Thursday, March 02, 2017

Yes, a violation of energy conservation can explain the cosmological constant

Chad Orzel recently pointed me towards an article in Physics World according to which “Dark energy emerges when energy conservation is violated.” Quoted in the Physics World article are George Ellis, who enthusiastically notes that the idea is “no more fanciful than many other ideas being explored in theoretical physics at present,” and Lee Smolin, according to whom it’s “speculative, but in the best way.” Chad clearly found this somewhat too polite to be convincing and asked me for some open words:

I had seen the headline flashing by earlier but ignored it because – forgive me – it’s obvious energy non-conservation can mimic a cosmological constant.

Reason is that usually, in General Relativity, the expansion of space-time is described by two equations, known as the Friedmann-equations. They relate the velocity and acceleration of the universe’s normalized distance measures – called the ‘scale factor’ – with the average energy density and pressure of matter and radiation in the universe. If you put in energy-density and pressure, you can calculate how the universe expands. That, basically, is what cosmologists do for a living.

The two Friedmann-equations, however, are not independent of each other because General Relativity presumes that the various forms of energy-densities are locally conserved. That means if you take only the first Friedmann-equation and use energy-conservation, you get the second Friedmann-equation, which contains the cosmological constant. If you turn this statement around it means that if you throw out energy conservation, you can produce an accelerated expansion.

It’s an idea I’ve toyed with years ago, but it’s not a particularly appealing solution to the cosmological constant problem. The issue is you can’t just selectively throw out some equations from a theory because you don’t like them. You have to make everything work in a mathematically consistent way. In particular, it doesn’t make sense to throw out local energy-conservation if you used this assumption to derive the theory to begin with.

Upon closer inspection, the Physics World piece summarizes the paper:
which got published in PRL a few weeks ago, but has been on the arxiv for almost a year. Indeed, when I looked at it, I recalled I had read the paper and found it very interesting. I didn’t write about it here because the point they make is quite technical. But since Chad asked, here we go.

Modifying General Relativity is chronically hard because the derivation of the theory is so straight-forward that much violence is needed to avoid Einstein’s Field Equations. It took Einstein a decade to get the equations right, but if you know your differential geometry it’s a three-liner really. This isn’t to belittle Einstein’s achievement – the mathematical apparatus wasn’t then fully developed and he was guessing its way around underived theorems – but merely to emphasize that General Relativity is easy to get but hard to amend.

One of the few known ways to consistently amend General Relativity is ‘unimodular gravity,’ which works as follows.

In General Relativity the central dynamical quantity is the metric tensor (or just “metric”) which you need to measure the ratio of distances relative to each other. From the metric tensor and its first and second derivative you can calculate the curvature of space-time.

General Relativity can be derived from an optimization principle by asking: “From all the possible metrics, which is the one that minimizes curvature given certain sources of energy?” This leads you to Einstein’s Field Equations. In unimodular gravity in contrast, you don’t look at all possible metrics but only those with a fixed metric determinant, which means you don’t allow a rescaling of volumes. (A very readable introduction to unimodular gravity by George Ellis can be found here.)

Unimodular gravity does not result in Einstein’s Field Equations, but only in a reduced version thereof because the variation of the metric is limited. The result is that in unimodular gravity, energy is not automatically locally conserved. Because of the limited variation of the metric that is allowed in unimodular gravity, the theory has fewer symmetries. And, as Emmy Noether taught us, symmetries give rise to conservation laws. Therefore, unimodular gravity has fewer conservation laws.

I must emphasize that this is not the ‘usual’ non-conservation of total energy one already has in General Relativity, but a new violation of local energy-densities does that not occur in General Relativity.

If, however, you then add energy-conservation to unimodular gravity, you get back Einstein’s field equations, though this re-derivation comes with a twist: The cosmological constant now appears as an integration constant. For some people this solves a problem, but personally I don’t see what difference it makes just where the constant comes from – its value is unexplained either way. Therefore, I’ve never found unimodular gravity particularly interesting, thinking, if you get back General Relativity you could as well have used General Relativity to begin with.

But in the new paper the authors correctly point out that you don’t necessarily have to add energy conservation to the equations you get in unimodular gravity. And if you don’t, you don’t get back general relativity, but a modification of general relativity in which energy conservation is violated – in a mathematically consistent way.

Now, the authors don’t look at all allowed violations of energy-conservation in their paper and I think smartly so, because most of them will probably result in a complete mess, by which I mean be crudely in conflict with observation. They instead look at a particularly simple type of energy conservation and show that this effectively mimics a cosmological constant.

They then argue that on the average such a type of energy-violation might arise from certain quantum gravitational effects, which is not entirely implausible. If space-time isn’t fundamental, but is an emergent description that arises from an underlying discrete structure, it isn’t a priori obvious what happens to conservation laws.

The framework proposed in the new paper, therefore, could be useful to quantify the observable effects that arise from this. To demonstrate this, the authors look at the example of 1) diffusion from causal sets and 2) spontaneous collapse models in quantum mechanics. In both cases, they show, one can use the general description derived in the paper to find constraints on the parameters in this model. I find this very useful because it is a simple, new way to test approaches to quantum gravity using cosmological data.

Of course this leaves many open questions. Most importantly, while the authors offer some general arguments for why such violations of energy conservation would be too small to be noticeable in any other way than from the accelerated expansion of the universe, they have no actual proof for this. In addition, they have only looked at this modification from the side of General Relativity, but I would like to also know what happens to Quantum Field Theory when waving good-bye to energy conservation. We want to make sure this doesn’t ruin the standard model’s fit of any high-precision data. Also, their predictions crucially depend on their assumption about when energy violation begins, which strikes me as quite arbitrary and lacking a physical motivation.

In summary, I think it’s a so-far very theoretical but also interesting idea. I don’t even find it all that speculative. It is also clear, however, that it will require much more work to convince anybody this doesn’t lead to conflicts with observation.


  1. Thanks Dr. H. Another good read. Doesn't QFT allow for local and temporary violations of energy conservation? Perhaps as with the process resulting in black hole evaporation some of these "temporary" violations become locked in?

  2. Hi Sabine,

    many thanks, this post is interesting and very refreshing (which is so rare in physics :).

    The paper is of particular interest to me because I think it has direct relation with the one I pointed to you a few weeks ago. The point is that writing the non-conservation equation for a start (and then maybe understanding what it means), you get straight to the densities of matter, dark matter and dark energy. Plus other goodies like MOND.

    But of course it requires to start with classical GR and not a straight jump to quantum fields.


  3. Sabine,

    "It is also clear, however, that it will require much more work to convince anybody this doesn’t lead to conflicts with observation."

    It seems to me that "much more work" would be welcomed by GR theoretical physicists.

  4. Baryogenesis, non-classical gravitations, dark matter, and SUSY are empirical failures. Add dark energy. If physics must be fantastical, perhaps physics should be different, Aristotle versus Galileo.
    ...Empress Eugénie's Circle is a suspension bridge with no anchorages.

  5. One thing that confuses me is that, in Einstein's seminal 1916 'Grundlage' paper, the condition [root (-g) = 1] is imposed in quite a few instances. It looks like he is using uni-modular gravity, but I presume he's just using the condition as a way of simplifying things, is that right?
    Regards, Cormac

  6. Professor,

    I'm not much of a historian, so I can't answer your question, sorry - maybe some of our readers can answer your question. What I recall is that Einstein tried various equations and ways to arrive at these equations and he was missing a constraint (the Bianchi identities), so it's quite plausible he'd have tried this.

    Having said that, note that the Schwarzschild metric in standard coordinates has deg g = r^2 sin(theta)^2, which you could set to one when you take cartesian coordinates. It's quite a natural way to fix the gauge which isn't the same as fixing the variation. Best, B.

  7. @Professor R: In section 14c, 'The Final Steps', of the superb Einstein biography 'Subtle is the Lord', Abraham Pais writes:

    "The remaining flaw was, of course, Einstein's unnecessary restriction to uni-modular transformations. The reasons which led him to introduce this constraint were not deep, I believe. He simply noted that this restricted class of transformations permits simplifications of tensor calculus."

    So your presumption seems to be correct.

  8. It's not entirely settled whether Unimodular gravity differs from GR's prediction at the quantum level. This goes back and forth endlessly in the literature.

    At the very least, its not clear what you gain when trying to solve the cosmological constant problem. There is still a finetuning problem, the difference is -they say- that there is only one number to explain, and not an entire renormalization tower of unknown physics which tends to drag you (order by order) towards a Planckian value.

  9. In 1918 Hermann Weyl derived a restriction-free form of the traceless Einstein equations by simply assuming R^2 in the Lagrangian rather than R. Since the energy-momentum tensor for the electromagnetic field is already traceless, such a straightforward method of getting traceless equations seems to be the way to go. It would thus appear that Weyl had effectively discovered unimodular gravity long before anyone else, and without assuming a fixed metric determinant.

  10. Sabine/others: Does unimodular gravity have different observational consequences in solar system and binary pulsars as compared to GR? Which of its PPN parameters differ from GR?

  11. Shantanu,

    As I said, unimodular gravity plus energy conservation reproduces GR so it has no observable consequences. And if you relax energy conservation you want to make really sure it doesn't have observable effects that spoil the fit to data. Not much is being said in the paper about this, more work is needed etc. Best,


  12. And, yes, what Haelfix says above is correct, there is a long back and forth in the literature about whether or not quantizing unimodular gravity helps with the cosmological constant problem by taming vacuum fluctuations, but the calculations in the paper above doesn't depend on the quantization. Best,


  13. Hi

    Don't know if you already commented that new article from the same authors : they now claim they can relate the value of the cosmological constant to the top mass
    arxiv 1711.05183.pdf

    Anyway i still have a problem with equation 7 and footnote 3 of the precursor article you are commenting on this page (arxiv: 1604.04183).

    In all other papers about UG, eventually rewritten as an Einstein equation the energy momentum is conserved as usual on the right side and the cosmological constant is a mere integration constant (which is interesting because it can be anything you want and you could absorb in that way the QFT vacuum energy ).

    But keeping it in the form of equation 7 with a non conserved energy momentum on the right hand side , then the cosmological constant term is not actually anymore a constant (it varies in time). What bothers me even more is i dont understand what gives you the right to consider that the equation being written in that way , the cosmological constant term on the left hand side (which depens on Q) is really what you can probe in cosmological tests : in actual cosmological tests the acceleration of the universe is the effect that appears as a departure from equations in which the energy-momentum of matter is conserved ... so it is only the departure from the equation written according footnote 3 (as usual for UG) that should be interpreted as a cosmological constant effect ... what did i miss ?!


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