Unigrav applied to problems of time and cosmo constant

[marcus]

Unimodular Gravity was proposed by Einstein in 1919. It has the same equations of motion as ordinary GR, so you couldn't tell the difference experimentally. But time and the cosmological constant are treated differently in uni-GR from how they are in usual-GR. This may actually turn out to be an advantage.

So some people are working on quantizing unigrav---to get, among other aims, unigrav version of LQG and LQC (the full theory and it's application to cosmology.)

One simple way of looking at unigrav is that it is just the same as usual-GR except that the determinant of the metric must equal -1---or anyway it must be constant.
And then you limit your diffeomorphisms to preserve that property.

It's not such a serious limitation. If you start with coords and a solution metric g_µν you can imagine stretching/squeezing the coords around each point to make the solution metric unimodular. Just mess with it some.

And that's not the only way to think about it. You don't have to require the determinant g be constant. There is some celebrated 1980s work by a Belgian and a Chilean (Henneaux and Teitelboim) which gave a general-covariant action for unigrav. So then any diffeomorphism was OK.

Teitelboim was not the guy's original name. He had temporarily taken the name Teitelboim as a safety precaution because of the dangerous dictatorship in Chile. His real name was Bunster. Both are excellent-sounding names, and without doubt Bunster (aka Teitelboim) is both a lucky and creative individual.

The Henneaux Teitelboim action is shown on page 3 of http://arxiv.org/abs/1007.0735. It is equation (2.1)

 

[marcus]

I think it's possible that we will switch to unigrav, because it gives the same observational results as usual-GR and it may solve some problems for us: how to treat time, where the cosmo constant comes from, or at least how to think of it.

Also usual-GR has the shocking problem of the 120 orders-of-magnitude, which is not a problem with unigrav. We hear a lot of screams and groans about how QFT predicts a constant vacuum energy density which is 120 orders higher than the actual energy density if it were caused by cosmo constant Lambda.
This is supposed to be something like the gravest and most embarrassing problem ever faced by physics. But if you use unigrav then this problem does not arise because any exactly constant energy density has no physical effect---it just cancels out.

So I think it's very likely that at least a fair number of people in the QG community will switch over to unigrav---and (who knows?) maybe other research communities will as well. The treatment of time, in unigrav, is really nice.

IMHO, the paper I mentioned in the previous post, by Chiou and Geiller, is one of the most valuable QG research papers to appear this quarter (namely July-September 2010). I think it has the potential to slightly change the course of Loop Cosmology.
In LQC it was always a bother that you had to use a scalar matter field as a clock. Chiou Geiller release the matter field from serving as a reference, because they have a different way to treat time. So the matter field can play a more interesting role---freed from its relational-time responsibilities. That's how it looks to me, anyway. A promising development. So I will copy the abstract for the Chiou Geiller paper here---maybe we can discuss it.

 

[marcus]

http://arxiv.org/abs/1007.0735
Unimodular Loop Quantum Cosmology
Dah-Wei Chiou, Marc Geiller
26 pages. Published in Physical Review D 82, 064012 (2010)
(Submitted on 5 Jul 2010)
"Unimodular gravity is based on a modification of the usual Einstein-Hilbert action that allows one to recover general relativity with a dynamical cosmological constant. It also has the interesting property of providing, as the momentum conjugate to the cosmological constant, an emergent clock variable. In this paper we investigate the cosmological reduction of unimodular gravity, and its quantization within the framework of flat homogeneous and isotropic loop quantum cosmology. It is shown that the unimodular clock can be used to construct the physical state space, and that the fundamental features of the previous models featuring scalar field clocks are reproduced. In particular, the classical singularity is replaced by a quantum bounce, which takes place in the same condition as obtained previously. We also find that requirement of semi-classicality demands the expectation value of the cosmological constant to be small (in Planck units). The relation to spin foam models is also studied, and we show that the use of the unimodular time variable leads to a unique vertex expansion."

 

[marcus]

Dahwei Chiou got his PhD from the physics department of UC Berkeley in 2006 and went from there to postdoc in Ashtekar's team at Penn State. Decisive move, otherwise would probably still be doing string theory. Then around 2009 he moved back to China, taking a position at Beijing-Normal, where they have a LQG group.
If what I think is right and this is an important paper, then it is an example of things working right---a young researcher, postdoc, opens up new territory.
I don't know who Marc Geiller is. I think he is a PhD student at "Paris Diderot"---the University of Paris Diderot campus in Arrondissement 13.

There is some background on unimodular gravity from a paper by Lee Smolin that came out in August 2010, a month after Chiou Geiller.
==quote http://arxiv.org/abs/1008.1759 page 1 ==

The unimodular formulation of general relativity was first proposed by Einstein in 1919 as an approach to the unification of gravity and matter[1] . It was studied by a number of authors in the 1980s and early 90s because of indications that it resolves two key problems in quantum gravity[2]-[9]. These are the cosmological constant problem and the problem of defining a physically meaningful time with which to measure evolution of quantum states in quantum cosmology, in the absence of a spatial boundary.

This is the second of two papers which report results which support and clarify the sense in which unimodular quantum gravity solves these two problems. In the first of these papers[10], I constructed the constrained phase space quantization of a formulation of unimodular gravity due to Henneaux and Teitelboim[8]. I showed that the quantum effective action is a functional of the unimodular spacetime metric g_µν with determinant fixed to:
(1)

$$\sqrt{det(\bar{g}_{\mu\nu})} = \epsilon_0$$

where epsilon₀ is a fixed nondynamical volume element. This means that the quantum effective equations of motion, which arise from varying the metric with (1) fixed have a symmetry
T_ab → T′_ab = Tab + ¯g_ab C
where C is a spacetime constant. This decouples the dynamics of the metric ¯g from any
contribution to the energy-momentum tensor, whether classical or quantum, of the form of a constant times the spacetime metric. This means that the puzzle of why huge contributions to T_µν of this form coming from the zero point energy of the fields,... are not sources of spacetime curvature...

==endquote==

 

[marcus]

I was misinformed about Claudio Bunster. Here is a Chilean magazine source:
http://www.australvaldivia.cl/prontus4_nots/site/artic/20050905/pags/20050905013121.html

The story of unimodular gravity is a "sleeper". It gives the same results as GR, it was proposed by Einstein in 1919.
It was reformulated, in I think a brilliant way, by Marc Henneaux and Claudio Teitelboim.

In a certain way it solves the problems of time and the cosmological constant.

And it has been largely ignored.

I think that it is possible that we will see unimodular gravity "wake up" over the next couple of years or so---possibly implemented in quantum gravity (LQC?, spinfoam?)

http://www.ulb.ac.be/sciences/ptm/pmif/membres/henneaux.html

The Henneaux Teitelboim (HT) paper came out in 1989
The cosmological constant and general covariance Physics Letters B 222, 195.

Look at the HT action (page 3 of Chiou Geiller paper, equation 2.1)

It is beautiful and also invariant under the full group of diffeomorphisms. The cosmo constant Lambda is a scalar field which serves as Lagrange multiplier for a vector density tau which provides an intrinsic time.
(intuitively every spacelike hypersurface has associated with it a time-measure which is "the volume of the past').

Beautiful idea. We have to get to know who these people are. Who has been instrumental so far in realizing the potential of the HT action?

 

[marcus]

Here are some of Smolin's references:

[4] Y.J. Ng, H. van Dam, Unimodular Theory Of Gravity And The Cosmological Constant
J.Math.Phys.32:1337-1340, 1991.

[5] W. G. Unruh, A Unimodular Theory Of Canonical Quantum Gravity
Phys.Rev.D40:1048, 1989.

[6] W. G. Unruh and R. M. Wald, Time And The Interpretation Of Canonical Quantum Gravity Phys.Rev.D40:2598, 1989.

[8] M. Henneaux and C. Teitelboim, The cosmological constant and general covariance
Phys. Lett. B, Vol. 222, No. 2, p. 195 - 199, 1989.

[9] L. Bombelli, W.E. Couch, R.J. Torrence, Time as space-time four volume and the Ashtekar variables
Phys.Rev.D44:2589-2592, 1991.

This is just a sampling, I omit some by Zee, Weinberg, Sorkin which you can find in Smolin's article.

The articles listed here are all about UNIMODULAR and they are about the cosmo constant and time----and those two things get multiplied together in the HT action:
one is the Lagrange multiplier of the other.

Unimodular hits two birds with one stone.

 

[bcrowell]

I'm thinking this should relate somehow to something else that has been confusing me for a while. Here http://arxiv.org/abs/physics/9905030 is a translation of Schwarzschild's original paper on the Schwarzschild metric. (The translators have unusual ideas about black holes, as shown in the abstract displayed on arxiv, but I have no reason to think that this impugns the accuracy of their translation.) Schwarzschild's equation (5), which he describes as the "equation of the determinant" is $|g_{\mu\nu}|=-1$. The paper is from 1916, which is 3 years before the date when Marcus says Einstein published unimodular gravity. I'd never really known what to make of this equation. (Well, the whole Schwarzschild paper is not done in a way that anyone in 2010 would consider pedagogically optimal). I'd sort of figured that it must be a primitive, confused attempt simply to say, "Hey, the signature is -+++, not +++ or ++++. This is a pseudo-Riemannian metric, not a Riemannian one." It hadn't occurred to me that this was a much more strict condition than a simple constraint on the signature.

Any idea how this might relate?

I'm intrigued by the whole idea that there are multiple formalisms of GR that are equivalent in most but not all cases. Similar situation with Ashtekar's formulation of GR when it comes to solutions that contain degeneracies of the metric...?

-Ben

 

[tom.stoer]

I think there is an older Smolin paper from 2008 or 2009 or so where he started to discuss unimodular gravity.

I think that most different formulations of GR are related (dual?) due to the huge symmetry of the theory. E.g. f(R) theories can be derived via auxiliary fields. In that sense unimodular gravity may be just a reformulatation of something else.

In the context of asymptotic safety all these different theories should arise in "full theory space" but are then renormalized such that (in certain regimes) the differences disappear. So one should have a look at this approach and then try to figure out how one can transport the main ideas to LQG. Therefore I do not only propose to quantize unimodular gravity according to the LQG approach; instead one should quantize "the full theory space" according to the LQG approach and try to understand what singles out the Einstein-Hilbert (Einstein-Cartan?) action from an QG perspective.

Some questions: What about coupling fermions to unimodular gravity? What about torsion? Or generally speaking Einstein-Cartan gravity which is the preferred low-energy limit of LQG?

 

[marcus]

I think there is an older Smolin paper from 2008 or 2009 or so where he started to discuss unimodular gravity...

That's right. He has an April 2009 paper about it. 0904.4841 (Physical Review D)

Also in May 2009 he gave a talk on it. Here are the slides
http://pirsa.org/pdf/files/8001b79a-af33-441f-8a23-2f8877744005.pdfH
(the first page is nearly blank, so you have to scroll down)

The video for the talk is here:
http://pirsa.org/09050091/
The quantization of unimodular gravity and the cosmological constant problems

Smolin's slide 8 quotes Weinberg's 1989 paper (based on a 1988 talk):
http://www-itp.particle.uni-karlsruhe.de/~sahlmann/gr+c_seminarII/pdfs/T3.pdf

For Weinberg's discussion of Unimodular starts around the bottom of page 12. Near the end, at the bottom of page 13, he edited in a reference to Henneaux Teitelboim (in square brackets)---he had a 1988 preprint of their 1989 paper. This gist is that he likes Unimodular as a solution to the (why it's not huge?) cosmo constant problem, but he doesn't know if it has a corresponding quantum theory. He says that is the key question.
Then in the end he edits in the reference to the HT 1988 preprint and, without revising his earlier cautious assessment, says that judging from HT it looks like there IS a quantum version, after all.

Smolin's April 2009 paper and May talk are largely devoted to confirming and carrying this through in detail. In Weinberg it is just a vague iffy suggestion---I would like Unimodular if it had a quantized version. Because it explains why the cosmo constant Lambda is huge.
Smolin wants you to know that for sure it DOES. So taking Weinberg seriously we would conclude that Unimodular is the right classical theory of gravity---and the other variants of GR are not right.

[But if you read the careful urbane tone of Weinberg's paper you realize that he does not quite want you to take what he says completely seriously.]

About the Cosmological Constant Problem (why isn't Lambda huge?), Weinberg quotes a children's nonsense poem at the beginning--a kind of Lewis Carroll or Mother Goose-type rhyme:

As I was going up the stair,
I met a man who wasn't there.
He wasn't there again today--
Oh how I wish he'd stay away!

 

[marcus]

... Therefore I do not only propose to quantize unimodular gravity according to the LQG approach; instead one should quantize "the full theory space" according to the LQG approach and try to understand what singles out the Einstein-Hilbert (Einstein-Cartan?) action from an QG perspective.
...

Another tack might be to try to understand what singles out the H-T action. These recent papers (and discussion going back to late 1980s too) suggest three considerations:

1. H-T action makes the huge, Planckscale, vacuum energy go away. It solves the longest-standing puzzle about the cosmological constant. Since vacuum energy is a quantum theory problem, this makes the H-T action particularly attractive as a basis for a quantum theory of gravity.

2. H-T action has a natural role for the cosmological constant Lambda to play. It must be there. (In a sense, conjugate to time.)

3. H-T action has a natural time, associated with each spacelike hypersurface. This is reminiscent of the situation in cosmology, where there is a "universe time"---a criterion for being at rest relative to the Background of ancient light, a natural foliation (if you like, according to Background temperature, or age of universe as perceived by stationary observers.) In cosmology time tends to be somewhat more "real" than it is in pure general relativity.

So the H-T action has these various features which appear to single it out.

Thanks for asking about fermions and Einstein-Cartan. I will try to find something on that.

 

[marcus]

...
Some questions: What about coupling fermions to unimodular gravity? What about torsion? Or generally speaking Einstein-Cartan gravity which is the preferred low-energy limit of LQG?

Tom, a preliminary search led me to this paper of Shaposhnikov and Zenhäusern. It seemed as if it might be interesting so I'll give the link and a sample quote even though it fails to answer the questions you asked:

http://arxiv.org/abs/0809.3395
Scale invariance, unimodular gravity and dark energy
Mikhail Shaposhnikov, Daniel Zenhäusern
9 pages, 1 figure, Phys.Lett.B671:187-192,2009
(Submitted on 19 Sep 2008)
"We demonstrate that the combination of the ideas of unimodular gravity, scale invariance, and the existence of an exactly massless dilaton leads to the evolution of the universe supported by present observations: inflation in the past, followed by the radiation and matter dominated stages and accelerated expansion at present. All mass scales in this type of theories come from one and the same source."

==quote Shaposhnikov Zenhäusern page 2==
The aim of this Letter is to show that the situation is completely different if general relativity in (4) is replaced by Unimodular Gravity (UG). UG is a very modest modification of Einstein’s theory: it adds a constraint g = −1 to the action principle defined by eq. (4) [9, 10, 11, 12, 13, 14, 15, 16]. ...
==endquote==

Shaposhnikov Zenhäusern are quite frank about there being some wishful thinking here (they use that precise phrase) and their arguments lead to conclusion which seem "too good to be true". But what I have seen from Shaposhnikov earlier makes me pay attention to his ideas. You may have a different opinion.

What I think is the case is that UG is OK with coupling fermions. But I don't know for sure and i will look around some more for confirmation.

====================================

So far no success in finding a discussion of UG torsion/fermions/Einstein-Cartan. I'll copy some abstracts partly just to keep track of them so I can examine them for leads. It is frustrating not to find the torsion-related issues addressed.

Here's one I do not have time to examine right now. Have to go out on errands.

http://arxiv.org/abs/hep-th/0501146
Can one tell Einstein's unimodular theory from Einstein's general relativity?
Enrique Alvarez
20 pages, JHEP 0503 (2005) 002
(Submitted on 19 Jan 2005)
"The so called unimodular theory of gravitation is compared with general relativity in the quadratic (Fierz-Pauli) regime, using a quite broad framework, and it is argued that quantum effects allow in principle to discriminate between both theories."

Briefly noted, some history and overview:
http://arXiv.org/abs/0809.1371
Semiclassical Unimodular Gravity
Bartomeu Fiol, Jaume Garriga

Ng and van Dam's earlier UG paper is not online but this later one is available and gives a recap:
http://arxiv.org/abs/hep-th/9911102
A small but nonzero cosmological constant
Jack Ng, Hendrik van Dam (University of North Carolina)
Int.J.Mod.Phys. D10 (2001) 49-56
(Submitted on 13 Nov 1999)
"Recent astrophysical observations seem to indicate that the cosmological constant is small but nonzero and positive. The old cosmological constant problem asks why it is so small; we must now ask, in addition, why it is nonzero (and is in the range found by recent observations), and why it is positive. In this essay, we try to kill these three metaphorical birds with one stone. That stone is the unimodular theory of gravity, which is the ordinary theory of gravity, except for the way the cosmological constant arises in the theory. We argue that the cosmological constant becomes dynamical, and eventually, in terms of the cosmic scale factor R(t), it takes the form Λ(t) = Λ(t₀)(R(t₀)/R(t))², but not before the epoch corresponding to the redshift parameter z ~ 1."

I was unable to find an online copy of this--can only give the abstract:

Time as spacetime four-volume and the Ashtekar variables
Luca Bombelli, W. E. Couch, and R. J. Torrence
Phys. Rev. D 44, 2589–2592 (15 October 1991)
"We consider a recently proposed theory of gravity, classically equivalent to Einstein's theory with the cosmological constant as an additional variable, in which spacetime volume plays the role of time. We develop a Hamiltonian formulation using Ashtekar's variables, set up the corresponding quantum theory, and show that the known loop state solutions of quantum general relativity are also solutions in the present theory. We conclude with some remarks on why we feel that this quantum theory deserves further study."

αβγδεζηθικλμνξοπρσςτυφχψω...ΓΔΘΛΞΠΣΦΨΩ...∏∑∫∂√ ...± ÷...←↓→↑↔~≈≠≡≤≥...½...∞...(⇐⇑⇒⇓⇔∴∃ℝℤℕℂ⋅)

 

[JustinLevy]

I'd like to address bcrowell's question as Marcus' initial post confused me a bit too at first.

Historically, there were many steps and papers leading to what we currently consider "canonical GR". It is my understanding that in Einstein's first paper which contained the field equations (1915), he restricted $|g_{\mu\nu}|=-1$. In the second paper (1916 I believe?) he relaxed this, for it was clear from the action principle (Hilbert's approach) that the field equations can handle any g_uv.

[Side Notes:
I've based this historical understanding on discussions with others since I can't read the original documents.
Einstein, Albert (November 25, 1915) http://nausikaa2.mpiwg-berlin.mpg.de/cgi-bin/toc/toc.x.cgi?dir=6E3MAXK4&step=thumb
it says "Das Koordinatensystem war dann nach der einfachen Regel zu spezialisieren, dab $\sqrt{-g}$ zu 1 gemacht wird, wodurch die Gleichungen der Theorie eine eminente Vereinfachung erfahren."
Which if I understand correctly is saying he will restrict himself to coordinate systems in which "$\sqrt{-g} \ is \ 1$" to simplify.

Einstein, Albert (1916). "The Foundation of the General Theory of Relativity". Annalen der Physik.
here he doesn't appear to require such restrictions on the coordinate systems to use the field equations (ie. the "simplification" above didn't really simplify the field equations; the field equations are true in general coordinate systems)
]


That is why I (and possibly bcrowell as well) was a little confused when Marcus summarized as:
"One simple way of looking at unigrav is that it is just the same as usual-GR except that the determinant of the metric must equal -1"

To make it clear, that is not really correct. The issue of these "extensions" is puting in other dynamical fields (ie. lambda is a scalar field instead of a constant, etc.), which can be seen in that arxiv paper linked in the openning post.

And now to respond to bcrowell's question:

I'm thinking this should relate somehow to something else that has been confusing me for a while. Here http://arxiv.org/abs/physics/9905030 is a translation of Schwarzschild's original paper on the Schwarzschild metric. (The translators have unusual ideas about black holes, as shown in the abstract displayed on arxiv, but I have no reason to think that this impugns the accuracy of their translation.) Schwarzschild's equation (5), which he describes as the "equation of the determinant" is $|g_{\mu\nu}|=-1$. The paper is from 1916, which is 3 years before the date when Marcus says Einstein published unimodular gravity. I'd never really known what to make of this equation. (Well, the whole Schwarzschild paper is not done in a way that anyone in 2010 would consider pedagogically optimal). I'd sort of figured that it must be a primitive, confused attempt simply to say, "Hey, the signature is -+++, not +++ or ++++. This is a pseudo-Riemannian metric, not a Riemannian one." It hadn't occurred to me that this was a much more strict condition than a simple constraint on the signature.

Any idea how this might relate?

Schwarzschild was using Einstein's approach from his first paper containing the field equations, and thus needed to restrict the coordinates so that |g_uv|=-1. This is why in his paper he used what appears to be a strange "trick" of defining some strange coordinate system ... this is to get the g_uv to have the right determinant.

Now adays we of course know he didn't have to do this. So it does look like a strange step. But the coordinate choice of course is mathematically valid. (And yes, in Schwarschild's original paper he mistook the coordinate singularities at the event horizon to be an indication that is where the mass is located and made a mistake. This was pointed out by Hilbert later and corrected. But the rest of his paper is fine.)

I hope that helps.

 

[bcrowell]

Thanks very much, JustinLevy, for your #12 -- very helpful!

-Ben

 

[marcus]

Thanks to everybody who has contributed so far to clarify the issues around UG.

The main issue, for me, is why didn't we already switch over to making UG our standard version of GR? So far I don't see any drawbacks to UG, and it is experimentally indistinguishable.

Enrique Alvarez has a nice paper about the potential for distinguishing (which would turn on "weighing the vacuum", as I take it)---a theoretical but impractical possibility. Any ideas?

It looks like it would cost nothing to make our preferred form of GR be Unimodular and it would help to resolve the main puzzles concerning Time and Lambda (the cosmological constant.) Anyone agree/disagree?

As a nice perspective on UG I will quote an excerpt of Alvarez' article.

== quote http://arxiv.org/abs/hep-th/0501146 ==
Although it does not seem to be generally known (see, however, a footnote in [16]), four
years after writing down the equations of general relativity, Einstein [7] also proposed
a different set of equations, what have subsequently been dubbed as corresponding to unimodular gravity. The... purpose was to obtain an alternative to Mie’s theory on the stability of the electron, and as such, it was unsuccessful...

But on the way, he realized already in 1919 that the unimodular theory is equivalent
to general relativity, with the cosmological constant appearing as an integration constant.

Let us quickly recall how this comes about.

The posited equations of motion are the tracefree part of Einstein’s general relativity
ones (written in dimension n):
R_µν − (1/n) Rg_µν = κ²(T_µν − (1/n) T g_µν)
(with κ² ≡ 8πG). It seems that there is less information here, because the trace has been left out, but this is deceptive: the contracted Bianchi identities guarantee that ...
==endquote==

He goes on and, in just a few lines, derives the familiar GR equation.

Alvarez does not get around to stating the Henneaux Teitelboim Lagrangian until page 13. It is his equation (75).
The neat thing about this HT action is that it is fully covariant. It is invariant under the whole diffeomorphism group, not just a restricted class of 'volume-preserving' ones.

Having some action like the HT action is necessary in order to quantize the theory. Einstein, when he invented UG in 1919, did not need a Lagrangian for his purposes so the question was not even considered. Basically there is a curious 70 year gap in the theory's history between Einstein 1919 and Henneaux-Teitelboim 1989.

 

[JustinLevy]

The main issue, for me, is why didn't we already switch over to making UG our standard version of GR? So far I don't see any drawbacks to UG, and it is experimentally indistinguishable.

This is probably the easiest to address.
There are many gravity theories deriving from action principles that, by appropriate parameter choice, fit all current data.

There's Brans-Dicke. There's the infinite family of f(R) theories. Etc.

GR is the standard classical theory, because it is the simplest. People have tried to quantize other gravity theories, but run into similar problems -- quantum field theory when the spacetime background itself is dynamic is difficult to make self-consistent. So the majority of work is trying to figure out tools for this, or figure out what can be added to make things more self-consistent (supersymmetry, etc.)

It looks like it would cost nothing to make our preferred form of GR be Unimodular and it would help to resolve the main puzzles concerning Time and Lambda (the cosmological constant.) Anyone agree/disagree?

I am not an expert in this field, so dissect my response if you need to, but I disagree.

In GR, the cosmological constant is just that: a constant. There is NO ambiguity.
Here instead

Consider instead:
The posited equations of motion are the tracefree part of Einstein’s general relativity
ones (written in dimension n):
R_µν − (1/n) Rg_µν = κ²(T_µν − (1/n) T g_µν)

Because this is tracefree, consider trying to solve for possible vacuum equations. Heck, consider trying to solve for the equivalent Schwarzschild solution. Unlike GR, you will find it is not possible to get a unique solution. Instead of the vacuum equation of GR, R = 4 Λ, in this new theory you will find R can be ANY value.

At least theories of "dark energy" replace Λ with some kind of scalar field whose state can be specified. Here instead, the curvature cannot be specified by the theory.

 

[tom.stoer]

I have to look at the LQG / SF approach for UG.

But as of today my favourite theory is Einstein-Cartan in connection formalism. The theory slightly differs from GR but is phenomenologically acceptable. Nevertheless I think that we will find out that there is a deeper relation between these different theories.

 

[JustinLevy]

But as of today my favourite theory is Einstein-Cartan in connection formalism. The theory slightly differs from GR but is phenomenologically acceptable.

These are more like questions than comments.

It was my understanding that EC comes from the same action as GR, but just allows more degrees of freedom (namely torsion). So if there is no torsion, then EC reduces exactly to GR.

Since the theories start from the same action, many attempts to quantize GR are actually, to get pedantic, quantizing the EC "extension" of GR. For example LQG historically was working towards a quantum theory of gravity starting with EC (it allowed torsion).

I've also heard that torsion is needed to allow non-scalar fields to couple with gravity (although some things I've heard suggest the real problem is just fermions). I'd be interested in understanding the degree to which this is true. Is this just the "natural" way, only straight forward way, to allow fermions to couple to gravity? Or is there a no-go theorem for torsionless GR + fermions? If so, that would rule out "Unigrav" already.

Again, these are more like questions than comments. Is my understanding of this correct?

 

[tom.stoer]

In EC theory a spin current acts as the source of torsion. But this torsion does not introduce new degrees of freedom as there is no propagating torsion field! Instead torsion can be expressed in terms of the spin current of the matter fields algebraically. That means that torsion vanishes outside matter; it does not propagate into the vacuum.

That's the reason why you can't detect it: inside matter (with spin) it's suppressed by another power of the gravitational constant, outside matter is exactly zero.

For an overview please check the following paper

http://arxiv.org/abs/gr-qc/0606062
Einstein-Cartan Theory
Andrzej Trautman
(Submitted on 14 Jun 2006)
Abstract: The Einstein--Cartan Theory (ECT) of gravity is a modification of General Relativity Theory (GRT), allowing space-time to have torsion, in addition to curvature, and relating torsion to the density of intrinsic angular momentum. This modification was put forward in 1922 by Elie Cartan, before the discovery of spin. Cartan was influenced by the work of the Cosserat brothers (1909), who considered besides an (asymmetric) force stress tensor also a moments stress tensor in a suitably generalized continuous medium.

 

[JustinLevy]

Yes, it doesn't propagate. But the field equations can be obtained from the same Einstein-Hilbert action, right? Without torsion, the connection can be obtained from the metric. So we just consider variations in the metric to obtain GR from the Einstein-Hilbert action. To get EC, we could consider varitions in the metric as well as variations in the torsion to get the EC field equations.

So yes, the torsion doesn't propagate, but to get different field equations out of the same action, we need to consider additional "degrees of freedom". It is extra degrees of freedom at least in this sense, no? That fact that it doesn't propagate comes from the resulting field equations, but we still needed to consider its variations to get these.

Well regardless, I guess I shouldn't have called them degrees of freedom... but there are at least more "states" that the connections describing spacetime can have now. So maybe the more appropriate way to state it is that for both we can consider the variation of the action with respect to the connections. We could then restrict the possible connections (no torsion) and essentially get the "Palatini variation" of GR.

So would it be more appropriate to say:
Both GR and EC-gravity have the same "degrees of freedom" (the connections), but GR adds an additional constraint on the connections to make it Riemannian geometry.

Or am I still missing the main point?

 

[tom.stoer]

So would it be more appropriate to say:
Both GR and EC-gravity have the same "degrees of freedom" (the connections), but GR adds an additional constraint on the connections to make it Riemannian geometry.

I think that's a good way to express ist.

 

[marcus]

To get back to your point about UG and EC:

The distinguishing feature of the various theories discussed in the literature under the general heading of Unimodular Gravity seems to be that any constant energy density is weightless.

In other words, a constant energy density does not couple gravitationally in these theories.

So, in particular, the very large (planckscale) estimated vacuum energy woud have no gravitational effect---because it is constant. The vacuum energy puzzle is what has typically been called the cosmological constant problem---or where several c.c. problems are discussed it is called the "first" or the main or the "direct" or the "old" c.c. problem. A number of prominent people going back to 1989 have discussed Unimodular Gravity (in various versions) as a possible solution to the cosmological constant problem.

I have been unable to find indications in the literature that some type of UG could not be implemented in Einstein-Cartan gravity. Can you point to something obvious that I'm missing here? Is there some reason to suspect that UG and EC don't mix?

 

[marcus]

If you want a more mathematically rigorous statement of the UG distinguishing characteristic it is:

"Contributions to the energy-momentum tensor proportional to the metric don’t couple to gravity!"

See slide #6 of Smolin's "Abhayfest" talk:
http://gravity.psu.edu/events/abhayfest/talks/Smolin.pdf

This was his contribution to a 60th birthday party symposium for Abhay Ashtekar June 2009.
The audio is also online and it includes Q&A with other LQG people. Smolin's Abhayfest talk is a nice clear introduction to UG and its relevance to the problems of time and Lambda. I will quote a few more slides:

==slide 8==
Unimodular gravity is not a new theory, it is a reformulation of GR.

Why isn’t this the solution to the first cosmological constant problem? Or, why isn’t the [observational] fact that Lambda is not Planck scale evidence that this is the right formulation of GR for quantum physics?

Weinberg discussed this in his 1989 review and said:
“In my view, the key question in deciding whether this is a plausible classical theory of gravitation is whether it can be obtained as the classical limit of any physically satisfactory [quantum] theory of gravitation."

We will study this problem and see that the answer is YES.

==endquote==

I have duplicated the highlighting from Smolin's slide and simply inserted the word [observational]. There seems to be clear observational evidence that UG is more correct than usual GR---UG does not suffer from this huge 10¹²⁰ discrepancy.

 

[atyy]

In the context of asymptotic safety all these different theories should arise in "full theory space" but are then renormalized such that (in certain regimes) the differences disappear. So one should have a look at this approach and then try to figure out how one can transport the main ideas to LQG. Therefore I do not only propose to quantize unimodular gravity according to the LQG approach; instead one should quantize "the full theory space" according to the LQG approach and try to understand what singles out the Einstein-Hilbert (Einstein-Cartan?) action from an QG perspective.

Yes, I think so too - at least from the old LQG perspective that gravity is not emergent.

 

[marcus]

I have to look at the LQG / SF approach for UG.

But as of today my favourite theory is Einstein-Cartan in connection formalism. The theory slightly differs from GR but is phenomenologically acceptable. Nevertheless I think that we will find out that there is a deeper relation between these different theories.

I think that is a good guess. Especially a deeper relation between Einstein-Cartan and Unimodular Gravity. Nature's idea of gravity is likely to be unimodular because we observe that the vacuum energy does not gravitate. And nature's gravity is also likely to resemble Einstein-Cartan (in quantum guise) because it must include fermionic matter. Intuitively, if nature's idea combines features of both these things, then there is apt to be a deeper relation between them.

I will repeat Smolin's question:

http://gravity.psu.edu/events/abhayfest/talks/Smolin.pdf ...
This was his contribution to a 60th birthday party symposium for Abhay Ashtekar June 2009.

Why isn’t the [observational] fact that Lambda is not Planck scale evidence that [unimodular gravity] is the right formulation of GR for quantum physics?
...

Or at least evidence that UG is less wrong. GR suffers from a huge 10¹²⁰ discrepancy because it thinks the vacuum energy should gravitate.

I think we are seeing a revival of interest in Unimodular because of its potential use as an heuristic guide.

 

[atyy]

Or at least evidence that UG is less wrong. GR suffers from a huge 10¹²⁰ discrepancy because it thinks the vacuum energy should gravitate.

Is this wrong prediction of 120 orders of magnitude a feature of LQC? Or of Asymptotic Safety?

It is a "problem" in handwavy inconsistent theories of QFT and quantum gravity, but is it a real problem - ie. does it show up in consistent quantizations of GR plus matter?

 

[marcus]

... but is it a real problem?

Obviously some people are now in the process of reformulating LQG (starting with the simple case of LQC) so it will have the unimodular property.

We won't know for a while how that works out. I gave a link already to the paper of Chiou and Geiller, in LQC. In that case it seemed to work out very successfully! They seem to gain something by having a natural "universe time" emerge, and being able to run the hamiltonian evolution on that time. (relational time in LQC has been workable but has made extra bother).

So in the Chiou Geiller case modifying the loop quantum theory to get unimodular LOOKS like pure gain. At least so far.

Now Smolin, in his August paper has pointed to how full spinfoam LQG might be modified to have the UG feature. Maybe this will work. Maybe not. We have to see. Smolin argues that it will be advantageous in the full theory case as well. He cites Chiou Geiller of course. And acknowledges discussions with Geiller, among others.

Again obviously, some people (quite reasonably in my view) take the main cosmological constant problem (the huge vacuum energy) as a "real problem". Others do not. We are talking heuristics---what different researchers choose to let guide the development of theory. LQG is developing comparatively rapidly at present, under various stimuli. We just have to see.

==========
BTW I expect that, over the next 18 months Rovelli will either post a paper showing interest in unimodular, or showing some reason to reject it. I have no sense of which way that will go. I would refuse to bet either way. His views tend to define the LQG mainstream. Smolin tends to be an outrider exploring off the beaten track.
What is significant, to me, is not the complementary behavior of these two major figures but the unanticipated behavior of the young guy, former postdoc working for Abhay Ashtekar, now in Beijing. If anything matters here it is the application, by Dah-wei Chiou, of unimodular to cosmology. That is what one can learn the most from, by concentrating on, I suspect.

 

[tom.stoer]

GR suffers from a huge 10¹²⁰ discrepancy because it thinks the vacuum energy should gravitate.

No, not really. This has not so much to do with GR. The huge value of the c.c. is not predicted by GR; in GR it's simply a constant you can chose as you like. The only question is why it's close tozero but not exactly zero.

It is still not clear whether the c.c. belongs to the left hand = gravity side of the Einstein equations (in which case it's a free parameter and its quantum corrections have to be calculated e.g. via a renormalization group approach) or to the right hand = matter side (in which case it's nt GR which predicts this huge number).

Even in the LQG community it's not so clear. Smoin proposed to use "non-local edges in spin networks" to mimic mismatch of micro and macro-causality which could explain the c.c. as an emergent phenomenon. On the other hand if you use framed spin graphs the c.c. is introduced via the quantum deformation, i.e. SU(2)_q. So it's not clear what it is.

There's another issue: what about other constants, e.g. from f(R) theories? Why are they small? You can't introduce a new theory for every new constant you want to explain. At some point we have to find a framework which predicts these coupling constants. I now that the c.c. is something special as it fits to the matter side as well, therefore it may have two different origins: 1) a parameter in GR, 2) quantum corrections in the matter sector. UG could solve the second problem for us, but is does not expalin what happens to all other possible constants arising in f(R) approaches.

 

[JustinLevy]

Marcus,
You keep flipping between classical and quantum issues, and making statements that I do not feel are really justified. Let's please approach this more systematically.

I would like to contradict two of your main claims:
1] the claim that unimodular gravity "solves" some problem with the cosmological constant
which comes at least in part from
2] the claim that in unimodular gravity, a stress energy tensor proportional to the metric does not gravitate (ie. non-zero vacuum energy)Let's start from a purely classical viewpoint.

If this is unigravity:
(eq 1 from the paper you linked http://arxiv.org/PS_cache/hep-th/pdf/0501/0501146v3.pdf )
$$R_{\mu\nu} - \frac{1}{n} R g_{\mu\nu} = \kappa^2 \left(T_{\mu\nu} - \frac{1}{n}T g_{\mu\nu} \right)$$
then let me comment again about its "cosmological constant" which you keep saying is so much better. It is actually much worse.

Given a stress energy tensor, try to solve for the Ricci curvature scalar and you will run into a severe problem that I already mentioned. The scalar curvature R is undefined. This is the case for any supplied stress energy tensor. The root problem is that the field equations are traceless so obviously cannot allow us to solve for R.

In GR, the cosmological constant is a parameter of the theory. The scalar curvature R can be uniquely obtained given a stress energy tensor. Instead in unimodular gravity, the best you can do is specify the scalar curvature up to an integration constant. There is NOT a unique answer to its field equations.

If you want unimodular gravity to actually be deterministic, you'd need to specify this integration constant as an actual parameter. Which puts us back to the same situation as in GR.

Furthermore, even with this integration constant, we get:
(eq 4 from the paper you linked http://arxiv.org/PS_cache/hep-th/pdf/0501/0501146v3.pdf )
$$\frac{n-2}{2d} R + \frac{2 \kappa^2}{d} T = constant$$
Which makes it abundantly clear that even a stress energy proportional to the metric will still gravitate (cause curvature).

It should be very clear that this doesn't solve anything with respect to the cosmological constant. If anything it is worse with respect to GR, because instead of getting a solution for R given a stress energy tensor, unimodular gravity cannot give a unique answer.

---

If you want to claim unigravity is just another formulation of GR, you can't also claim it makes different predictions for the same stress energy tensor. So if the vacuum has a non-zero stress energy tensor proportional to the metric, it must gravitate in both of them or neither of them.

Please stop making contradictory claims.

 

[marcus]

No, not really. This has not so much to do with GR. The huge value of the c.c. is not predicted by GR; in GR it's simply a constant you can chose as you like. The only question is why it's close tozero but not exactly zero.

It is still not clear whether the c.c. belongs to the left hand = gravity side of the Einstein equations (in which case it's a free parameter and its quantum corrections have to be calculated e.g. via a renormalization group approach) or to the right hand = matter side (in which case it's nt GR which predicts this huge number).

Even in the LQG community it's not so clear. Smoin proposed to use "non-local edges in spin networks" to mimic mismatch of micro and macro-causality which could explain the c.c. as an emergent phenomenon. On the other hand if you use framed spin graphs the c.c. is introduced via the quantum deformation, i.e. SU(2)_q. So it's not clear what it is.

There's another issue: what about other constants, e.g. from f(R) theories? Why are they small? You can't introduce a new theory for every new constant you want to explain. At some point we have to find a framework which predicts these coupling constants. I now that the c.c. is something special as it fits to the matter side as well, therefore it may have two different origins: 1) a parameter in GR, 2) quantum corrections in the matter sector. UG could solve the second problem for us, but is does not expalin what happens to all other possible constants arising in f(R) approaches.

Tom, you make a series of valid points. Much of each paragraph I take for granted without question.
Obviously if one looks at pure GR in isolation, the CC is just a constant. (Rovelli has a nice paper presenting a similar viewpoint "Why all these prejudices against a constant?") It is obviously not the fault of GR that QFT gets a big value for the vacuum energy!

And of course it is not clear if CC belongs on left or belongs on right!

And obviously there is disagreement among people we normally consider LQG community. And so on.
======================

I think if you examine what you have said here you will agree that what you are raising are issues of motivation, blame and responsibility. To paraphrase what I think is the sense:

Whose fault is it---GR or QFT? Which theory is to blame for this big discrepancy?

Is there a good motivation for LQG people to pursue this line of investigation (in addition to the other ideas they already work on)?

=======================

My primary aim is to report what I see going on, taking a position is secondary but can also help with the reporting.

My position in this case is as follows:

1. I think it is futile to imagine a permanent division between GR and QFT and to argue which is at fault, or who is responsible, or which side of the equation right or left. One admits there is a serious discrepancy regardless of which is responsible, and one tries different things.

2. I think there is good motivation to pursue the UG line of inquiry. This motivation is on several levels and is around two big issues: CC and TIME. One cannot understand the motivation for investigating UG unless one pays attention to the current discussion of time.

3. I have no idea how this will play out. I would not even estimate odds. I hope that, in the next 18 months, Rovelli will have something more to say about time and CC. I think UG is a major issue with potential to change the picture, so at this point I would advise against dismissing it.

BTW Rovelli posted papers this year about the CC ("why all these prejudices..." co-written with Bianchi) and about time (the temperature of space article, co-written with Smerlak). They are in tension/conflict with the line that Smolin is pursuing. Also Smolin has a book in preparation that proposes a revised view of time. I expect this book to appear also in the next 18 months and contribute to a kind of conceptual turbulence from which new ideas may surface.

 

[marcus]

Justin,

It's nice that you want to criticize some of the arguments and statements that I have been paraphrasing/quoting from the papers I have cited. I would like to call your attention to just these three and ask you to read them, and focus your comments on them. These are the central papers, as I see it, in what seems a significant revival of interest in UG (as it relates to the problems of both time and the cosmological constant.)

When you criticize, please quote exactly from the articles, and give page references. That way I can see what you are talking about. Do not paraphrase what you think they are saying, or what you think I am saying, since you can easily be confused as to the line of reasoning, and misrepresent.

I think sticking to these three will give a clear focus to what you say.

http://arxiv.org/abs/1007.0735
Unimodular Loop Quantum Cosmology
Dah-Wei Chiou, Marc Geiller
26 pages. Published in Physical Review D 82, 064012 (2010)
(Submitted on 5 Jul 2010)
"Unimodular gravity is based on a modification of the usual Einstein-Hilbert action that allows one to recover general relativity with a dynamical cosmological constant. It also has the interesting property of providing, as the momentum conjugate to the cosmological constant, an emergent clock variable. In this paper we investigate the cosmological reduction of unimodular gravity, and its quantization within the framework of flat homogeneous and isotropic loop quantum cosmology. It is shown that the unimodular clock can be used to construct the physical state space, and that the fundamental features of the previous models featuring scalar field clocks are reproduced. In particular, the classical singularity is replaced by a quantum bounce, which takes place in the same condition as obtained previously. We also find that requirement of semi-classicality demands the expectation value of the cosmological constant to be small (in Planck units). The relation to spin foam models is also studied, and we show that the use of the unimodular time variable leads to a unique vertex expansion."

http://arxiv.org/abs/1008.1759
Unimodular loop quantum gravity and the problems of time
Lee Smolin
14 pages
(Submitted on 10 Aug 2010)
"We develop the quantization of unimodular gravity in the Plebanski and Ashtekar formulations and show that the quantum effective action defined by a formal path integral is unimodular. This means that the effective quantum geometry does not couple to terms in the expectation value of the energy-momentum tensor proportional to the metric tensor. The path integral takes the same form as is used to define spin foam models, with the additional constraint that the determinant of the four metric is constrained to be a constant by a gauge fixing term. We also review the proposal of Unruh, Wald and Sorkin- that the hamiltonian quantization yields quantum evolution in a physical time variable equal to elapsed four volume-and discuss how this may be carried out in loop quantum gravity. This then extends the results of arXiv:0904.4841 to the context of loop quantum gravity."

http://arxiv.org/abs/0904.4841
The quantization of unimodular gravity and the cosmological constant problem
Lee Smolin
(Submitted on 30 Apr 2009)
"A quantization of unimodular gravity is described, which results in a quantum effective action which is also unimodular, ie a function of a metric with fixed determinant. A consequence is that contributions to the energy momentum tensor of the form of the metric times a spacetime constant, whether classical or quantum, are not sources of curvature in the equations of motion derived from the quantum effective action. This solves the first cosmological constant problem, which is suppressing the enormous contributions to the cosmological constant coming from quantum corrections. We discuss several forms of uniodular gravity and put two of them, including one proposed by Henneaux and Teitelboim, in constrained Hamiltonian form. The path integral is constructed from the latter. Furthermore, the second cosmological constant problem, which is why the measured value is so small, is also addressed by this theory. We argue that a mechanism first proposed by Ng and van Dam for suppressing the cosmological constant by quantum effects obtains at the semiclassical level."

 

[marcus]

It could be that the Bianchi Rovelli paper "Why all these prejudices against a constant?" actually disposes of the UG gambit. I'm not sure. In any case, the conclusions are well worth quoting:

==quote http://arxiv.org/pdf/1002.3966 conclusions==

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.

As mentioned in the introduction, it is good scientific practice to push the tests of the current theories as far as possible, and to keep studying possible alternatives. Hence it is necessary to test the ΛCDM standard model and study alternatives to it, as we do for all physical theories. But to claim that dark energy represents a profound mystery, is, in our opinion, nonsense. “Dark energy” is just a catch name for the observed acceleration of the universe, which is a phenomenon well described by currently accepted theories, and predicted by these theories, whose intensity is determined by a fundamental constant, now being measured. The measure of the acceleration only determines the value of a constant that was not previously measured. We have only discovered that a constant that so far (strangely) appeared to be vanishing, in fact is not vanishing. Our universe is full of mystery, but there is no mystery here.

To claim that “the greatest mystery of humanity today is the prospect that 75% of the universe is made up of a substance known as ‘dark energy’ about which we have almost no knowledge at all” is indefensible. Why then all the hype about the mystery of the dark energy? Maybe because great mysteries help getting attention and funding. But a sober and scientifically sound account of what we understand and what we do not understand is preferable for science, on the long run.

==endquote==

 

[JustinLevy]

When you criticize, please quote exactly from the articles, and give page references. That way I can see what you are talking about. Do not paraphrase what you think they are saying, or what you think I am saying, since you can easily be confused as to the line of reasoning, and misrepresent.

I think sticking to these three will give a clear focus to what you say.

Marcus,
This is the problem here. I am trying to get you to think and discuss the material instead of just grab and make all kinds of statements that are contradicting each other.

I already cited an article and equations (one that you previously brought up in fact). Instead of responding to the math and discussion, you just respond with more abstracts. That is the problem here. Please, take the time to think and discuss the material. Stop just avoiding contradictions by redirecting to more papers.

So fine, I'll abandon the previous papers I commented on, and focus on one of the three you listed there. But no more redirecting, okay? Let's focus on this one:
http://arxiv.org/abs/1007.0735

--------------
The action specifying the theory (eq 2.1):
$$S = \frac{1}{2\kappa} \int_\mathcal{M} d^4x \left[ \sqrt{-g} (R-2\Lambda+\mathcal{L}_m) + 2\Lambda \partial_\mu \tau^\mu \right]$$
Here, Λ is not a parameter like in GR (ie. it is not specified once and for all, but is a scalar field).
While it is not a huge issue (since it doesn't change anything regarding claims of whether a constant energy density gravitates or not), I think there is a typo in the equation. For based on their next comments in the paper, I believe they mean:
$$S = \frac{1}{2\kappa} \int_\mathcal{M} d^4x \left[ \sqrt{-g} (R-2\Lambda+ 2 \kappa \mathcal{L}_m) + 2\Lambda \partial_\mu \tau^\mu \right]$$
which is needed to make the field equations work out as they claim.

With that typo fixed (just missing that constant), it is mathematically straight forward to verify, as the paper says, "variations with respect to the metric yield Einstein’s field equations".
$$R_{\mu \nu} - \frac{1}{2} g_{\mu \nu} R + \Lambda g_{\mu \nu} = \kappa T_{\mu \nu}$$

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

Compare this to your claim:

The distinguishing feature of the various theories discussed in the literature under the general heading of Unimodular Gravity seems to be that any constant energy density is weightless.

In other words, a constant energy density does not couple gravitationally in these theories.

That is definitely wrong for the unimodular gravity theory in this paper. Can we at least agree on the math this far?

 

[marcus]

Justin, I am interested in what specific criticisms you might have of those three articles.
I would like to hear them, since those are the three UG articles that I am most interested in.
I asked that you not bring in my statements (which may or may not reflect what is going on with those three), but focus on those papers specifically.

You have chosen to look for the moment at the one by Chiou Geiller. What specifically, if anything, do you find amiss with that article, and its conclusions?

Or if you don't find anything wrong, can you point out anything of interest? I'd find other people's reactions to these new UG papers helpful, so I'd like to hear anything you can offer, whether positive or negative.

 

[marcus]

This is not intended to be part of my conversation with Justin, which I hope will focus on three recent UG papers of special interest. This is to clarify an earlier statement of mine to Tom Stoer:

...The distinguishing feature of the various theories discussed in the literature under the general heading of Unimodular Gravity seems to be that any constant energy density is weightless.

In other words, a constant energy density does not couple gravitationally in these theories.
...

Technically it's a statement about the energy-momentum tensor and what I meant by "constant" is equal to some constant times the metric---in other words "proportional to the metric" by some constant proportion. To make this more precise I followed up in the next post:

If you want a more mathematically rigorous statement of the UG distinguishing characteristic it is:

"Contributions to the energy-momentum tensor proportional to the metric don’t couple to gravity!"

See slide #6 of Smolin's "Abhayfest" talk:
http://gravity.psu.edu/events/abhayfest/talks/Smolin.pdf
...

 

[JustinLevy]

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.