforums.com/showthread.php?t=509650

 

[Finbar]

Well, not really. It has an explanation from (1) statistical physics plus (2) the assumption that the universe used to be in a low-entropy state. You can't do it without ingredient #2.


Fine-tuning is required no matter what. We observe that the big bang had low entropy compared to a maximum-entropy big bang. This is a ridiculous amount of fine-tuning, and it's simply an observed fact.


...in which case loop quantum cosmology is trivially wrong and not worth pursuing? Seems unlikely that its practitioners would never have considered this issue.

I agree. But reversing the 2nd law and requiring fine tuning seems worse than simply the fine tuning.

Nothing is trivial here. LQC works with much symmetry, it's just a toy model.Toy models can be useful but they're not reality.

 

[bcrowell]

But reversing the 2nd law and requiring fine tuning seems worse than simply the fine tuning.

But then that doesn't address the point of my original post, which is that this seems to invalidate loop quantum cosmology.

 

[bcrowell]

Possibly relevant:

http://arxiv.org/abs/arXiv:0812.3544
http://physics.stackexchange.com/questions/5156/question-regarding-the-validity-of-the-big-bounce

 

[Haelfix]

No one has solved the reason for the low entropy initial conditions of cosmology. The problem exists for almost every single proposal. Loop or other.

Taken at face value, it rules out almost all of cosmology.

 

[TrickyDicky]

No one has solved the reason for the low entropy initial conditions of cosmology. The problem exists for almost every single proposal. Loop or other.

Taken at face value, it rules out almost all of cosmology.

Good point.
Don't want to sound sarcastic, but if GR had no problem not following strictly the previously "sacred" first law of thermodynamics, what prevents it from not strictly following the second too?
I always considered both laws of thermodynamics in the same pack, but that seemed to be just me, last time I argued this here I was told that they are independent of each other and the second one was more important than the first if one was to choose which one should be disobeyed by a theory like GR. I can't say I'm totally convinced of that, though.

 

[marcus]

No one has solved the reason for the low entropy initial conditions of cosmology. The problem exists for almost every single proposal. Loop or other.

Taken at face value, it rules out almost all of cosmology.

Good point.
Don't want to sound sarcastic, but if GR had no problem not following strictly the previously "sacred" first law of thermodynamics, what prevents it from not strictly following the second too?
I always considered both laws of thermodynamics in the same pack, but that seemed to be just me, last time I argued this here I was told that they are independent of each other and the second one was more important than the first if one was to choose which one should be disobeyed by a theory like GR. I can't say I'm totally convinced of that, though.

Compliments, both, on several good points. Penrose argues against LQC bounce, but the essence of the bounce is that gravity becomes repellent due to quantum corrections at near-Planck density--that's why there is a bounce.

If gravity becomes repellent, what happens to BH entropy? If the collapsing universe, prior to bounce, consists mainly of black holes, and its entropy is predominantly BH entropy, then how does one define the global entropy as gravity becomes increasingly repellent going into the bounce?

There seem to be problems with the definition of entropy underlying the 2nd law, when one tries to apply it in this context.

 

[Naty1]

..does quantum gravity remove singularities..

We are, alas, unlikely to resolve the question here.
Well, I know I won't be!
Marcus:

[Penrose argues against LQC bounce, but the essence of the bounce is that gravity becomes repellent due to quantum corrections at near-Planck density--that's why there is a bounce./QUOTE]

I agree that's conventional wisdom, but when a 'correction' starts a universe, color me 'suspicious'.

Presumably he has his cosmic cyclic cosmology (CCC) model in mind here

yes...there is a lecture online by Penrose, which I can't find that was referenced in another thread in these forums, in which Penrose describes the transition from a high entropy end to a low entropy start of another universe...lots of his neat transparancies with overlays...Here are some related references I saved from other discussions in these forums...no, they don't give a definitive answer.

from Roger Penrose THE ROAD TO REALITY...PG 766

Let us now think of a universe evolving so that an initially uniform distribution of material [with some density fluctuations] gradually clumps gravitionally, so that eventually parts of it collapse into black holes. The initial uniformity corresponds to a mainly Ricci-curvature [matter] distribution, but as more and more material collects gravitationally, we get increasing amounts of Weyl curvature...The Weyl curvature finally diverges to infinity as the black-hole singularities are reached. If we think of the material as having been originally spewed out from the Big Bang in an almost completely uniform way, then we start with a Weyle curvature that is...[essentially] zero. Indeed, a feature of the FLRW models is that the Weyl curvature vanishes completely. ...For a universe to start out closely FLRW we we expect the Weyl curvature to be extremely small, as compared with the Ricci curvature, the latter actually diverging at the Big Bang. This picture strongly suggests what the geometrical difference is between the initial Big Bang singularity- of exceedingly low entropy- and the generic black hole singularities, of very high entropy.

BEFORE THE BIG BANG:
AN OUTRAGEOUS NEW PERSPECTIVE AND ITS IMPLICATIONS FOR
PARTICLE PHYSICS
Roger Penrose
Mathematical Institute, 24-29 St Giles’, Oxford OX1 3LB, U.K.

. It may be seen that, with time symmetrical
dynamical laws, the mere smallness of the
early universe does not provide a restriction on its degrees
of freedom. For we may contemplate a universe model in
the final stages of collapse. It must do something, in
accordance with its dynamical laws, and we expect it to
collapse to some sort of complicated space-time
singularity, a singularity encompassing as many degrees
of freedom as were already present in its earlier nonsingular
collapsing phase. Time-reversing this situation,
we see that an initial singular state could also contain as
many degrees of freedom as such a collapsing one. But in
our actual universe, almost all of those degrees of
freedom were somehow not activated.

from
http://arxiv.org/PS_cache/arxiv/pdf/1009/1009.1136v1.pdf

The BKL picture was originally developed as an attempt to understand the cosmology of the very early Universe near an initial spacelike singularity. Near such a singularity, light rays are typically very strongly focused by the gravitational field, leading to the collapse of light cones and the shrinking of particle horizons. This “asymptotic silence” [43] is the key ingredient in the ultralocal behavior of the equations of motion, from which the rest of the BKL results follow.
Small-scale quantum gravity has no such spacelike singularity, so if a similar mechanism
is at work, something else must account for the focusing of null geodesics. An obvious candidate is “spacetime foam,” small-scale quantum fluctuations of geometry. Seeing whether such an explanation can work is very difficult;...

and a 2009 perspective from Steve Carlip that may offer interesting possibilities:The Small Scale Structure of Spacetime

http://arxiv.org/PS_cache/arxiv/pdf/1009/1009.1136v1.pdfSeveral lines of evidence hint that quantum gravity at very small distances may be effectively two-dimensional.

Stephen Hawking and George Ellis prefaced their seminal book, The Large Scale
Structure of Space-Time...

At much smaller scales, on the other hand, the proper description is far less obvious….Indeed, it is not completely clear that “space” and “time” are even the appropriate categories for such a description….Over the past several years, evidence for another basic feature of small-scale spacetime has been accumulating: it is becoming increasingly plausible that spacetime near the Planck scale is effectively two-dimensional. No single piece of evidence for this behavior is in itself very convincing, and most of the results are fairly new and tentative. But we now have
hints from a number of independent calculations, based on different approaches to quantum gravity, that all point in the same direction...

all rather mysterious!

 

[tom.stoer]

The big difference is that Penrose uses entropy in his reasoning whereas LQC doesn't. Both approaches are incomplete: Penrose has no detailed model at all, LQG is a detailed model but with too many simplifications.

 

[julian]

"The nature of space and time" is an old book. I think he put similar views forward in "The Emperor's new mind". You know that Penrose has made a bit of a U-turn and now argues that in the thermal death of one universe (in which he presumes there are no non-zero rest mass particles) there is no way of building clocks or reference systems to provide a notion of time intervals or length intervals and so the universe is induistinguishable from zero volume big bang situation (this is the view he puts forward in his new book "Cycles of time"). I have been wondering how he reconciles these seemingly contradictory views. Glad bcrowell brought it up. Would like to understand better.

 

[marcus]

The big difference is that Penrose uses entropy in his reasoning whereas LQC doesn't...

I heard Penrose give this argument in March 2006 to an audience of math and physics people at the MSRI. He was charming and had great slides but the argument was handwaving and not convincing. You cannot use entropy in a rigorous math argument unless you can define it and he was not able to define the global entropy through the course of the LQC bounce. So he used vague suggestive language and did not claim certainty.

It is really interesting to consider how the entropy of a BH could be defined and could evolve when gravity becomes repellent! On the face of it, considered naively, the entropy should change sign:

Suppose we take the Bekenstein-Hawking effective description at face value: S = A/(4G_Newton) and the effective value of G_Newton goes temporarily negative. Then unless the black hole has dissipated by then it would seem to have negative entropy

This is not how one would argue in reality, just meant to be suggestive. In LHC gravity becomes repellent at extreme density. that is what causes the bounce. So all I can say is that this makes the definition of entropy itself an extremely interesting problem (in the context of LQC models).

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.

 

[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]

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

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.

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."

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]

This summarizes the challenges very well!

Thanks

and in addition there is one more: TIME.

Yes; that's closely related to H ~ 0

 

[marcus]

Yes; that's closely related to H ~ 0

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]

fine - but that does not address gravitational entropy

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 realized 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]

... 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...

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]

... and I think there is another issue: the LQC models ... always in a pure state and have entropy zero

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)

 

[Demystifier]

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

I have two comments on that.

First, all models for a bouncing Universe I have ever seen involve a rather SMALL number of the degrees of freedom. On the other hand the second "law" (which would be better called the second RULE, because there is always a small probability of its violation) is valid only for systems with LARGE number of the degrees of freedom. Therefore, such toy models with a small number of degrees of freedom cannot be directly applied to tackle the problem of the second law.

Second, if one studies a model of a bouncing universe with a LARGE number of degrees of freedom, one can find a bouncing solution by FINE TUNING the initial conditions at the bouncing point. Namely, the entropy can be easily chosen to be small at one particular time at which the universe has the smallest size. But for most choices of such initial conditions, the time evolution in both time directions will reveal that entropy will increase in BOTH time directions. In other words, the entropy at the bouncing point will have the minimal value, and the arrow of time "before" bouncing will have the opposite direction from the arrow "after" the bouncing.

For an explicit example of a numerical simulation (not really a bouncing universe, but a system with a minimal entropy at one particular time) see e.g. Fig. 4 in
http://arxiv.org/abs/1011.4173v5 [Found. Phys. 42, 1165-1185 (2012)]

 

[Demystifier]

... 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

That is irrelevant. Even if the number of dof's is infinite, the whole universe should be in the pure state, and therefore its von Neumann entropy is zero and does not change with time. But von Neumann entropy is not the only meaningful entropy in QM. The entropy one expects to increase with time necessarily involves some kind of COARSE GRAINING, corresponding to the inability to see all the details of a complex system. This means that some microscopically distinct states are viewed as macroscopically identical, so entropy will increase in the sense that the system will evolve towards macro-states that can be realized by a larger number of different micro-states.

 

[bcrowell]

For an explicit example of a numerical simulation (not really a bouncing universe, but a system with a minimal entropy at one particular time) see e.g. Fig. 4 in
http://arxiv.org/abs/1011.4173v5 [Found. Phys. 42, 1165-1185 (2012)]

Nice! But their discussion assumes an interaction, whereas we know that different regions of the universe are causally disconnected.

It seems like all we know fundamentally is that:

(1) If there is low entropy in some region, the most likely thing is that it's preceded and followed by higher entropy.
(2) Regions of spacetime that are causally connected should have arrows of time that agree.
(3) Our universe is nearly flat, so it's probably either spatially infinite or at least much larger than the observable universe.
(4) Our own past light cone appears to have had an arrow of time going back to at least the era of big bang nucleosynthesis. This requires extreme fine-tuning.
(5) Our own thermodynamic arrow of time currently points away from the big bang.

Based on these observations, it seems to me that the most general picture we can construct is a universe in which nearly all of spacetime is in thermal equilibrium, but there is one or more causally separated islands that are not at equilibrium. In each of these islands, there is some time of minimum entropy, which may or may not coincide with the big bang (or bounce). In our own island, this time seems to have been either at the big bang or before nucleosynthesis.

It probably doesn't make sense to explain our island, or any others that exist, as thermal fluctuations, because then it would be overwhelmingly more probable for us to be Boltzmann brains rather than real observers inhabiting a large, non-equilibrium region of spacetime. This means that extreme fine-tuning is required. We have no physical law or principle that explains this fine-tuning, so we can't say whether (1) there should be more than one island, (2) an island's minimum entropy should coincide with the big bang, or (3) our own island actually encompasses the whole universe. In principle we can test all three of these empirically, but 1 and 3 can only be tested by waiting for cosmological lengths of time and continuing to make observations. 2 can be tested in our own island by seeing whether cosmological models correctly explain the very early universe with the normal second law of thermodynamics.

I don't think a bounce really changes this picture very much. Penrose's argument seems to be based on the assumption that the second law is fundamental and universal, but that doesn't seem to me like a natural point of view.

 

[Finbar]

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.

The difference has nothing to do with the presence of some surrounding space that doesn't collapse. The difference is in the symmetries and reduction to a finite number of degrees of freedom in LQC.

 

[marcus]

...models for a bouncing Universe I have ever seen involve a rather SMALL number of the degrees of freedom. ...

Then please see for example the paper I just referred to in my last post (#34)---Agullo Ashtekar Nelson November 2012.
Infinite dimensional Hilbert space of states. Basically defines the new face of LQC.

Anyone at all interested in bounce cosmology (the bulk of that being by the LQC community) should probably memorize the arxiv number 1211.1354 and spend some time reading the paper. It's 50 pages. They have a followup/companion paper in prep.
"[2] I. Agullo, A. Ashtekar and W. Nelson, The pre-inflationary dynamics of loop quantum cosmology: Confronting quantum gravity with observations, (in preparation)"

 

[marcus]

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.
...

But it DOES lead, both with extensive equation modeling and with numerical simulation. To adequate start of big bang and sufficient inflation with high probability and no fine-tuning. See Ashtekar's paper on the probability of inflation within LQC context.

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.
...

My mention of the difference between BB and BH is just a footnote to my general rebuttal of what you said. I'm sure you are well aware of the difference. You seem knowledgeable--or IMHO have written knowledgeably about this stuff in the past---but now seem out of touch with current research.

The difference has nothing to do with the presence of some surrounding space that doesn't collapse. The difference is in the symmetries and reduction to a finite number of degrees of freedom in LQC.

I'm not sure what the "difference" is that you are saying has nothing to do with this or that, but you are not up-to-date regarding "finite number of degrees of freedom". See 1211.1354

 

[marcus]

That is irrelevant. Even if the number of dof's is infinite, the whole universe should be in the pure state, and therefore its von Neumann entropy is zero and does not change with time...

Demy, that is very interesting! You could be talking about the new LQC+Fock hybrid with infinite d.o.f. where the bounce is highly robust and does not require fine tuning. Why then do you say the U should be in a pure state!

I am used to seeing "squeezed" states in the LQC literature, it seems routine to employ mixed states in bounce cosmology analysis. What am I missing? Is there some PHILOSOPHICAL reason you have in mind for why the U should be in a pure state?

Correct me if I am wrong but I think the U can ONLY be in a mixed state simply because no observer can see all of it Cosmologists have this distance called "particle horizon" estimated at 46 billion ly which is the distance to the farthest matter which we could in principle have gotten a signal from. But the whole thing (assuming finite) is estimated to be at least several times larger.

You know the Bohr proverb about phyiscal science: it's not about what IS but instead what we can SAY about it. In that spirit, all we can say is a mixed state. And that therefore is the state.

So your statement that the U must be in a pure state is really interesting to me and I wish you would explain.

 

[marcus]

Loop is transforming and there's a short list research fronts to watch, where the change is happening. Thanks Ben C, Demy, Tom, Finbar for starting and/or contributing to this discussion which has underscored the importance of the hybrid LQC work by Agullo Ashtekar Nelson! I definitely have to add it to the watch list. Several of the following could turn out to be among the most important QG research papers of 2012:
hybrid LQC
An Extension of the Quantum Theory of Cosmological Perturbations to the Planck Era (1211.1354)
The pre-inflationary dynamics of loop quantum cosmology: Confronting quantum gravity with observations (in prep)
GR Thermo
General relativistic statistical mechanics (1209.0065)
Horizon entanglement entropy and universality of the graviton coupling (Bianchi's ILQGS talk and 1211.0522)
tensorialGFT (Carrozza's ILQGS talk and 1207.6734)
holonomySF (Hellmann's ILQGS talk and 1208.3388)
twistorLQG (Speziale's ILQGS talk and 1207.6348)
dust (Wise's ILQGS talk and 1210.0019)

 

[Finbar]

Correct me if I am wrong but I think the U can ONLY be in a mixed state simply because no observer can see all of it

Isn't it rather that the universe at the big bang can be viewed as being in a pure state since all the matter was in causal contact. Then once the universe evolves we can only consider "observable" universes which are subsystems of the whole system. Theses observable universes are described as mixed states obtained by tracing over the unobservable universe. So while it is correct to say that the current observable universe is a mixed state the universes at the big bang can be considered a pure state.

So maybe I'll do a u-turn and tentatively buy the "bounce" cosmology. We could think that the state of the universe at the bounce is a pure state. Then the universe evolves to the current day at which point each observer can only see a finite amount of the universe which will be described as a mixed state. Finally the universe then collapses at which point all the universe comes back together and a pure state is again recovered. Now the issue is why the final pure state would look anything like the initial pure state.

 

[tom.stoer]

Demystifier is right that a system in a pure state with finitely (or infinitly many) d.o.f. will remain in a pure state under (unitary) time-evolution and will therefore never have entropy > 0.

Pleae note that the underlined words are not well-defined or not known in general in the case of QG ;-)

 

[marcus]

...
So maybe I'll do a u-turn and tentatively buy the "bounce" cosmology. ... Now the issue is why the final pure state would look anything like the initial pure state.

There are certainly some extremely fascinating unresolved issues!
I can only speculate as a non-expert. My intuition says that a collapsing phase of the U filled with a gas of black holes would reach (assuming the LQC model where gravity becomes repellent at high density) a stage where the horizons of all BH rapidly shrink, releasing Hawking radiation.

So what goes into the bounce is a universe full of Planck-scale gamma photons and Planck-scale black holes, which can interconvert.

The picture is somewhat analogous to a pair-instability supernova where the photons have become so energetic they become indistinguishable from electron-positron pairs. But that is only a crude analogy. I am groping for a picture of what it could be like when radiation and geometry interconvert the one form of energy into the other and back again, very rapidly, at Planck scale.

Just a speculative picture FWIW.

If I were Ivan Agullo or Eugenio Bianchi or Bill Nelson (who has taken a postdoc at Nijmegen in the Netherlands and will be giving talk(s) at Stockholm this month, I gather) I think I would be working towards a model of the bounce of a classical universe which collapses into a Planck soup of that kind of stuff (maybe ) and bounces.

The one thing that Agullo Ashtekar Nelson say in 1211.1354 is that their followup paper is based on numerical simulations of the bounce (with perturbations) in a hybrid LQC-Fock picture. They stress the numerical simulations. that rings a bell with me. The results should be very very interesting even if one only tentatively semi-accepts the preliminary model.

 

[Demystifier]

I am used to seeing "squeezed" states in the LQC literature, it seems routine to employ mixed states in bounce cosmology analysis. What am I missing?

From that sentence it looks as if you are missing the fact that squeezed state is a pure state, not a mixed state.

Is there some PHILOSOPHICAL reason you have in mind for why the U should be in a pure state?

Strictly logically the Universe does not necessarily need to be in a pure state, but in the theory of decoherence it is usually assumed so. If you are not familiar with the basic ideas of decoherence, see e.g.
http://arxiv.org/abs/quant-ph/9803052
for a brief introduction.

 

[marcus]

From that sentence it looks as if you are missing the fact that squeezed state is a pure state, not a mixed state.


Strictly logically the Universe does not necessarily need to be in a pure state, but in the theory of decoherence it is usually assumed so. If you are not familiar with the basic ideas of decoherence, see e.g.
http://arxiv.org/abs/quant-ph/9803052
for a brief introduction.

Thanks for the correction! I'm also used to seeing peaked semiclassical states in LQG, are these also pure?
I'm glad that you find that it is not strictly logically necessary for the U to be in a pure state!
I will look at your link.

Ah, Claus Kiefer! He had an article just recently about decoherence in LQG, I'll get the link.
The quantum state of geometry decoheres through interaction with FERMIONS. The triad variable of LQG chooses an orientation, is forced by the presence of fermions to classicalize.
http://arxiv.org/abs/1210.0418
Interpretation of the triad orientations in loop quantum cosmology
Claus Kiefer, Christian Schell
(Submitted on 1 Oct 2012)
Loop quantum cosmology allows for arbitrary superpositions of the triad variable. We show here how these superpositions can become indistinguishable from a classical mixture by the interaction with fermions. We calculate the reduced density matrix for a locally rotationally symmetric Bianchi I model and show that the purity factor for the triads decreases by decoherence. In this way, the Universe assumes a definite orientation.
12 pages, 1 figure

I don't remember if you already commented on this paper (it came up in another thread.)
If you did I'd like very much to see your comment and would appreciate a link to your post about it. If you haven't yet I hope you will. It's interesting to see Kiefer focusing on one of the outstanding problems in LQG the orientation symmetry of the main variable (not present in theories based on the metric).

My take on it is that when Kiefer or others start with the U in a pure state and have it progressively decohere, this does not mean that in reality the U would necessarily have to start pure. The analysis just shows how it could start in a purER state and become LESS pure.
The analysis is a fortiori. It is just a convenient simplification to imagine that the system starts in a pure state, the important thing is progressive decoherence starting from whatever level of (im)purity or mixedness. I can imagine you might disagree.

 

[Finbar]

Thanks for the correction! I'm also used to seeing peaked semiclassical states in LQG, are these also pure?

Pure states are states that can represented as state vectors in the Hilbert space.

 

[marcus]

Pure states are states that can represented as state vectors in the Hilbert space.

I believe that's right, Finbar. And mixed states are probabilistic superpositions of pure states, are they not?
What I'm unsure about is the meaning of "squeezed" states. Do you have a good brief explanation, or a link for that?

If not, it probably is just a side issue, so no matter.

What interested me just now was Demy's mentioning decoherence and the work of Claus Kiefer, a longtime central figure in Quantum Gravity. Just last month Kiefer posted this paper on decoherence in LQG. I'd really appreciate your comments!

http://arxiv.org/abs/1210.0418
Interpretation of the triad orientations in loop quantum cosmology
Claus Kiefer, Christian Schell
(Submitted on 1 Oct 2012)
Loop quantum cosmology allows for arbitrary superpositions of the triad variable. We show here how these superpositions can become indistinguishable from a classical mixture by the interaction with fermions. We calculate the reduced density matrix for a locally rotationally symmetric Bianchi I model and show that the purity factor for the triads decreases by decoherence. In this way, the Universe assumes a definite orientation.
12 pages, 1 figure

It seems that purity and mixedness are not absolute properties but are on a range. Maybe all states should be thought of as a density matrix rho and the degree of purity would be the trace of the square of rho.
==quote page 7 Kiefer Schell==
A measure for the purity of the total state (15) is the trace of ρ_red², which is equal to one for a pure state and smaller than one for a mixed state; it is directly related to the linear entropy S_lin = 1 − ρ_red² [5]. One could also discuss the von Neumann entropy −k_Btr (ρ_red ln ρ_red), but for the present purpose it is sufficient to restrict to S_lin.
==endquote==

This seems like a mathematically natural way to go. How does it strike you, Finbar? Does this appeal to you, or is perhaps how you already think of quantum states? On a continuum of mixedness?

 

[Finbar]

I believe that's right, Finbar. And mixed states are probabilistic superpositions of pure states, are they not?
What I'm unsure about is the meaning of "squeezed" states. Do you have a good brief explanation, or a link for that?

A mixed state is not a superposition. Because any superposition of pure states is itself a pure state. A mixed state has an uncertainty beyond that of a quantum mechanical superposition. A mixed state requires a probability distribution over pure states. So one is formally doing quantum statistical physics once you deal with mixed states. One can also think of classical statistical states where we have a probability distribution over classical states.

Not sure what a squeezed state is. Something to do with the saturated uncertainty relation?