Unigrav applied to problems of time and cosmo constant

[marcus]

So please, let's discuss specifics.

given the action:
$$S = \frac{1}{2\kappa} \int_\mathcal{M} d^4x \left[ \sqrt{-g} (R-2\Lambda+ 2 \kappa \mathcal{L}_m) + 2\Lambda \partial_\mu \tau^\mu \right]$$
Here, Λ is not a parameter like in GR (ie. it is not specified once and for all, but is a scalar field).
...

For starters, that is not what Smolin calls the unimodular action. He gives S^uni, based on a constant determinant.

And that is so to speak chapter 1 of the story. Chapter 2 introduces the Henneaux Teitelboim action which he calls S^HT, then he does some derivation and ends up deriving the non-coupling condition. Page 7 eq.24. Later (after gauge fixing) he actually gets back to the unimodular action.

Smolin's April paper 0904.4841 is more self contained. It is a better introduction. Chiou-Geiller jump right in at chapter 2. They don't give S^uni. They start right off with the Henneaux Teitelboim action S^HT. It's easy to get confused by their paper if you don't read Smolin's along with it.

 

[atyy]

Is unimodular gravity background independent? (I hope the answer is no;)

 

[marcus]

Is unimodular gravity background independent? (I hope the answer is no;)

Well keep in mind that the story is told in two chapters. The basic idea unimodular is restricting to metrics with determinant = 1, or better said, equal to some constant density.
That means the theory is not even diffeormorphism invariant . You are restricted to diffeomorphisms with unit Jacobian---volume preserving maps.

So you don't work on a fixed background geometry, but it is pretty restricted. That is the original idea. uni+modular = equal to one+determinant.

There's a wonderful concise presentation of the whole business that Smolin prepared for the Abhayfest. You might like it. Please do have a look.
http://gravity.psu.edu/events/abhayfest/talks/Smolin.pdf

The table of contents goes:

1. The basic idea of unimodular gravity
2. Henneaux and Teitelboim, Plebanski
3. Hamiltonian formulation
4. The path integral quantum theory is unimodular
...
...

The HT reformulation of GR is background independent.
And it takes some additional definitions and work to derive the unimodular feature
(the non-gravitating condition we associate with the basic UG idea.) That in part is what chapter 4 is about---carrying the result over from the classical to the quantum context.

 

[atyy]

Hmmm, let me think about it. I have to say I'm skeptical. I can believe the HT reformulation has equations of motion which are generally covariant. However, I find it hard to believe they are background independent.

 

[marcus]

Atyy, since I think the Abhayfest slides are a valuable concise presentation, I'll try to reproduce some of the equations.

The basic idea of unimodular gravity:
$$S^{uni} = \int_\mathcal{M} \epsilon_0 \left(- \frac{1}{8 \pi G} \bar{g}^{ab}R_{ab} + \mathcal{L}^{matter}(\bar{g}_{ab}, \psi) \right)$$

det(g) has been constrained to be equal to a fixed volume element:

$$\sqrt{-g} = \epsilon_0$$

The diffeomorphism group is reduced to volume preserving diffeo’s:

$$\partial_a (\epsilon_0 v^a) = 0$$

The eq’s of motion are just the tracefree part of Einstein:

$$R_{ab} - \frac{1}{4} \bar{g}_{ab} R = 4 \pi G (T_{ab} - \frac{1}{4} \bar{g}_{ab}T)$$

This has decoupling symmetry:

$$T_{ab} \rightarrow T'_{ab} = T_{ab} + g_{ab}C$$

 

[atyy]

Marcus,
Come on, this is getting frustrating for me. Do you want to learn or not?

I AM trying to focus on the papers. But you keep ignoring any math and discussion, and maintaining statements that contradict the paper.

If you really want to learn and discuss the theory presented in that paper, we need to agree on some basic math results in this theory to have any hope of discussing further specifics.

So please, let's discuss specifics.

given the action:
$$S = \frac{1}{2\kappa} \int_\mathcal{M} d^4x \left[ \sqrt{-g} (R-2\Lambda+ 2 \kappa \mathcal{L}_m) + 2\Lambda \partial_\mu \tau^\mu \right]$$
Here, Λ is not a parameter like in GR (ie. it is not specified once and for all, but is a scalar field).

Variation with respect to the metric leads to the field equations:
$$R_{\mu \nu} - \frac{1}{2} g_{\mu \nu} R + \Lambda g_{\mu \nu} = \kappa T_{\mu \nu}$$

And now it is easy to show that, just as in normal GR,
$$R = 4 \Lambda - \kappa T$$

Variation of the action with respect to tau leads to the equation of motion,
$$\partial_\mu \Lambda = 0$$
So while Λ is not specified (it is a scalar field, not a parameter like in GR), the only allowed configuration is for Λ to be a constant everywhere.

So classically, unimodular gravity is mathematically equivalent to GR once Λ is specified.

So, at least this version of unimodular gravity does not predict a constant energy density doesn't gravitate. This can be seen not only by just calculating the curvature directly, but also it should be obvious this had to be the case since they are equivalent classically.

I'd very much appreciate answers to these direct questions:
1] Do you agree with my math and discussion of unimodular gravity in this post? (If not, what do you disagree with?)

2] In particular, do you see that this version of unimodular gravity and GR are equivalent classically?


I'm sorry I have to be so blunt with questions. But I'm trying to build up specifics that we can agree on, so we can discuss further specifics in the papers.

This seems right. Apparently, one also must vary with respect to lambda to get the unimodular condition.

 

[marcus]

Does anyone see any faults with this? I assume not--it's really basic. And this "basic idea of UG" is what I was talking about earlier. It seemed to provoke a challenge! What you see here at the end is the equation that says "constant multiples of the metric don't gravitate." Adding a constant multiple g_abC to the energy-momentum tensor T_ab does not change anything.

That is the "non-coupling condition", as Smolin calls it. He later derives this from the HT formulation.

The basic idea of unimodular gravity:
$$S^{uni} = \int_\mathcal{M} \epsilon_0 \left(- \frac{1}{8 \pi G} \bar{g}^{ab}R_{ab} + \mathcal{L}^{matter}(\bar{g}_{ab}, \psi) \right)$$

det(g) has been constrained to be equal to a fixed volume element:

$$\sqrt{-g} = \epsilon_0$$

The diffeomorphism group is reduced to volume preserving diffeo’s:

$$\partial_a (\epsilon_0 v^a) = 0$$

The eq’s of motion are just the tracefree part of Einstein:

$$R_{ab} - \frac{1}{4} \bar{g}_{ab} R = 4 \pi G (T_{ab} - \frac{1}{4} \bar{g}_{ab}T)$$

This has decoupling symmetry:

$$T_{ab} \rightarrow T'_{ab} = T_{ab} + g_{ab}C$$

A good presentation of the HT formulation is on pages 6 and 7 of http://arxiv.org/abs/0904.4841 , equations (17) through (24).

You can see that equation (18) on page 6 is the Henneaux-Teitelboim action and already the decoupling symmetry is beginning to emerge at equation (24). As Smolin says it is the same thing "said differently". The final derivation doesn't come until page 12 section 4.2 Gauge Fixing, in equation (64) where it says
"...so we return to the action of the original form of unimodular gravity, S^uni".

The initial passage, eqs. (17) to (24), is instructive and you can see that it almost recovers the basic unimodular idea, said differently. So maybe I should reproduce some of the equations. The key thing is that two new fields are introduced, a 3-form a_abc and a scalar field φ which serves as a lagrange multiplier.

 

[atyy]

http://arxiv.org/abs/1008.1196
"What about experiments? The experimental predictions of the two theories are the same, so no experiment can tell the difference between them, except for one fundamental feature: the EFE (confirmed in the solar system and by binary pulsar measurements to high accuracy) together with the QFT prediction for the vacuum energy density (confirmed by Casimir force measurements) give the wrong answer by many orders of magnitude; the TFE does not suffer this problem. In this respect, the TFE are strongly preferred by experiment."

But the conclusion about experimental confirmation of zero-point energy is probably incorrect because of http://arxiv.org/abs/hep-th/0503158 .

Here's interesting criticism http://arxiv.org/abs/0805.2183
"The integration constant C reappears as the cosmological constant when this equation is inserted back into the traceless part of Einstein’s equation. Thus, unimodular gravity does not solve the problem but makes some people “feel more comfortable” because in theoretical physics, supposedly, we have the license to set integration constants to whatever we want."

 

[marcus]

http://arxiv.org/abs/1008.1196
"What about experiments? The experimental predictions of the two theories are the same, so no experiment can tell the difference between them, except for one fundamental feature...

Beautiful find. Thanks! And everybody knows George Ellis is a major figure. I figured Unimodular was apt to get big when I saw a half-dozen recent papers by Enrique Alvarez---he's been at CERN, Harvard, Princeton, now senior faculty at Uni Madrid. I'd say on the order of 1000 lifetime cites (much of it earlier in string research). He calls stuff by different names, like "transverse" but UG has caught his attention. Alvarez is a particle theorist---Ellis is better known on the GR side. I will copy the abstract and take a look at the paper.

http://arxiv.org/abs/1008.1196
The gravitational effect of the vacuum
George F. R. Ellis, Jeff Murugan, Henk van Elst
(Submitted on 6 Aug 2010)
"The quantum field theoretic prediction for the vacuum energy density leads to a value for the effective cosmological constant that is incorrect by between 60 to 120 orders of magnitude. We review an old proposal of replacing Einstein's Field Equations by their trace-free part (the Trace-Free Einstein Equations), together with an independent assumption of energy--momentum conservation by matter fields. We confirm that while this does not solve the fundamental issue of why the cosmological constant has the value it has, it is indeed a viable theory that resolves the problem of the discrepancy between the vacuum energy density and the observed value of the cosmological constant. We also point out that this proposal may have a valid quantum field theory basis in terms of a spin-2 field theory for the graviton interaction with matter."

Really good paper. Makes the argument that we should use the Tracefree Einstein equations---the TFE---instead of the EFE. Balances that with counterargument and alternative, but makes the argument clearly that one should use UG instead of GR. And this guy is probably the most influential relativist/cosmologist today. Or one of a handful.

 

[JustinLevy]

For starters, that is not what Smolin calls the unimodular action. He gives S^uni, based on a constant determinant.

And that is so to speak chapter 1 of the story. Chapter 2 introduces the Henneaux Teitelboim action which he calls S^HT, then he does some derivation and ends up deriving the non-coupling condition. Page 7 eq.24. Later (after gauge fixing) he actually gets back to the unimodular action.

Smolin's April paper 0904.4841 is more self contained. It is a better introduction. Chiou-Geiller jump right in at chapter 2. They don't give S^uni. They start right off with the Henneaux Teitelboim action S^HT. It's easy to get confused by their paper if you don't read Smolin's along with it.

Marcus, this is exactly what I was complaining about. You keep avoiding discussing specifics, and always instead redirect to another paper. In addition you continue to make false claims about the theory in the paper you brought up and I was discussing here:

http://arxiv.org/abs/1007.0735
Unimodular Loop Quantum Cosmology

So please stop and discuss the theory.

There is nothing that bothers me more than an intelligent person actively ignoring simple results/discussion so that they can continue to state false information. If your goal is not to learn about the physics in these papers, then why even bother. You are an intelligent person Marcus, please please pause to learn here.

I implore you to please go back and answer my two direct questions at the end of post #35 regarding the action in that paper.

 

[JustinLevy]

Does anyone see any faults with this? I assume not--it's really basic. And this "basic idea of UG" is what I was talking about earlier. It seemed to provoke a challenge! What you see here at the end is the equation that says "constant multiples of the metric don't gravitate." Adding a constant multiple g_abC to the energy-momentum tensor T_ab does not change anything.

Yes, yes I do have faults with this. The very conclusions you are drawing from those equations are not correct. Smolin's paper I feel is sloppier (not as "honest" about some of the steps he's skipping that are required, and also making some misleading claims and overstating some things). That is why I was trying to discuss the other paper first.

I don't want to create parallel discussions, but in the spirit of discussing specifics, let me address at least one (please don't make me regret this by focusing on this now instead of answering my questions. I still would like you to answer my previous questions.):

The eq’s of motion are just the tracefree part of Einstein:

$$R_{ab} - \frac{1}{4} \bar{g}_{ab} R = 4 \pi G (T_{ab} - \frac{1}{4} \bar{g}_{ab} T)$$ [typo corrected]

This has decoupling symmetry:

$$T_{ab} \rightarrow T'_{ab} = T_{ab} + g_{ab}C$$

Two points:
----
1] Incorrect conclusions are being read into that symmetry of the tracefree equations. To drive this point home, note that the tracefree equations are invariant to the change:
$$T_{ab} \rightarrow T'_{ab} = T_{ab} + g_{ab} f(x^\mu)$$
where f(x) is some scalar function of spacetime.
Does this mean we can claim the tracefree equation says the pressure term p g_ab in the stress energy of a perfect fluid does not gravitate? No, of course not.

One could ask, Why not? After all the symmetry is there in the tracefree equations. Well I'll discuss the error in my next point.

----
2] The error is your claim that the equations of motion are just the tracefree part of Einstein's equations. If this were the case, then unimodular gravity is not deterministic as the solution is underdetermined.

In order for the theory to be deterministic, there needs to be enough constraint to allow one to solve for the spacetime curvature. This can be done for example by including two more constraints:
- First -
Stating explicitly that the stress energy tensor is divergence free (energy-momentum is locally conserved). This can be derived in GR, but must be explicitly stated if all one has is the tracefree equations, for it cannot be derived in this case. Now one can obtain a relation for the divergence of R and T (the non-tracefree equation).
- Second -
Specifying the "integration constant" Λ of the resulting non-tracefree equation.

Once that is done, you will see that changing the stress energy by a constant times the metric does change the equations of motion.

 

[Haelfix]

The biggest problem with the Unimodular theory as I understand it, is that it is very hard to figure out if it is or isn't different than the EH theory as a nonperturbative *quantum* theory (classically it coincides onshell).

Most certainly it is different perturbatively, which was apparent back in the 80s already and the point of the Henneaux and Teitelbaum papers (incidentally they are amongst the most well known and respected theoretical physicists of the last 30 year and wrote the book on Canonical quantization in gauge and string theories).

Now as to what happens nonperturbatively there are essentially three good guesses and it all hinges on whether the unimodularity condition is anomalous or not.

1) If the unimodularity condition is anomalous, the most probable thing is that just like Pauli-Fiersz gravity, the resulting nonperturbative theory is identical to whatever it is that the nonpertubative EH theory is. This is probably the best guess, albeit the one with the least punch, b/c the cosmological constant is simply regenerated by renormalization group flow and you haven't ameliorated the situation any and are back to finetuning 120 decimal places and knowing all about every matter field from here all the way down to the Planck scale.

2) If the unimodularity condition is anomalous, but not equivalent to EH nonperturbatively. This is the least likely, b/c you don't solve anything and you don't ameliorate the cc problem but should be included as a possibility.

3) If the unimodularity condition is nonanomalous, then you are left with a quantum theory, different than EH. It will of course have the nice properties described where you only have to worry about 1 number (and not infinitely many unknown physical quantities) and where you avoid some ridiculous mismatch between quantum theory and GR.

The problem is that no one really knows which of the three solutions is correct or not, or been able to demonstrate anything concretely beyond what the original H T papers showed, which is why the good idea has languished for sometime and reinvented every few years.

 

[marcus]

Thanks, this seems like a constructive comment and shows openness to the idea. I also want to confirm yr testimonial to Henneaux and Teitelboim. Seen a lot of respect for their contributions over the years.

The biggest problem with the Unimodular theory as I understand it, is that it is very hard to figure out if it is or isn't different than the EH theory as a nonperturbative *quantum* theory (classically it coincides onshell).

Most certainly it is different perturbatively, which was apparent back in the 80s already and the point of the Henneaux and Teitelbaum papers (incidentally they are amongst the most well known and respected theoretical physicists of the last 30 year ...
...
3) If the unimodularity condition is nonanomalous, then you are left with a quantum theory, different than EH. It will of course have the nice properties described where you only have to worry about 1 number (and not infinitely many unknown physical quantities) and where you avoid some ridiculous mismatch between quantum theory and GR...

I've highlighted the upbeat possibility. And the major difficulty you point out---understanding the unimodular alternative nonperturbatively. After all, a nonperturbative quantum treatment of UG is precisely what people (Smolin, Chiou, Geiller) are working on these days.

I think a nonperturbative formulation will go some ways towards settling the issue.

 

[marcus]

Haelfix, your taking the trouble to point out that Teitelboim is an enormously respected figure (both in and out of the string community, I would judge) strkes a sympathetic chord with me. It really makes a practical difference when major figures sign on to an idea. It helps to encourage younger people to work on it, when people with established reputations point up the merits.

So back around 1989-1991 there was this flurry of paper by stars like Henneaux and Teitelboim and Steven Weinberg and Bill Unruh, and I would also count Jack Ng and Hendrik van Dam.

Now 20 years later we are seeing a UG revival with people like George Ellis, Enrique Alvarez, and Lee Smolin signing on. It doesn't mean the idea is right, but it re-establishes the fact that it is interesting. Not to be automatically dismissed.
I was delighted by this paper that Atyy called attention to:

Beautiful find. Thanks! And everybody knows George Ellis is a major figure. I figured Unimodular was apt to get big when I saw a half-dozen recent papers by Enrique Alvarez---he's been at CERN, Harvard, Princeton, now senior faculty at Uni Madrid. I'd say on the order of 1000 lifetime cites (much of it earlier in string research). He calls stuff by different names, like "transverse" but UG has caught his attention. Alvarez is a particle theorist---Ellis is better known on the GR side. I will copy the abstract and take a look at the paper.

http://arxiv.org/abs/1008.1196
The gravitational effect of the vacuum
George F. R. Ellis, Jeff Murugan, Henk van Elst
(Submitted on 6 Aug 2010)
"The quantum field theoretic prediction for the vacuum energy density leads to a value for the effective cosmological constant that is incorrect by between 60 to 120 orders of magnitude. We review an old proposal of replacing Einstein's Field Equations by their trace-free part (the Trace-Free Einstein Equations), together with an independent assumption of energy--momentum conservation by matter fields. We confirm that while this does not solve the fundamental issue of why the cosmological constant has the value it has, it is indeed a viable theory that resolves the problem of the discrepancy between the vacuum energy density and the observed value of the cosmological constant. We also point out that this proposal may have a valid quantum field theory basis in terms of a spin-2 field theory for the graviton interaction with matter."

Really good paper. Makes the argument that we should use the Tracefree Einstein equations---the TFE---instead of the EFE. Balances that with counterargument and alternative, but makes the argument clearly that one should use UG instead of GR. And this guy is probably the most influential relativist/cosmologist today. Or one of a handful.

Ellis really signs on forcefully. His handle on the UG idea is "Trace-Free Einstein Equations" (TFE as opposed to the usual EFE). Here is a quote from the conclusions:

==Ellis et al right at the end==
Thus, a good assumption to make is that the true effective gravitational field equations may be the TFE not the EFE, and any huge Λ_vac is powerless to affect cosmology, or indeed the solar system... The huge zero point energy will not affect spacetime curvature. The EFE will be as usual but with ... an integration constant that may be small, or may be zero.

Overall, this proposal does not solve the issue of why the cosmological constant has the value it has today; but it does resolve the issue of why it does not have the huge value implied by the obvious use of the QFT prediction for the vacuum energy in conjunction with the EFE (the patently incorrect result obtained in this way is a major crisis for theoretical physics, because it suggests a profound contradiction between QFT and GR).

The present proposal also indicates a route to investigate in terms of quantum gravity theory: whatever full theory of quantum gravity is eventually arrived at, whether based in the string theory approach, loop quantum gravity, causal dynamical triangulations, or whatever [16], it should have as its field theory limit a spin-2 theory where the trace-free nature of the graviton leads to an effective trace-free version of Einstein’s Field Equations.
==endquote==

Readers inclined to quibble with the term TFE should be advised to actually read equations (25) and (27) of Ellis paper. What he means by it involves two conditions not just (as might be supposed) the one (25).

 

[marcus]

Thermal time and unimodular time compatible?

So far this thread has been aimed at getting an intuitive feel for unimodular gravity (UG) and some allied formulations---which are related to time and issues over vacuum energy and cosmo constant.

The UG implications for our concept of time are especially significant. In the HT formulation, a global time and the cosmological "constant" arise as conjugate variables. Maybe one should speak of the "cosmological variable".

The UG global time variable is not to be confused with an observer time. It is associated with space-like hypersurfaces. Each such hypersurface has a time quantity associated with it---intuitively speaking, one can think of it as the age of the universe as measured by a certain class of proper observers living on that hypersurface. Or as the volume of the 4D past that the hypersurface rests on.

Readers who want more detail on this can see equations (3)-(5) of http://arxiv.org/abs/1008.1759 and the ample explanation that follows.

What I want to do in this thread now is move on to consider what I think is the most important question about UG global time and Rovelli's Thermal Time.

Thermal Time is also a global time. The thermal time hypothesis (TTH) was proposed by Rovelli and Alain Connes (the lead proponent of Noncommutative Geometry as grounds for the Standard Model.) As I recall the original Rovelli Connes paper on TTH was 1994.
Thermal Time is completely emergent, it is a macroscopic bulk phenomenon analogous, in a sense, to temperature. Indeed geometry (the GR metric) has a definable temperature related to location in the gravitational field, and that geometric temperature and the thermal time are related.

The most recent paper on the TTH was Rovelli-Smerlak. That paper was on our 2nd quarter 2010 MIP ("most important paper) poll. And it got the largest number of votes.

http://arxiv.org/abs/1005.2985
Thermal time and the Tolman-Ehrenfest effect: temperature as the "speed of time"
Carlo Rovelli, Matteo Smerlak
4 pages
(Submitted on 17 May 2010)
"The thermal time hypothesis has been introduced as a possible basis for a fully general-relativistic thermodynamics. Here we use the notion of thermal time to study thermal equilibrium on stationary spacetimes. Notably, we show that the Tolman-Ehrenfest effect (the variation of temperature in space so that T\sqrt{g_{00}} remains constant) can be reappraised as a manifestation of this fact: at thermal equilibrium, temperature is locally the rate of flow of thermal time with respect to proper time - pictorially, "the speed of (thermal) time". Our derivation of the Tolman-Ehrenfest effect makes no reference to the physical mechanisms underlying thermalization, thus illustrating the import of the notion of thermal time."

Here is the MIP poll, where this paper was a strong favorite:
https://www.physicsforums.com/showthread.php?t=413838
To see the full results
https://www.physicsforums.com/poll.php?do=showresults&pollid=1831
Oddly enough, it did not occur to me to vote for the Rovelli Smerlak paper myself, but 6 other people did. Hats off to their foresight, it was a good choice.

 

[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
$$S = \int_\mathcal{M} d^4x \sqrt{-g} \left[ (\frac{1}{2\kappa}R + \mathcal{L}_m) + (R^{abcd}R_{abcd} \phi + g^{ab}g_{ab}\psi - 10 \psi) \right]$$
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]
...