Thursday, December 12, 2013

The Comeback of Massive Gravity

Massive gravity, a modification of general relativity in which gravitons have mass, has an interesting history. Massive gravity was long believed to be internally inconsistent, but physicists at Stockholm University now claim to have constructed a consistent theory for massive gravity. This theory is a viable alternative to general relativity and can address some of its problems.

In Einstein’s theory of general relativity gravitational waves spread with the speed of light, and the quanta of the gravitational field, the gravitons, are expected to do the same*. To be more precise, gravitons move with the speed of massless particles because they are assumed to be massless. But whether or not a particle is indeed massless is in the end a question of experiment.

Neutrinos were long believed to be massless, but we know today that at least two of them have tiny non-zero masses (whose absolute value has not yet been determined). The mass of the photon is known to be zero to extremely high precision on experimental grounds. But what about gravity? This is a timely question because a small mass would lead to a long-distance modification of general relativity, and present observational evidence left physicists with some puzzles at these long distances, notably dark energy and dark matter.

However, to be able to even properly ask whether gravitons have masses, we need a consistent theory for massive gravity. But making gravitons massive is a challenge for the theoretical physicist. In fact, it was long believed to be impossible.

The problems start when you want to introduce a mass-term into general relativity. For vector fields, you can take a contraction of fields of the form AνAν to stand in front of the mass term. In general relativity the field is the metric tensor, and the only full contractions that you can create without using derivatives are constant: they create a cosmological constant, not a graviton mass. If you want a mass-term in general relativity you need a second two-tensor, that is a field which looks like a metric but isn’t the metric. Theories of this type are also known as ‘bi-metric’. Massive gravity is thus intimately related to bi-metric gravity.

But that’s only the beginning of the problems, a beginning that dates back more than 70 years.

In 1939, Fierz and Pauli wrote down a theory of massive gravity in the perturbative limit. They found that for the theory to be consistent – meaning free of ‘ghosts’ that lead to unphysical instabilities – the parameters in the mass-terms must have specific values. With these values, the theory is viable.

In 1970 however, van Dam and Veltman and, independently, Zukharov, showed that in the Fierz-Pauli approach, the limit in which the mass of the graviton is taken to zero is not continuous and does, contrary to naïve expectations, not reproduce general relativity. Any graviton mass, regardless how small, leads to deviations that can contribute factors of order one to observables, which is in conflict with observation. The Fierz-Pauli theory now seemed theoretically fine, but experimentally ruled out.

Two years later, in 1972, Vainshtein argued that this discontinuity is due to the treatment of the gravitational degrees of freedom in the linearization procedure and can be cured in a full, non-linear, version of massive gravity. Unfortunately, in the same year, Deser and Boulware claimed that any non-linear completion of the Fierz-Pauli approach reintroduces the ghost. So now massive gravity was experimentally fine but theoretically sick.

Nothing much happened in this area for more than 30 years. Then, in the early 2000s, the wormy can was opened again by Arkani-Hamed et al and Creminelli et al, but they essentially confirmed the Deser-Boulware problem.

The situation began to look brighter in 2010, when de Rahm, Gabadadze and Tolley proposed a theory of massive gravity that did not suffer from the ghost-problem in a certain limit. Needless to say, after massive gravity had been thought dead and buried for 40 years, nobody really believed this would work. The de Rahm-Gabadadze approach did not make many friends because the second metric was treated as a fixed background field, and the theory was shown to allow for superluminal propagation (and, more recently, acausality).

However, starting in 2011, Fawad Hassan and Rachel Rosen from Stockholm University (ie next door), succeeded in formulating a theory of massive gravity that does not suffer from the ghost instability. The key to success was a generalization of the de Rahm-Gabadadze approach in which the second metric is also fully dynamic, and the interaction terms between the two metrics take on a specific form. The specific form of the interaction terms is chosen such that it generates a constraint which removes the ghost field. The resulting theory is to best present knowledge fully consistent and symmetric between the two metrics.

(Which, incidentally, explains my involvement with the subject, as I published a paper with a fully dynamic, symmetric, bi-metric theory in 2008, though I wasn’t interested in the massive case and don’t have interaction terms. The main result of my paper is that I ended up in defense committees of Fawad’s students.)

In the last years, the Stockholm group has produced a series of very interesting papers that not only formalizes their approach and shows its consistency, but they also derived specific solutions. This is not a small feat as it is already difficult to find solutions in general relativity if you have only one metric and having two doesn’t make the situation easier. Indeed, not many solutions are presently known, and the known ones have quite strong symmetry assumptions. (More students in the pipe...)

Meanwhile, others have studied how well this modification of general relativity fares as an alternative to ΛCDM. It has been found that massive gravity can fit all cosmological data without the need to introduce an additional cosmological constant. But before you get too excited about this, note that massive gravity has more free parameters than ΛCDM, that being the coupling constants in the interaction terms.

What is missing right now though is a smoking-gun signal, some observation that would allow to distinguish massive gravity from standard general relativity and could be used to distinguish between both. This is presently a very active area of research and one that I’m sure we’ll hear more about.

* To be precise, in the classical theory we should be speaking of gravitational waves instead. The frequent confusion between gravitational waves and gravitons, the latter of which only exist in quantized gravity, is a bad habit but forgivable. Far worse are people who say ‘gravity wave’ when they refer to a gravitational wave. A gravity wave is a type of cloud formation and has nothing to do with linearized gravity.


  1. What I find confusing when reading about multi-metric theories is the degrees of freedom count, since it is not always mentioned up front. GR has two massless physical polarizations. What about these massive gravity proposals?

    Also, speaking of multi-metric theories. Long ago, Cutler and Wald (1987) wrote a paper that tried to classify consistent non-linear interactions of multiple spin-2 fields. Do the new massive gravity proposals constitute a special case of that classification, or do they fall outside it?

  2. Igor,

    Of course! Carefully counting the degrees of freedom is key to the whole problem. Basically, it looks a priori as you have too many propagating modes, but then the interaction terms are constructed such as to create an additional constraint. There is a longer elaboration on this in this paper, section 3. Best,


  3. Fawad Hassan and the Stockholm gang wrote a popular article about massive gravity in Populär Astronomi last year; maybe it's the first of its kind. Here's a link to the pdf version: Tyngre tyngdkraft tar oss bortom Einstein.

  4. Does massive gravity do away with horizons or singularities?

  5. The massive graviton is the natural consequence of mass-energy equivalence: if the gravitons is formed with curved space-time, this curvature has some energy density assigned, so it must have some mass density assigned too.

    /* Does massive gravity do away with horizons or singularities? */
    Not horizons but singularities. It essentially says, the space-time curvature has some material density and gravity field. But during collapse of massive bodies the highest curvature of space-time and gravity field isn't at the center of massive bodies, but at their surface and when this massive body will collapse, under certain moment the mass density of gravity field would violate the buoyancy condition. Such a massive object wouldn't collapse anymore and it will change into undulating quantum waves. This object will be probably dense enough for being surrounded with event horizon, but still less dense than any pin-point singularity. It's physical surface undulating beneath the event horizon is probably equivalent to firewall concept, postulated recently.

  6. Zephir,

    "...this curvature has some energy density assigned..."

    There is no such thing as the energy density of the curvature. This is an old result from GR --- one cannot define the concept of energy for a gravitational field, locally. It can be done only in a suitable global sense (spacetime with asymptotic global time-translation symmetry).

    HTH, :-)

  7. Sabine, this is why I find it confusing. Is it necessary to read the whole paper to find the answer? It should have been somewhere in the abstract. So what's the answer: 2 polarizations, 5 polarizations, some other number?

  8. Igor,

    If you read the abstract of the paper, you can find out that the authors are giving the proof that massive gravity with the auxiliary scalar fields (the Stuckelberg formulation) has the same number of degrees of freedom as does the massive gravity without those scalar fields.

    They don't need to actually calculate the number of degrees of freedom in order to provide that proof.

    HTH, :-)

  9. If I understand,in GR it is impossible to replace CC by a field (function of space,time). Thus one has to deal with the situation that there are two different quantities with a similar purpose: (1)CC at this time to help with repulsive expansion and (2)Inflaton at the inflation time to cause rapid exponential expansion. Does the massive gravity theory point to a solution of this problem, or one has to live with these two as separate issues?


    The graviton exists with non-zero rest mass, Big G (exponentially?) decreases with (large) distance. Observed galactic cluster dynamics takes a hit. If deep relativistic gravitons, then graviton Einstein rings. Intense gravitators (large mass pulsars) would anomalously cool through graviton evaporation.

    SUSY, then the gravitino, then theory to create it (TeV/c^2 is occupied. PeV/c^2 ?) and render it empirically sterile - certainly during post-Big Bang nucleosynthesis that otherwise would be gone for a Burton, and most certainly now. A see-saw mechanism for gravitons....

  11. @vmarko
    Scalar gravitation empirically fails unless it has a massive carrier to severely constrain the scalar coupling distance,

    Science 340(6131) 448 (2013)
    Physics Today 66(7) 14 (2013)

    How many more curve-fitting particles must clothe the Emperor?

  12. /* cannot define the concept of energy for a gravitational field, locally..*/

    Of course it's hyperdimensional stuff violating the equivalence principle in 4D. So you need to consider more general coordinate system (analogous to second quantization in QM) for it. The resulting massive field is the source of another curvature and massive field, so that this approach is actually implicit and infinitely nested.

  13. Malte,

    Thanks for the link, I didn't know about this! Yes, I hope that maybe we'll see an article in English at some point. Best,


  14. Arun,

    In my understanding the causal structure is somewhat unclear in these scenarios. I suspect that in the end it depends on how you couple the matter. Best,


  15. Igor,

    In the case with a fixed background it's 5. The link that I gave elaborates on why the 6th isn't propagating. Best,


  16. kashyap,

    "in GR it is impossible to replace CC by a field (function of space,time)"

    That's wrong. You can replace the CC by a field. This generalization is called 'dark energy'. The CC is a special kind of dark energy. This earlier post should clarify this. Best,


  17. Hi Bee, Thanks for the reply. I take it that models other than CC can be arranged to give covariance and the reason they are not popular is that they do not add anything phenomenologically (experimentally). Is this correct? Is Lambda a constant for 13.8 B years? How about the second question-does Lambda (or some quitessence field) have anything to do with inflation?


  18. Wouldn't the missing smoking-gun signal be "gravitons".

    Until they are observed, of how much value are these conjectures.
    Fun to play with maybe, but one must be careful to remember that all this is pure speculation until "gravitons" are observed.

    Gravity waves appear almost anywhere fluid dynamics applies.

  19. " This is presently a very active area of research and one that I’m sure we’ll hear more about."

    I'm not so sure. The [very serious] problems you mentioned have been piling up and nobody has made them go away.

  20. /*..until they are observed, of how much value are these conjectures..*/

    In AWT the gravitons are already routinely observed with CMBR noise. Even at the water surface you cannot have transverse wave without some longitudinal wave attached to it and vice-versa. They're coupled with every photon and they're the reason why gamma ray photons remove the matter from stars during supernovae explosions. The massive character of photons is responsible from clustering of photons inside of distant gamma ray bursts, which do follow the Lorentz invariance in this way. Somewhat ironically, just the violation of equivalence principle with massive gravitons is required for restoring of Lorentz symmetry for these photons and vice-versa. It's not difficult to imagine it, but from perspective of mainstream theories rigor it's a pure mess.

  21. This comment has been removed by the author.

  22. @MarkusM: Yep, the rest mass of graviton would represent the energy of photon of wavelength, which wouldn't allowed to move it, i.e. the photon fitting the diameter of the observable Universe. The dynamic/relativistic mass of graviton would correspond the energy of CMBR photon (Landauer's limit represents an energy of approximately 0.0178 eV).

  23. Rastus: Which problem are you referring to that hasn't gone away?

  24. Markus,

    Yes, that's where the number comes from, and it's also the order of magnitude expected for the graviton mass. Best,


  25. Try to think about it in conceptual way: if the stars would radiate the energy in harmonic waves, there would be no apparent way, how they could radiate their matter during this. The loss of matter with radiation comes just from the fact, the light waves are fragmented into many tiny packets of higher space-time curvature, i.e. the photons. The existence of gravitons is therefore hidden right there. In linearized, Einstein–Maxwell theory on flat spacetime, an oscillating electric dipole is the source of a spin-2 field.

    The true is, if the light would fulfill the Maxwell's theory exactly, it would spread like pure transverse waves with no effective mass, which is just the situation, for which the special relativity has been derived originally - but the light doesn't spread so.

    So there's no need to afraid of massive photons, because the photons are concept of quantum mechanics and the general relativity has actually nothing to say about these quantum artifacts - it just deals with Maxwellian flat waves.

    The situation is slightly complicated with the fact, the space-time itself is full of CMBR photons, so that the photons of CMBR wavelengths are still behaving like flat Maxwell waves and the photons of larger wavelength are essentially a tachyons inside of such environment and they do decompose fast - but the above principle remains the very same.

  26. Is it known if there is a well-posed IVP (Cauchy problem) in the bimetric version of drgt?

  27. I wonder if it is a good idea to use the phrase "the graviton". It makes it sound like the hypothetical particle actually exists, whereas there is not a shred of evidence that it does.

    Using the phrase "the putative graviton" may be cumbersome (but not really that much considering the risks of encouraging false beliefs) but I think we would be better off if it was crystal clear when we are discussing things that are pure speculation, and when we are discussing things that have some empirical evidence supporting them. Need I mention 44 years of faith-based string theory?

  28. The concept of massive gravity doesn't require the graviton, just the mass-energy equivalence. The existence of graviton follows just from consequential application of this principle, which is probably impossible to achieve in analytical way. And not all gravity is mediated via quantized gravitons, with massive gravitons the more. The concept of massive graviton is implicit (what mediates the mass of graviton - another graviton? Etc..)

    And vice versa, the existence of graviton still doesn't mean, that the gravity must be considered massive. Even photons can be considered massless, because they do propagate at "infinite distance". We are just neglecting their finite life-time due to quantum decoherence. If the graviton or photon would oscillate, it is really the same massless particle like at the moment of its origin? Aren't we rather speaking of infinite many particles here?

    Such a questions are conceptual things, which must be judged arbitrarily. IMO the future physics will be very implicit and independent of absolute truth ("if we consider this, this and this, then the result will be this - but not otherwise").

  29. Leo,

    I don't know but I am guessing there is. They have two metrics, but only one manifold. As long as you have at least one direction on which both agree is timelike I'd think it should work. Best,


  30. Robert,

    Please read the footnote of my post. Gravitons are the quanta of the graviational field. You can define them in perturbatively quantized gravity, which is a theory that we know exists, irrespective of the question what its uv completion looks like. Whether they 'really' exist is a question of experiment, and you know that I am cautiously optimistic that we will eventually be able to experimentally confirm quantum gravity.

    Referring to every theoretical concept that has not been experimentally verified with an adjective documenting this lack of observation would be infinitely cumbersome. Best,



  31. Cumbersome? Yes, a bit depending on how many tooth fairies one discusses in one sitting.

    "Infinitely cumbersome? - No.

    Scientists used to be a lot more careful in their discussions, carefully distinguishing the empirically-based from pure speculation.

    I think the problem has developed because we now deal with so many purely speculative concepts, models, particles, interactions, etc.

    A lovely example was posted to today: arXiv:1312.3636v1 [astro-p.CO]. Pure fantasy from start to finish and everything discussed as if it were tested reality.

  32. It breaks diff invariance. GR people will not be happy. Their precious background independence is lost:-)


    In a KK the zero mode is massless and the tower is massive. How you get rid of the zero mode?

  33. Giotis,

    It doesn't break diff invariance, not sure why you think so. I also don't know what KK tower you're referring to. Best,


  34. How is that possible? A massive graviton breaks diff invariance.

    For the other question I was trying to figure out how you can get a massive graviton (at lowest order) from a maseless graviton in higher dimension via KK compactification/reduction.

  35. How does it do that?

    As to the KK case. If you can do a massive graviton in flat space, I don't know what prevents you from adding compactified dimensions that give you excitations over that, but not sure that's what you mean.

  36. There is the standard way to go from the linearized action to the full non linear action with general covariance. This method works only with a maseless graviton.

    I will try to find references for this when I find the time.

  37. Giotis,

    Maybe you should actually read the papers I was referring to in my post, because you clearly haven't. For the non-linear generalization you introduce the second metric. The full action can be found eg in this paper, Eq (3). Best,



  38. OK I checked a bit...

    diff invariance is broken of course but it seems they restore it using the Stukelberg trick.

    You could just say that instead of asking why I think diff is broken.

    Anyway it seems to me that the Stukelberg trick is just that, a trick, since this way you can restore the gauge symmetry in any theory.

    But indeed I have to read more about this theory...

  39. My third sentence doesn’t make sense.

    What I meant to write is:

    “You could have just said that instead of asking why I think diff is broken”

  40. Giotis,
    Sorry for being vague, I was trying to gauge (excuse the choice of word :p) how much you had read of the papers rather than guessing the reason for your conviction. Best,

  41. Bee,
    In these massive graviton theories, do gravitational waves travel slower than speed of light? If so, is it
    consistent with the lower limit on speed of gravity
    calculated by Caves(1980) and Moore-Nelson(2001)
    and ?

    Or is it that the above argument cannot be applied?

    Also there are limits on massive graviton from binary pulsar(
    I presume the required graviton mass needed to
    explain dark energy is much smaller than this bound?

  42. "I presume the required graviton mass needed to
    explain dark energy is much smaller than this bound?"

    Where did you get the idea that a massive graviton has anything to do with dark energy?

  43. Philip,
    I thought one of the motivations of these models
    is an alternative to standard Lambda CDM?

  44. "I thought one of the motivations of these models
    is an alternative to standard Lambda CDM?"

    Maybe. That's why I want to know where the idea came from.

    Of course, not all alternatives are worth pursuing. Sometimes the cure is worth than the disease. Actually, I am among those who think that there is no "problem" at all. See, for example, this paper.

  45. Shantanu,

    It's an interesting question, but not one there is an easy answer to. To begin with, in the bi-metric gravity setting there are two propagating fields, one massless and one massive, but the metrics are (if I recall correctly) actually superpositions of these (ie not mass eigenstates). So I don't think the constraints can be applied just by looking at the numbers. Second, the paper you refer to actually has a constraint on the speed, not on the mass. If the mass is non-zero the effect on the speed depends on the energy, thus the constraint is dependent on the situation (I didn't read the paper). Third, my gut feeling would be that for the typical mass-scales expected for the graviton (of the order of the Hubble-radius), the effect on the propagation speed is way beyond experimental precision in the regime where you can measure it. But maybe it's something worth looking into. I am not aware that somebody did it before. Best,


  46. Phillip,

    Yes, one of the motivations to look into this is to have an alternative to LambdaCDM. You want there to be modifications on supergalactic scales, but not screw up solar system physics. That roughly tells you the ballpark for the graviton mass. There are in my counting 3 different cosmological constant problems, which one are you referring to? Best,


  47. "There are in my counting 3 different cosmological constant problems, which one are you referring to?"

    Let's see:

    1. What is dark energy?

    2. Why is its energy density roughly comparable to that of matter (coincidence problem)?

    3. Why is its energy density much smaller than QFT estimates on the back of the proverbial envelope (classical "cosmological constant problem").

    Perhaps there are others.

    The paper argues that none of the three is really a problem, at least not a problem with the cosmological constant.

    I really hate the term "dark energy". It was chosen to make things seem more sensationalistic than they are. Yes, one should check whether the equation of state w really is exactly -1 and does not change with time, but since no evidence indicates otherwise, I think we should continue to think of it as the cosmological constant until proven otherwise.

  48. There is a slight inaccuracy in crediting the authors. The key element to avoid the BD ghost in massive gravity was introduced by de Rham-Gabadadze-Tolley, ie, the theory was constructed by dRGT, hence the prevalence use of the term "the dRGT model" in the active community.

    Hassan and Rosen proved that the dRGT model is free of the BD ghost. They also showed that promoting the second metric to be dynamic, ie, a bi-metric dRGT-like model, does not re-introduced any BD ghost.

  49. Phillip Helbig wrote: "I really hate the term "dark energy"."

    What term do you like better? It's good to have a term referring to the observed phenomena; "cosmological constant" doesn't work for that, because it refers to a specific theoretical attempt to explain these phenomena - with the built-in assumption that something is constant.

    I've heard "invisible tension" as an attempted better term, but it's not catching on. I think "dark energy" has won, permanently.



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