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Blackforest's question about George Ellis inflation paper

  1. Jun 27, 2013 #1


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    Blackforest asked about an interesting paper recently posted by G.F.R. Ellis, that MTd2 spotted and added to our bibliography.
    I hope we can have some discussion of the Ellis paper. Unimodular gravity is a variant of the usual GR which is restricted to metrics which do not feel "vacuum energy". In other respects it reproduces all the usual GR behavior.

    So it is a way of avoiding the fact that Quantum Field Theory (developed on flat Minkowski space with no gravity, thus in a very different context) predicts an embarrassingly huge vacuum energy which we do not observe. In unimodular GR this energy (if it existed) would have no effect.

    Ellis "trace-free" is, he says, essentially the same as unimodular GR. So the good news is it does not suffer from the absurd QFT vacuum energy. But then what happens in the supposed inflation era when one WANTS an inflaton field to behave somewhat like this high vacuum energy?

    Ellis addresses this question.
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  3. Jun 27, 2013 #2


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    Hey! This paper of Ellis is beautiful! Thanks MTd2 and Blackforest for making sure we read it.
    Lee Smolin had an 2009 paper on the trace-free version of GR, aka unimodular gravity. I remember starting a thread about it. It neutralizes the gross QFT overestimate of vacuum energy. Ellis shows that you still get inflation as long as the "inflaton" scalar field is changing. So the expansion effect of the scalar field only disappears when it becomes constant---then the Friedman equation will ignore it, as it does any constant vacuum energy.
    Ellis paper is beautifully written, he makes his explanations and equation steps clear with loving care. To use a phrase of Frank Wilczek, this "deserves to be true". I mean it already is true mathematics---it deserves to be true about nature.

    Smolin's paper gives motivational context, so you can see why Ellis paper matters. Google
    "smolin unimodular"

    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.
    Comments: 22 pages, no figures

    Unimodular loop quantum gravity and the problems of time
    Lee Smolin
    (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.
    Comments: 14 pages
    Last edited: Jun 27, 2013
  4. Jun 27, 2013 #3


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    Amusing side aspect to this: some devoted string folks WANT there to be a cosmological constant problem because then you can have MULTIVERSE TO THE RESCUE and the anthropic principle can explain why we have this very small positive cosmological constant. This is a roundabout argument for string theory because it gives us a Landscape of some unimaginably large number of different versions of physics.

    Ellis explains that he was at the Cambridge "Infinities and Cosmology" workshop in 2013 and Michael Douglas was unhappy with tracefree gravity, noting that it gets rid of the cosmological constant problem, which otherwise, by motivating multiverse proposals, can serve as "an important motivation for the correctness of string theory" :biggrin: These are Ellis words on the top of page 2.

    So as Ellis recounts on page 1, Michael Douglas, a great expert on the String Landscape, was arguing against tracefree gravity: it could not be right because (according to him) it could not undergo inflation. The present Ellis paper represents Ellis spelling it out very clearly that on the contrary it can undergo inflation.

    In the first paragraph of the paper Ellis consoles Douglas (and likeminded folk) by recalling an alternate motivation for string theory, namely its promise to unify all the forces of nature in a TOE :smile: So there is no need to lament the loss of the cosmological constant problem (because other multiverse/string motivations can be found). One sees what a nice person Ellis is.

    In case anyone is curious, here is a link to "Infinities and Cosmology" workshop:
    Last edited: Jun 27, 2013
  5. Jun 28, 2013 #4
    Thanks so much for the considerable efforts in developping the context. So far I follow the debate, with that article, the ball stays now on the "stringists" side. So let us wait for the next set. I unfortunalety didn't get here a direct answer to my question concerning, not the multiverse but, the multimass densities. To contrebalance the discrepancy between inertial and gravitational mass dentities one his mathematically obliged to "play" with the volumes... Does it indirectly imply that we have an inertial and a gravitational volume:confused:? I certainly miss something and return to my schoolbooks!:smile:
  6. Jun 28, 2013 #5


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    As I understand it, what Ellis calls gravitational and inertial mass densities in Eq 21 are simply his names for the different terms appearing in Eq (6) and (7), which follow from the EFE (Eq 1), the perfect fluid stress energy tensor (Eq 5), and an homogeneous isotropic expanding universe (Eq 4). There is no violation of the principle of equivalence, which technically means that the stress-energy tensor is the "generally covariant" version of its form in special relativity. So basically to check that the equivalence principle is satisfied, don't worry too much about his naming convention, instead look at Eq (5) and see if it closely resembles Wikipedia's form for a perfect fluid in flat spacetime.

    Given this form, at a point, the stress energy tensor in curved spacetime can be made to look exactly as in flat spacetime, ie the gravitational field seems to disappear. This is the general relativistic version of "gravitational mass being the same as inertial mass", because in Newtonian gravity the equality of gravitational and inertial masses means all things fall at the same rate, so over a very small region, a group of falling objects are stationary relative to each other, and it seems as if the gravitational field has disapeared.

    Ellis wishes to give different names such as "gravitational" and "inertial" to the terms in Eq (6) and (7) which assume the EFE, because he wishes to stress that the corresponding Eq (24) and (25) are different in the case of the TFE.
    Last edited: Jun 28, 2013
  7. Jun 28, 2013 #6


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    Wow, very very very interesting. The idea is so simple and elegant on first thought it looks like a tautology.
  8. Jun 29, 2013 #7
    I agree, but would add another few very's ! If all is correct and, as Ellis says, "cosmology no longer has a vacuum energy problem" (because gravity is unimodular), another implication of unimodular gravity should be considered.

    This is, I think, that "a length scale Lmin>0 exists below which no physical process can probe any smaller length scale L<Lmin", as Tom Stoer writes in post 36 of another interesting thread here, which poses the question "Is space-time discrete or continuum?"

    Have I got this right? I'm not familiar with the unimodular gravity that Einstein discussed in 1919.
  9. Jul 1, 2013 #8
    As I mentioned in starting the discussion: the equation of state for the vacuum (let us call it “the Eos”) and the expansion actually are two enigmas for the scientific community (whatever our personal opinion is). The work which is presently discussed does not provide any explanation for the Eos (this is not a critic). The latter is accepted for what it is, as a fact, and, in fact, as a peculiar case.

    My three doubts concerning the TFE:
    A) This is what I personally find to be a first embarrassing point. Although that Eos may be seen as a special feature of a more general equation, it does not take into account the fact that empty regions are far to be an exception in our universe. But ok, this is not a scientific argument; this is merely a statement: the most frequent type of volumes in our universe is described by the simplest Eos.

    B) Now, the second embarrassing point with unimodular gravity is the implicit logic involved in it. Let me explain. All the work which has been made, starting somewhere with Gauss, Riemann and E. B. Christoffel, going then via J.C. Maxwell, Morley and Michelson and finally ending with A. Einstein yields the EFE. The latter are the most general achievement of a long road and they are supposed to describe gravity. I think anybody agrees with that.

    Now, if one affirms that the EFE are just containing the gravitation as a special feature and if that special feature is correctly described with just a part of the EFE, namely the TFE, then (next logical interrogation): “What do the EFE really describe which is containing TFE, thus gravity?” Already here, one gets the feeling that the EFE would describe more than was expected.

    It is known that the TFE allows a junction with the gauge field theories. Well, that is good for the (quest of) unity but it does not answer the previous question: “What does the EFE really describe?” If (with all respect) G.F.R. Ellis is right, since the EFE have the reputation to be correct (numerous experiments confirm that affirmation), then the EFE are not exactly what we until now though they were, but quite more... but what*?

    C) The third embarrassing point was contained in my first intervention and concerns the fact that (the zero mode of) the vacuum energy seems to violate the equivalence principle. I did my own research and discovered a recent article [hep-th/1301.5130v3, 21 March 2013] and a reference in it eliminating my objection (= the unimodular gravity approach avoids the problem) because (citation of the resume of the first reference herein; there are): “Some models (…) in which the strength of the gravitational coupling of the potential energy relative to the same coupling for the kinetic energy is, in a precise sense, adjustable. The gauge symmetry of these models consists of those coordinate changes with unit Jacobian”…]

    Nevertheless, 3 questions stay in my mind:
    1°) “Why should a part of the energy not gravitate?” This is finally the consequence of the proposed approach. The latter seems to be governed by a logic which can be roughly compared with the following: “I have a problem with the dust in my flat, then I put the dust under the carpet and my wife will not see it!” Either the vacuum energies related to the different scales exist because they are the logical consequences of the diverse theories predicting them (and we were able to measure them in some ad hoc experiment confirming their existence) and the equivalence principle is respected... then all kinds of energy should gravitate... or they don’t all exist...

    2°) Unimodular gravity is related to area preserving diffeomorphisms. Does it give us some indications related to my second doubt* (The EFE contains more than gravity but what?)?

    3°) Is the unimodular gravity approach the unique answer to the third objection (the violation of the equivalence principle)?
  10. Jul 2, 2013 #9
    This stays in my mind as well.

    The tiny Casimir effect, which has nevertheless been measured, shows that part of the Vacuum energy exists in amounts that agree with quantum electrodynamics, the original, quintessential quantum field theory. We have also known since 1918 that electromagnetic waves carrying energy/mass gravitate (Eddington's eclipse expedition observations), just as Einstein's ordinary general relativity requires.

    Can't we then conclude that the electromagnetic part of vacuum energy has been proven to gravitate? And if so, why shouldn't its other parts? Aren't they also described by quantum field theories that are cousins to electrodynamics?

    I guess there's a lot that I don't follow about Ellis' elegant analysis, and/or about existing physics!
  11. Jul 4, 2013 #10
    On a completely minor note, I found it interesting to see the relative contributions to the vacuum energy density, cosmological constant, calculated via QFT: [ these appear as equations 9-12] 99.9999…....% of the energy is from Planck scale fluctuations; contributions from electroweak and QCD are insignificant at 63 and 72 decimal places smaller, respectively.

    I had never before seen the relative contributions.
  12. Jul 5, 2013 #11
    On an even more minor note, might one then conclude from Ellis' paper that virtual mass/energy (made of vacuum fluctuations, or virtual particle-antiparticle pairs) that dominates the cosmo-constant, doesn't gravitate? Or perhaps that the antí-mass/energy component falls up?
  13. Jul 5, 2013 #12
    Funny you mention that because I wondered if Ellis was negating dark energy as the cosmological constant, but neither is what he says. I had to go back and reread portions.

    First he explains how even though the scalar potential V (ϕ) doesn't appear, he can introduce it indirectly via it's derivative. Then he says it's effect is the same as the cosmological constant:

    Footnote Edit: You can blame Marcus for this kind of thing if you like; I do! From time to time, just when you think you might be beginning to understand something, he comes along with an interesting paper like this and causes a lot of additional work!!!
  14. Oct 5, 2013 #13
    Ellis' paper seems to me a real sleeper -- bits have become stuck in my mind and grow more
    interesting with every encounter. Ellis shows that 'quantum zero-point energy' doesn't gravitate when taken in the context of an everywhere isotropic and homogeneous fluid model universe. Which may explain why 'dark energy', which is believed to help flatten the geometry of our universe, is much, much too small to be simply identified with quantum zero-point energy. Ellis explains, concerning this energy, that:

    Would universe-wide quantum zero-point energy then be detectable only if it varied from location to location in spacetime, rather like gravitational potential energy in 19th century physics? But could it vary over space sections? Or with time? If so, could it render 'space' or the vacuum of spacetime somehow elastic?
  15. Oct 5, 2013 #14


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  16. Oct 6, 2013 #15
    Interesting paper but one thing I don't see mentioned is that the TFE are not generally covariant as they favor unimodular coordinates over the rest, is this principle not considered important in gravitational theories anymore?
  17. Oct 6, 2013 #16


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    Thanks for the comment! To give some background clarifying the issue I'll quote from Smolin's 2009 paper on unimodular, in a moment. But first to answer directly, I don't see how anyone can speak for the collective research community but my impression is that general covariance is certainly taken very seriously!

    Nevertheless even though a principle is considered of fundamental importance, researchers are free to consider conscious departures from it! In this case one still has diffeomorphism invariance just under a restricted class of diffeomorphisms---those preserving a volume element.

    And certainly this is controversial! Only SOME researchers will wish to pursue the consequences of this limited departure from general covariance. In fact Einstein offered the unimodular version of his theory back in 1919 (as I think you probably know T.D.) but it received only limited attention over the years. Why? Smolin's paper gives some reasons it is controversial and some historical background.

    ==quote page 3 of http://arxiv.org/abs/0904.4841 ==
    Here we note that a theory making use of this mechanism is almost as old as general relativity. This is unimodular gravity, first written down by Einstein[4] in 1919. Unimodular gravity[4]-[12] modifies general relativity by imposing a constraint that the metric of spacetime, gab, have a fixed determinant. This has the effect of reducing the gauge symmetry from full spacetime diffeomorphism invariance to invariance only under diffeomorphisms that preserve this non-dynamical fixed volume element. In spite of these differences, the field equations are the same as general relativity. Only now the cosmological constant, Λ, is a constant of integration rather than a parameter of the lagrangian.

    If one asks why this kind of approach to the cosmological constant problem has not been more fully considered, part of the reason is that the quantization of unimodular gravity has remained obscure. For example, as discussed by Unruh in [8], there are additional constraints that complicate the construction of the quantum theory. But as the first cosmological constant problem concerns suppressing large quantum corrections, it must be solved in the context of a quantum theory. This means that the symmetry (1) has to be satisfied by the full quantum equations of motion, which follow from the quantum effective action. However, as Weinberg pointed out, it is not clear whether there is any theory whose quantization yields a quantum effective action which is a functional of the unimodular metric[6]. This question is resolved affirmatively here.

    To give a well defined construction of the path integral for unimodular gravity, we follow Henneaux and Teitelboim[11] in making a background density, which is not always written down in discussions of unimodular gravity, but must be there for the action principle to be sensible, into a dynamical field...
    Last edited: Oct 6, 2013
  18. Oct 6, 2013 #17
    Thanks, Marcus.
    I did read something(way back when) about 1919 Eisnstein's unimodular gravity, but to give a little more historical background this came from even further back, from november 1915 when Einstein was struggling to find the correct EFE, just before he found them he published the equations in unimodular form, this form was the one used by Schwarzschild to obtain the first exact solution in vacuum. These equations, by allowing only unimodular coordinate transformations produce a solution that could not be extended and led him to believe his solution was singular at the Schwarzschild radius, not allowing the current notion of a black hole defined by its event horizon. So here is an instance were limiting the general covariance of the theory could have practical consequences.
  19. Oct 8, 2013 #18
    But the universe, presumably, was "a priori" not isotropic and homogeneous at the begining (Big Bang). This is sheding some shadow on the generality of the scenario.

    Does at the end Ellis article propose a concrete link between the general relativity and the quantum theory? I don't believe it. As I understand it: the conclusion only is that the general relativity is "too general" for the observed data concerning that point (vacuum energy)... I am not really convinced I must say.

    Why not? But if yes, this is putting the discussion on another "philosophical" ground and pushing it into the direction of a "everywhere and at any time" creationism (the elasticity of the universe explaining the birth of gravitation and matter?). Oups... and what with our Big Bang?
  20. Oct 8, 2013 #19


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    Why does this cast a shadow? It is not necessary to assume "isotropic and homogeneous".

    The point is that if the TFE is the true equation of space-time geometry (rather than the standard GR equation) then vacuum energy is powerless to effect geometry.

    It is powerless (according to TFE) whether or not the geometry is "isotropic and homogeneous."

    We are not talking about a "scenario", Blackforest, but simply about two equations: TFE or usual GR. Which is the true one? We do not know yet.

    It's interesting, so I will quote the conclusions paragraph of the Ellis paper:

    ==quote Ellis http://arxiv.org/abs/1306.3021 ==
    If the true gravitational field equations are the TFE (14), implying that only the trace-free part of the energy–momentum tensor Tab of matter is gravitating, then the effective cosmological constant Λˆeff is a constant of integration that is arbitrarily disposable (as in classical General Relativity), and, hence, is independent of any fundamental value assigned to the vacuum energy. This solves the major problem of a huge contradiction between the calculated vacuum energy and the cosmologically observed effective value of the cosmological constant. Any huge Λvac is powerless to affect cosmology, or indeed the solar system, as the vacuum energy will not affect spacetime geometry.
    It has been suggested that there are problems with this proposal, because on the face of it, it also disempowers the scalar field potential from having any effect on inflationary dynamics. This paper has shown that concern is unjustified: that [scalar field] potential is indeed able to influence inflationary dynamics as in the standard theory. It is just its zero point value that does not matter - as is the case in general for potential energy.

    The solar system is not homogeneous and isotropic, and yet (if the TFE is the right equation of gravity/geometry) a vacuum energy is powerless to affect the solar system.

    We have already had some discussion here of Unimodular Gravity (the earlier papers of Smolin and of Enrique Alvarez, which Ellis cites.) They make it clear that the powerlessness result, about vacuum energy, is very general.
    This is not the main issue Ellis is addressing. His main point is that the same does not apply to a slow-roll scalar field. I don't completely understand his argument. Why is a slowly varying scalar field not like vacuum energy? Maybe if you study his equations you can get the gist of it and explain it to us.
  21. Oct 8, 2013 #20
    The article is developped around the equation (19) which is (citation) the stress energy tensor of a perfect fluid.

    If you google in internet around "vacuum, equation of state..." you may find articles relating observations proving that the trace of the energy tensor does not vanish.

    Please give me a reasonable delay for that... I should be sleeping now: its midnight here :smile:
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