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Unigrav applied to problems of time and cosmo constant

  1. Sep 19, 2010 #1


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    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 gµν 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)
    Last edited: Sep 19, 2010
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  3. Sep 19, 2010 #2


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    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.
  4. Sep 19, 2010 #3


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    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."
  5. Sep 19, 2010 #4


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    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 gµν with determinant fixed to:

    [tex]\sqrt{det(\bar{g}_{\mu\nu})} = \epsilon_0[/tex]

    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
    Tab → T′ab = Tab + ¯gab 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 Tµν of this form coming from the zero point energy of the fields,... are not sources of spacetime curvature...

    Last edited: Sep 19, 2010
  6. Sep 21, 2010 #5


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    I was misinformed about Claudio Bunster. Here is a Chilean magazine source:

    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?)


    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?
    Last edited: Sep 22, 2010
  7. Sep 22, 2010 #6


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    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.
    Last edited: Sep 22, 2010
  8. Sep 22, 2010 #7


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    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 [itex]|g_{\mu\nu}|=-1[/itex]. 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...?

  9. Sep 22, 2010 #8


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    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?
  10. Sep 22, 2010 #9


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    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 [Broken]
    (the first page is nearly blank, so you have to scroll down)

    The video for the talk is here:
    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 [Broken]

    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!
    Last edited by a moderator: May 4, 2017
  11. Sep 22, 2010 #10


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    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.
  12. Sep 22, 2010 #11


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    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:

    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]. ...

    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.

    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:
    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:
    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⋅)
    Last edited: Sep 22, 2010
  13. Sep 22, 2010 #12
    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 [itex]|g_{\mu\nu}|=-1[/itex]. 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 [itex]\sqrt{-g}[/itex] 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 "[itex]\sqrt{-g} \ is \ 1[/itex]" 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:
    Schwarzchild 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.
    Last edited: Sep 22, 2010
  14. Sep 22, 2010 #13


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

  15. Sep 23, 2010 #14


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    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):
    Rµν − (1/n) Rgµν = κ2(Tµν − (1/n) T gµν)
    (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 ...

    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.
    Last edited: Sep 23, 2010
  16. Sep 23, 2010 #15
    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.)

    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
    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.
    Last edited: Sep 23, 2010
  17. Sep 24, 2010 #16


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    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.
  18. Sep 24, 2010 #17
    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?
    Last edited: Sep 24, 2010
  19. Sep 24, 2010 #18


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    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

    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.
  20. Sep 24, 2010 #19
    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?
  21. Sep 24, 2010 #20


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