Unigrav applied to problems of time and cosmo constant

[marcus]

This raises what I think is an important question. We have two kinds of global time both of which are applicable to cosmology.
We have some people (Rovelli, Connes, Smerlak and possibly others) who are interested in thermal time.

We have some other people (Smolin, Ellis, Chiou, Geiller and numerous previous authors) with an interest in unimodular time.

Are the two concepts in any way compatible?

And it is even more interesting, because cosmologists already have a global time which they use. The Friedmann model (standard in cosmo, the basis of the prevailing LambdaCDM model) has a natural foliation into spacelike hypersurfaces. You can think of this as based on observers who are at rest in the Hubble flow, or with respect to ancient matter---the source of the CMB---or at rest relative to the Background. They call it by various names: Friedmann time, Universe time... It is the natural global time for working cosmologists, and it functions in the conventional statement of basic mathematical relations like the Hubble Law.

So think about the hypersurface consisting of observers at rest relative Background all of whom measure the same age of the universe, or the same Background temperature (adjusting for different depths in gravitational field, or approximately anyway )

Is that global time going to be compatible with either Rovelli's thermal or Smolin's unimodular? By the way Rovelli says YES IT IS, as regards his thermal global time proposal.
He explained that in a popular essay on it posted around 2009. Under assumptions relevant to cosmology, thermal time agrees with Friedmann time--the global "universe time" already used in cosmo models.

Just to keep track of the basic links relevant to this new stage of discussion, here are a few:

http://arxiv.org/abs/1008.1759
Unimodular loop quantum gravity and the problems of time
Lee Smolin

http://arxiv.org/abs/1005.2985
Thermal time and the Tolman-Ehrenfest effect: temperature as the "speed of time"
Carlo Rovelli, Matteo Smerlak

A popular essay which can provide intuitive understanding for TTH:
http://arxiv.org/abs/0903.3832
"Forget Time!"
[In TTH, time is emergent, non-existent at funda level, so in that sense forgetable]

On brief acquaintance I would say this paper could be a handy reference for definitions of terms you see in the appendix at the end of the Rovelli-Smerlak paper.
http://arxiv.org/abs/1007.4094

 

[marcus]

To drive home the importance of this compatibility issue, here is something from page 8 of "Forget Time!", right before equation (19):

== quote http://arxiv.org/abs/0903.3832 ==

The “thermal time hypothesis” is the idea that what we call “time” is the thermal
time of the statistical state in which the world happens to be, when described in terms of the macroscopic parameters we have chosen.

Time is, that is to say, the expression of our ignorance of the full microstate¹⁰.

The thermal time hypothesis works surprisingly well in a number of cases. For example, if we start from radiation filled covariant cosmological model, with no preferred time variable and write a statistical state representing the cosmological background radiation, then the thermal time of this state turns out to be precisely the Friedmann time [21].

Furthermore, this hypothesis extends in a very natural way to the quantum context, and even more naturally to the quantum field theoretical context, where it leads also to a general abstract state-independent notion of time flow...

==endquote==

For fuller explanation check out the full article. For mathematical detail see the more recent http://arxiv.org/abs/1005.2985 and references therein.

One obvious point to draw is as follows: The Friedmann time used in cosmology is a time ('age of universe') assigned to space-like hypersurfaces. Rovelli and Alain Connes' thermal time can agree with that.

Now Smolin (and Henneaux and Teitelboim's) unimodular time is also a time that you can assign to space-like hypersurfaces. It would look like a serious problem if there were no bridge between these two kinds of time.

BTW both depend on (either the classical or quantum) state---in the classical setup they depend on the metric, that is on a solution---and in quantum setup they appear to depend on the quantum state (Smolin working on formulating this, Rovelli Connes already very explicit how the dependence goes--see last paragraph of 1005.2985.)

 

[MTd2]

http://www.physast.uga.edu/ag/uploads/UR.pdf

 

[marcus]

Thanks! David F's work on unimodular goes back to 1971. So he is more with the first batch (1989-1991) or even antecedent to it.

You might be interested in this:
http://arxiv.org/abs/gr-qc/9406019

 

[MTd2]

That guy found the Lagrangian for the theory, and that is an introduction to the theory, so I guess all confusion can be settled by reading that introduction. I have yet to read it. The introduction is very interesting, to give an intuitive idea:

"Unimodular relativity is an alternative theory of gravity considered by Einstein in 1919
without a Lagrangian and put into Lagrangian form by Anderson and Finkelstein. The space–time of unimodular relativity is a measure manifold, a manifold provided by nature with a fixed absolute physical measure field %(x) to be found by direct measurement, subject to no dynamical development. The sole structural variable is a conformal metric tensor f%&, subject to dynamical equations. The measure of a space–time region may be regarded as indirectly counting the modules of which it is composed, in the way that the volume of a lake indirectly counts its water molecules. Both space–time measure and liquid measure indicate a modular structure below the limit of resolution of the present instruments."

I have yet it to read, but this guy has also very crazy and beautiful ideas:

http://arxiv.org/abs/1007.1923

http://arxiv.org/PS_cache/hep-th/pdf/9604/9604187v1.pdf

There are others. I think this guy worked on acestors of causal networks and such.

You can find more of his papers here:

http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=FIND+EA+FINKELSTEIN,+D&FORMAT=www&SEQUENCE=ds(d )

http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=ea+Finkelstein,+David+R

http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=ea+Finkelstein,+D+R

http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=ea+Finkelstein,+David+Ritz

 

[atyy]

The HT reformulation of GR is background independent.

I don't understand why you say the HT reformulation of unimodular gravity is background independent. I understand the HT unimodular Lagrangian is generally covariant, but is general covariance alone equal to background independence?

As you know, I have nothing against not being background independent. But some others do

 

[marcus]

I don't understand why you say the HT reformulation of unimodular gravity is background independent...

As I understand it, Atyy, a theory's definition can either depend on specifying a background geometry (where other needed stuff can live) or not.
In whatever I've read about the HT formulation, I haven't noticed any background geometry being talked about. One doesn't seem to be needed, any more than with ordinary GR.

So I conclude HT is definable independent of any background geometry. Like GR.

Let me know if I missed spotting a geometric setup in the HT picture! Please point it out with a page reference. My eyes sometimes fail to catch stuff.

BTW I'm gradually realizing that the 1994 Connes Rovelli is great.

Thermal Time is also covered in sections 3.4 and 5.5 of Rovelli's book, but in a somewhat more condensed way. The Connes Rovelli paper includes some additional helpful discussion, it seems to me, and it is all in one place. You might be interested:
http://arxiv.org/abs/gr-qc/9406019

The C* algebra treatment of a general quantum theory is beautiful. The theory then lives more in the algebra of observables, and less in the configurations and wave functions. A "state" becomes a positive linear functional on the algebra.

 

[atyy]

Ok, I read the Alvarez treatment. So in the end we get the Einstein equations with a cosmological constant, so that is as before background independent. But that's provided covariant energy conservation is enforced separately - how does that occur in the HT formulation?

 

[marcus]

Ok, I read the Alvarez treatment...

Could you mean the Ellis et al? There is just one Ellis et al paper on this. Alvarez has written several and my impression is he varies the terminology. If it was really an Alvarez paper, please give me the link.

 

[atyy]

Could you mean the Ellis et al? There is just one Ellis et al paper on this. Alvarez has written a bunch and my impression is he uses different terminology. If it was really an Alvarez paper, please give me the link.

The one in your post #11 http://arxiv.org/abs/hep-th/0501146 .

I think Ellis says the same thing.

 

[marcus]

...
I think Ellis says the same thing.

Then let's look at the Ellis. It is current and the Alvarez is from 2005. I take it you are thinking that the Ellis et al approach might not be background independent?

 

[atyy]

Then let's look at the Ellis. It is current and the Alvarez is from 2005. I take it you are thinking that the Ellis et al approach might not be background independent?

No, I now think the classical equations resulting are background independent - since they are after all the standard EFE plus cosmological constant.

However, both Alvarez and Ellis get the EFE+cc only if covariant energy conservation is assumed separately. In the EH action, or a generally covariant action based only on the metric, we get covariant energy conservation automatically. So I'm wondering how covariant energy conservation will come about if we use the HT action.

 

[marcus]

... So I'm wondering how covariant energy conservation will come about if we use the HT action.

Good. In that case maybe we should look at one of the Smolin papers. They discuss HT action at some length.
(Correct me if I'm wrong but I don't recall very much about HT in either that 2005 Alvarez or the 2010 Ellis.)

Atyy, I reviewed the discussion of the HT approach on page 6 of 0904.4841 and see nothing that corresponds to the conservation condition that was required in the TFE approach. It comes at things from a different direction. So it looks like we don't have to worry about how the conservation condition is statisfied, or about background independence either.

I think I will knock off for the evening. (Currently I'm most interested in the relation of unimodular time and thermal time. I posted some about that. It would be fascinating if they were related somehow.)

 

[JustinLevy]

marcus,
I wish you would have taken the time to answer my questions so we can discuss these papers in depth. Science is not a theology where it is handed down from on high. Please stop just rummaging through papers, and quoting things. Also please stop commenting on who's well respected. If things are contradicting, stop and investigate. (For example, some people claim Unimodular gravity is classically equivalent to GR, while other papers claim Unimodular gravity makes any term of the stress energy tensor proportional to the metric not gravitate. These claims contradict.)


atyy,

I think you may be mixing up the lagrangians. There are two of them that could be considered unimodular gravity.

First:
S = \int_\mathcal{M} d^4x \ \epsilon_0 \left(- \frac{1}{8 \pi G} \bar{g}^{ab}R_{ab} + \mathcal{L}^{matter}(\bar{g}_{ab}, \psi) \right)
where
\epsilon_0 = \sqrt{-g}
is a fixed volume element. The only variations of g considered are those at preserve this.
The equations of motion are then the trace free einstein field equations:
R_{ab} - \frac{1}{4} \bar{g}_{ab} R = 4 \pi G (T_{ab} - \frac{1}{4} \bar{g}_{ab}T)

Then there is the Henneaux-Teitelboim (HT) lagrangian:
S = \frac{1}{2\kappa} \int_\mathcal{M} d^4x \left[ \sqrt{-g} (R-2\Lambda+ 2 \kappa \mathcal{L}_m) + 2\Lambda \partial_\mu \tau^\mu \right]
now all variations of g can be considered, and instead the equations of motion end up restricting the \sqrt{-g}

This was formulated specifically to allow all variations of g when calculating the equations of motion. Then the HT lagrangian does give energy conservation just as in GR. However the first lagrangian, which gives only the tracefree equations, does not. It truly has to be given as another assumption, or considered a "coincidence".

 

[atyy]

Then there is the HT lagrangian:
S = \frac{1}{2\kappa} \int_\mathcal{M} d^4x \left[ \sqrt{-g} (R-2\Lambda+ 2 \kappa \mathcal{L}_m) + 2\Lambda \partial_\mu \tau^\mu \right]
now all variations of g can be considered, and instead the equations of motion end up restricting the \sqrt{-g}

This was formulated specifically to allow all variations of g when calculating the equations of motion. Then the HT lagrangian does give energy conservation just as in GR. The first, which gives only the tracefree equations, do not. It truly has to be given as another assumption.

Yes, that seems to be the case - very clever formulation!

So the tau field is not observable?

 

[JustinLevy]

So the tau field is not observable?

No, and for that reason I consider it more of a math trick than a physical field.

After all, only the divergence of tau shows up in any of the equations of motion. So then the question is: Can we at least measure the divergence of tau? The answer is still no since the "equation of motion" it ends up in is, from varying \Lambda:
\sqrt{-g} = \partial_\mu \tau^\mu

Which if that was measurable, would mean we could essentially experimentally "disprove" a coordinate system. Which of course makes no sense.

I think one paper described the extra fields as just a lagrange multiplier to enforce a constraint. That is probably the best way to think about it.

 

[atyy]

No, and for that reason I consider it more of a math trick than a physical field.

After all, only the divergence of tau shows up in any of the equations of motion. So then the question is: Can we at least measure the divergence of tau? The answer is still no since the "equation of motion" it ends up in is:
\sqrt{-g} = \partial_\mu \tau^\mu

Which if that was measurable, would mean we could essentially experimentally "disprove" a coordinate system. Which of course makes no sense.

I think one paper described it as just a lagrange multiplier to enforce a constraint. That is probably the best way to think about it.

Hmmm, I wonder if we could write a generally covariant Lagrangian and enforce flatness with a lagrange multiplier, and maybe 10 dimensions too ...

And then apply loop quantization ... ;)

 

[JustinLevy]

Hmmm, I wonder if we could write a generally covariant Lagrangian and enforce flatness with a lagrange multiplier, and maybe 10 dimensions too ...

And then apply loop quantization ... ;)

Hehe... you're evil ;)
here's a crude attempt at the first part with flatness and 10 dimensions
S = \int_\mathcal{M} d^4x \sqrt{-g} \left[ (\frac{1}{2\kappa}R + \mathcal{L}_m) +<br /> (R^{abcd}R_{abcd} \phi + g^{ab}g_{ab}\psi - 10 \psi) \right]
With the "dynamical" fields \phi,\psi

If we were to jokingly take this "seriously", I could puff it up as: By adding the contributions of these psi and phi field terms to GR, we find that the only allowed values of phi and psi fields work to cancel any curvature caused by the matter fields. Futhermore, we find the only allowed spacetime dimension is 10, in agreement with string theory!


The fact that unigrav is essentially just a lagrange multiplier should be more of a warning to people. In other words, unigrav is taking beautiful GR, and limiting it, and claiming the very limits you put in (constant volume element) are instead a wonderful "result" (constant volume element, helping one to folliate spacetime). The other claim involving the stress-energy terms proportional to the metric not gravitating, I of course still dispute.

EDIT:
To counter some of my joking harshness there, it is of course a whole other issue what happens when we try to quantize such theories. Classical equivalence, when extra fields are involved, does not necessarily yield equivalence after quantizing. See Haelfix comments in post #47, for some notes on this regarding unigrav.

 

[atyy]

And yes, I do agree that at first sight, it seems quite ugly, although very clever too!

EDIT:
To counter some of my joking harshness there, it is of course a whole other issue what happens when we try to quantize such theories. Classical equivalence, when extra fields are involved, does not necessarily yield equivalence after quantizing. See Haelfix comments in post #47, for some notes on this regarding unigrav.

Soberly, yes, one can't rule out unimodular gravity, and it'd be interesting if it holds up quantum mechanically.

 

[marcus]

Consider the opposing viewpoint

I think I may already have mentioned a paper by Bianchi Rovelli called "Why all these prejudices against a constant" in this thread. In it they argue that the cosmological constant is not problematical---disposing of various puzzles people often bring up (vacuum energy discrepancy, naturalness, coincidence) as fake problems.

So you could say this is the antithesis of Smolin's position. Smolin considers that time and Lambda present real puzzles and he offers Unimodular to address them. As it happens, Rovelli has addressed the issue of time as well, and proposed a different way to resolve it. (The Thermal Time Hypothesis according to which time emerges statistically, somewhat analogously to thermodynamic quantities.) This seems to make Unimodular Gravity redundant in respect to both time and Lambda---weakening Smolin's two motivations for it. I suppose it's possible that Rovelli might assign low priority to Unimodular Gravity. (I don't have any idea what he actually thinks of it, never having seen any reference by him to it.)

http://arxiv.org/abs/1002.3966
Why all these prejudices against a constant?
Eugenio Bianchi, Carlo Rovelli
9 pages, 4 figures
(Submitted on 21 Feb 2010)
"The expansion of the observed universe appears to be accelerating. A simple explanation of this phenomenon is provided by the non-vanishing of the cosmological constant in the Einstein equations. Arguments are commonly presented to the effect that this simple explanation is not viable or not sufficient, and therefore we are facing the 'great mystery' of the 'nature of a dark energy'. We argue that these arguments are unconvincing, or ill-founded."

It basically gives reasons why the cosmo constant is not a problem. On the other hand, here's a landmark paper giving Rovelli's and Alain Connes' reasons for suggesting that time emerges statistically. http://arxiv.org/abs/gr-qc/9406019 According to the Thermal Time Hypothesis (TTH) time emerges in a way that is roughly analogous to thermodynamic quantities.

Earlier in this thread I posted links to a couple of other papers that discuss this view of time.

http://arxiv.org/abs/1005.2985
Thermal time and the Tolman-Ehrenfest effect: temperature as the "speed of time"
Carlo Rovelli, Matteo Smerlak

A popular essay which can provide intuitive understanding for TTH:
http://arxiv.org/abs/0903.3832
"Forget Time!"
[In TTH, time is emergent, non-existent at funda level, so in that sense forgetable]
...

 

[marcus]

The message of the "Why all these prejudices?" paper is an unusual one. So rarely heard in the scientific community that I had better quote the conclusions to make sure that I and others understand what's being said.

==from conclusions of http://arxiv.org/abs/1002.3966 ==

First, the cosmological constant term is a completely natural part of the Einstein equations. Einstein probably considered it well before thinking about cosmology. His “blunder” was not to add such a term to the equations: his blunder was to fail to see that the equations, with or without this term, predict expansion. The term was never seen as unreasonable, or ugly, or a blunder, by the general relativity research community. It received little attention only because the real value of λ is small and its effect was not observed until (as it appears) recently.

Second, there is no coincidence problem if we consider equiprobability properly, and do not postulate an unreasonably strong cosmological principle, already known to fail.

Third, we do not yet fully understand interacting quantum field theory, its renormalization and its interaction with gravity when spacetime is not Minkowski (that is, in our real universe). But these QFT difficulties have little bearing on the existence of a non vanishing cosmological constant in low-energy physics, because it is a mistake to identify the cosmological constant with the vacuum energy density.
==endquote==

These conclusions are not simply stated, they are argued in the paper. Quantitative discussion of why the fact that, for example, it should not be deemed an especially remarkable coincidence that we live in an era when ordinary matter density and putative "dark energy" density are comparable---within a factor of 20 of each other. Or why, for example, it is a mistake to identify cosmo constant with the QFT vacuum energy density. IMHO you get the complete point of view only if you read the supporting arguments.

 

[atyy]

http://math.ucr.edu/home/baez/vacuum.html

 

[marcus]

http://math.ucr.edu/home/baez/vacuum.html

Nice pedagogical accessory to the Bianchi Rovelli article.
Baez always good at explaining.

 

[atyy]

Hmmm, I wonder if we could write a generally covariant Lagrangian and enforce flatness with a lagrange multiplier, and maybe 10 dimensions too ...

And then apply loop quantization ... ;)

Hehe... you're evil ;)
here's a crude attempt at the first part with flatness and 10 dimensions
S = \int_\mathcal{M} d^4x \sqrt{-g} \left[ (\frac{1}{2\kappa}R + \mathcal{L}_m) +<br /> (R^{abcd}R_{abcd} \phi + g^{ab}g_{ab}\psi - 10 \psi) \right]
With the "dynamical" fields \phi,\psi

If we were to jokingly take this "seriously", I could puff it up as: By adding the contributions of these psi and phi field terms to GR, we find that the only allowed values of phi and psi fields work to cancel any curvature caused by the matter fields. Futhermore, we find the only allowed spacetime dimension is 10, in agreement with string theory!


The fact that unigrav is essentially just a lagrange multiplier should be more of a warning to people. In other words, unigrav is taking beautiful GR, and limiting it, and claiming the very limits you put in (constant volume element) are instead a wonderful "result" (constant volume element, helping one to folliate spacetime). The other claim involving the stress-energy terms proportional to the metric not gravitating, I of course still dispute.

EDIT:
To counter some of my joking harshness there, it is of course a whole other issue what happens when we try to quantize such theories. Classical equivalence, when extra fields are involved, does not necessarily yield equivalence after quantizing. See Haelfix comments in post #47, for some notes on this regarding unigrav.

Damn - it's been done! (I noticed via marcus's posting of the latest Thiemann paper.)

http://arxiv.org/abs/0805.0208
http://arxiv.org/abs/1001.3505

 

[marcus]

...(I noticed via marcus's posting of the latest Thiemann paper.)
http://arxiv.org/abs/0805.0208
http://arxiv.org/abs/1001.3505

Thiemann calls the papers by Ladha and Varadarajan "seminal" and appears to draw on them in a substantial way. The second L&V paper looks interesting. I will copy the abstract:
http://arxiv.org/abs/1001.3505
Polymer quantization of the free scalar field and its classical limit
Alok Laddha, Madhavan Varadarajan
58 pages
(Submitted on 20 Jan 2010)
"Building on prior work, a generally covariant reformulation of free scalar field theory on the flat Lorentzian cylinder is quantized using Loop Quantum Gravity (LQG) type 'polymer' representations. This quantization of the continuum classical theory yields a quantum theory which lives on a discrete spacetime lattice. We explicitly construct a state in the polymer Hilbert space which reproduces the standard Fock vacuum- two point functions for long wavelength modes of the scalar field. Our construction indicates that the continuum classical theory emerges under coarse graining. All our considerations are free of the 'triangulation' ambiguities which plague attempts to define quantum dynamics in LQG. Our work constitutes the first complete LQG type quantization of a generally covariant field theory together with a semi-classical analysis of the true degrees of freedom and thus provides a perfect infinite dimensional toy model to study open issues in LQG, particularly those pertaining to the definition of quantum dynamics."

As so often happens, L&V do not make clear what version of LQG dynamics they are talking about when they refer to "triangulation ambiguities which plague..." They seem to assume that whatever version they have in mind must be the official version. As far as I know current LQG dynamics (non-embedded spin foam, see for example http://arxiv.org/abs/1010.1939 ) has no triangulation ambs. What the devil would they be triangulating? There's no manifold.

But even if L&V are not fully in touch which the larger LQG picture, and are focused on some definite version of the dynamics (e.g. one of the formulations investigated by Thiemann, a canonical quantization of gr?) that's just context and may be irrelevant. What they are doing sounds quite interesting regardless of how they see it fitting into the program.

 

[atyy]

http://arxiv.org/abs/1010.2535
"On the other hand, diffeomorphism-invariance alone cannot be enough to yield features analogous to AdS/CFT. The point here is that any local theory (e.g., a single free scalar field) can be written in diffeomorphism-invariant form through a process known as parametrization. But it is clear that free (unparametrized) scalar fields are not in themselves holographic since time evolution mixes boundary observables at anyone time t with independent bulk observables (say, those space-like separated from the cut of the boundary defined by the time t). As a result, boundary observables at one time cannot
generally be written in terms of boundary observables at any other time."

"The canonical formalism for parametrized field theories on manifolds without boundary was studied in [5], [6]."

[5] K. Kuchar, “Geometry of hyperspace. I, ” J. Math. Phys. 17, 777 (1976) ; “Kinematics of tensor fields in hyperspace. II, ” J. Math. Phys. 17, 792 (1976) ; “Dynamics of tensor fields in hyperspace. III, ” J. Math. Phys. 17, 801 (1976) ; “Geometrodynamics with tensor sources. IV,” J. Math. Phys. 18, 1589 (1977)

[6] C. J. Isham and K. V. Kuchar, “Representations Of Space-Time Diffeomorphisms. 1. Canonical Parametrized Field Theories,” Annals Phys. 164, 288 (1985).