In Einstein's general relativity, gravitational interactions are mediated by a massless spin-2 particle - the so-called

*graviton*. This is what gives it its hallmark properties: the long range and the universality. One obvious way to screw with Einstein is to add mass to the graviton, as entertained already in 1939 by Fierz and Pauli. The Particle Data Group quotes the constraint

*m ≤*6*10^−32 eV, so we are talking about the De Broglie wavelength comparable to the size of the observable universe. Yet even that teeny mass may cause massive troubles. In 1970 the Fierz-Pauli theory was killed by the van Dam-Veltman-Zakharov (vDVZ) discontinuity. The problem stems from the fact that a massive spin-2 particle has 5 polarization states (0,±1,±2) unlike a massless one which has only two (±2). It turns out that the polarization-0 state couples to matter with the similar strength as the usual polarization ±2 modes, even in the limit where the mass goes to zero, and thus mediates an additional force which differs from the usual gravity. One finds that, in massive gravity, light bending would be 25% smaller, in conflict with the very precise observations of stars' deflection around the Sun. vDV concluded that "the graviton has rigorously zero mass". Dead for the first time...

The second coming was heralded soon after by Vainshtein, who noticed that the troublesome polarization-0 mode can be shut off in the proximity of stars and planets. This can happen in the presence of graviton self-interactions of a certain type. Technically, what happens is that the polarization-0 mode develops a background value around massive sources which, through the derivative self-interactions, renormalizes its kinetic term and effectively diminishes its interaction strength with matter. See here for a nice review and more technical details. Thanks to the Vainshtein mechanism, the usual predictions of general relativity are recovered around large massive source, which is exactly where we can best measure gravitational effects. The possible self-interactions leading a healthy theory without ghosts have been classified, and go under the name of the dRGT massive gravity.

There is however one inevitable consequence of the Vainshtein mechanism. The graviton self-interaction strength grows with energy, and at some point becomes inconsistent with the unitarity limits that every quantum theory should obey. This means that massive gravity is necessarily an effective theory with a limited validity range and has to be replaced by a more fundamental theory at some cutoff scale 𝞚. This is of course nothing new for gravity: the usual Einstein gravity is also an effective theory valid at most up to the Planck scale MPl～10^19 GeV. But for massive gravity the cutoff depends on the graviton mass and is much smaller for realistic theories. At best,

So the massive gravity theory in its usual form cannot be used at distance scales shorter than ～300 km. For particle physicists that would be a disaster, but for cosmologists this is fine, as one can still predict the behavior of galaxies, stars, and planets. While the theory certainly cannot be used to describe the results of table top experiments, it is relevant for the movement of celestial bodies in the Solar System. Indeed, lunar laser ranging experiments or precision studies of Jupiter's orbit are interesting probes of the graviton mass.

Now comes the latest twist in the story. Some time ago this paper showed that not everything is allowed in effective theories. Assuming the full theory is unitary, causal and local implies non-trivial constraints on the possible interactions in the low-energy effective theory. These techniques are suitable to constrain, via dispersion relations, derivative interactions of the kind required by the Vainshtein mechanism. Applying them to the dRGT gravity one finds that it is inconsistent to assume the theory is valid all the way up to 𝞚max. Instead, it must be replaced by a more fundamental theory already at a much lower cutoff scale, parameterized as 𝞚 = g*^1/3 𝞚max (the parameter g* is interpreted as the coupling strength of the more fundamental theory). The allowed parameter space in the g*-m plane is showed in this plot:

Massive gravity must live in the lower left corner, outside the gray area excluded theoretically and where the graviton mass satisfies the experimental upper limit

*m*～10^−32 eV. This implies g* ≼ 10^-10, and thus the validity range of the theory is some 3 order of magnitude lower than 𝞚max. In other words, massive gravity is not a consistent effective theory at distance scales below ～1 million km, and thus cannot be used to describe the motion of falling apples, GPS satellites or even the Moon. In this sense, it's not much of a competition to, say, Newton. Dead for the second time.

Is this the end of the story? For the third coming we would need a more general theory with additional light particles beyond the massive graviton, which is consistent theoretically in a larger energy range, realizes the Vainshtein mechanism, and is in agreement with the current experimental observations. This is hard but not impossible to imagine. Whatever the outcome, what I like in this story is the role of theory in driving the progress, which is rarely seen these days. In the process, we have understood a lot of interesting physics whose relevance goes well beyond one specific theory. So the trip was certainly worth it, even if we find ourselves back at the departure point.

## 27 comments:

I’m going to ask what may well be a stupid layman’s question.

You say that “In Einstein's general relativity theory, gravitational interactions are mediated by a massless spin-2 particle”. I kind of understand why you may want to incorporate gravity into a system where forces are mediated by particles but when I was taught GR – many, many years ago – gravitons were never mentioned. Are gravitons really essential or can GR happily operate (unmodified and where quantum effects can be ignored) without anything other than the field equations?

It's true that few GR books present this point of view. You can certainly go very far without mentioning gravitons, much like you can do classical electromagnetism without knowing about photons. However, the graviton picture is very illuminating, at least for particle physicists. If you postulate the existence of a massless spin-2 particle coupled to matter, then the entire GR (universality, equivalence principle, Einstein field equations, etc) follows automatically just from consistency conditions of the quantum theory at the leading order in 1/Mpl expansion. In the classical regime the spin-2 graviton picture is completely equivalent to the usual GR equations of motion, but it is more general and more fundamental as it also covers quantum phenomena (as long as the relevant energy is below the Planck scale).

Thanks for a great post. I really appreciate the nice summaries you give of topical areas in physics.

Gravitons are indeed present in General Relativity! Of course, it is something not usual in the usual approach to GR teaching yet! Bronstein, russian pioneer, derived the gravitational wave quadrupole formula using basically quantum mechanical rules and gravitons. The point is that General Relativity is an effective theory, like the SM, when we are far away from Planck energy. Thus, as everyone knows, you can linearize the theory and got gravitational waves (even you can consider non-linear gravitational waves) and moreover, as long as you give up certain assumptions, you can get quantum corrections to the gravitational potential too. Of course, this subject is not very known. Even I don't think many people know that there are interesting processes like gravitons turning into photons and another particles that could be very interesting to study for astrophysics and cosmology....

Vainshtein working on something introduced by Einstein sounds like a delayed April fools' joke.

"Whatever the outcome, what I like in this story is the role of theory in driving the progress, which is rarely seen these days."

You're right in a way. On the other hand, it would have been even better if people realized right from the outset that massive gravity, bigravity, and a lot of other things like that are obvious bullshit.

It amazes me that, as GR has gone from triumph to triumph over the last few years, there has been an explosion of papers claiming that it is wrong, all the way down to blatant crackpottery like "entropic gravity". Why not just admit the obvious and move on?

Albert Vielesteine:

Disproving the crackpot theories led to better understanding of the correct theory. Without the crackpot theories, may be this knowledge would never be gained.

It is the general theory of relativity, not the theory of general relativity. The theory is general, not the relativity. GR is an acceptable shorthand, but if you are using the full name get he words in the right order. They do not commute.

"Disproving the crackpot theories led to better understanding of the correct theory."

Sorry, no. All we have learned is that a variety of obviously stupid theories are in fact wrong. That is no surprise.

Graviton mass can be bounded using data from binary black hole mergers, with current estimates placing the bound at about m(g) < 10 ^(-23) eV, see

https://arxiv.org/abs/1706.01812

Any modified theory of gravity, including Massive Gravity must be compatible with these results (and future refinements coming out of gravitational wave astronomy).

Suppose that GR eventually has to be modified, perhaps in the manner of MOND, but that no graviton particle with the necessary properties exists. Presumably, string theory predicts the graviton, in the sense that it predicts that every force is accompanied by a particle? So would that invalidate string theory?

String Theory predicts that, there is one particle, that is a loop (no extremety) that its mass is 0, its charge is 0 and its spin is 2 --> it is the expected proporties of the graviton.

Maybe in the landscape of string theory, there could be massive graviton... there are 10^500 possibilities...

Great article.

In the Galileon case, note that small perturbations on top of a spherically symmetric scalar field profile sourced by a point mass (i.e., toy model for the Sun) can propagate at superluminal speeds. This, to the old-fashioned me, already indicates the theory ought to be thrown into the dustbin.

Additionally, since the Galileon action is schematically $(\partial \Pi)^2$ times a power series in $\partial^2 \Pi/\Lambda^3$, where $\Pi$ is the Galileon itself and $\Lambda$ is its mass scale; and since, when evaluated on this same spherically symmetric profile sourced by our Sun, $\partial^2 \Pi/\Lambda^3 \sim 10^7-10^8$ for $\Lambda \sim 1/10^3$ km; it is quite possible that quantum corrections such as $\sum_n \chi_n \Lambda^4 (\partial^2 \Phi/\Lambda^3)^n \sim \sum_n \chi_n \Lambda^4 (10^7 - 10^8)^n$ (there are other terms) would not admit even an asymptotic series, which in turn indicates such fifth force theories that are supposed to exhibit Vainshtein screening do not, in fact, work as intended: quantum corrections are overwhelming precisely in the same nonlinear regime the screening is supposed to kick in. I'm not an expert, though, on what fraction of these objections apply to massive/dRGT gravity proper, even though I understand Galileons are supposed to arise from its decoupling limit in some manner.

@Yi-Zen Chu, actually, it's known that the one loop corrections in the galileon theory do resum nicely despite the large value of $Z=\partial^2\Pi$, since they depend only on Z'/Z and Z''/Z, so that large value of Z and small gradients (i.e. small $\partial^3\Pi$ relative to \partial^\Pi) are ok. For higher loops the resummation is not known, although long ago it was speculated a power counting in https://arxiv.org/abs/hep-th/0404159 that would perhaps make sense to all loop order.

The constraints from superluminality are also certainly worrisome indeed (although people always complain about them); but the constraints from the dispersion relations that produced the plot in jester's post are stronger.

Is there a possible flaw in the vDVZ disconuity argument? I have suggested that the empirical successes of MIlgrom's Modified Newtonian Dynamics (MOND) might be explained by MOND-chameleon particles having variable effective mass depending upon nearby gravitational acceleration. These hypothetical MOND-chameleon particles would be the dark matter MOND-compatible analogue of the hypothetical dark energy chameleon particles.

Chameleon particle, Wikipedia

If gravitons were massive spin-2 particles having variable effective mass depending upon various local environmental circumstances, then the vDVZ discontinuity argument might be invalidated.

Anonymous: Thanks for the input! Could you point me to the re-summed form? The quantum corrections I wrote down was in fact one of those written down by Nicolis and Rattazzi (i.e., the hep-th/0404159 you cited above). There was a follow-up explicit computation in arXiv:1310.0187 that appeared to confirm their form. However, is it not true these terms they wrote down arise from pure Galileon self-interactions; whereas Galileons -- being the 5th force "modifier" of the equivalence principle respecting gravitational force -- ought to couple to everything in sight, and therefore would receive all sorts of unknown quantum corrections, including those that would break the Galileon symmetry itself? That is, does the re-summed form take into account Galileon-matter interactions? Additionally, even if the re-summed form is known, doesn't it have to change under Renormalization Group flow -- is that understood?

I hope Jester does not view such discussions as off-topic.

Yi-Zen Chu: take e.g. Eq.57 and 58 in Nicolis-Rattazzi, as you can see the log-diverging terms depend only on m^2 which is a function of Z'/Z and Z''/Z. Those are the calculable RG running effects from pure galileon interactions.

The power-diverging terms are not calculable i.e. they are UV sensitive, parametrising the effect of the UV completion rather than runnning effects. In a sense they tell you how the coupling to matter will enter in loops. So, it's true that coupling to matter (and perhaps higher than one- galileon loops) could destroy the consistency of the theory, but I think it's fair to say that this has not been fully established in one sense or in another. Moreover, the non-trivial invariant galileon terms are not even renormalized even by matter loops that respect the galileon symmetry as it was pointed out in https://arxiv.org/abs/1606.02295. Finally, let me remark that the bounds Jester is pointinng out are IR sensitive as they apply to massive gravity (which contains a galileon mode) but not to Horndeski-like theory that have a massless graviton plus a galileon.

"I kind of understand why you may want to incorporate gravity into a system where forces are mediated by particles but when I was taught GR – many, many years ago – gravitons were never mentioned. Are gravitons really essential or can GR happily operate (unmodified and where quantum effects can be ignored) without anything other than the field equations?"

True GR can operate just fine without gravitons. And, while massless spin-2 gravitons are equivalent to GR in a suitable classical limit, there are some distinct way that theories that have massless spin-2 gravitons are different from GR:

1. In GR, gravitational potential energy is not localized, in a graviton theory it is.

2. In GR, energy is conserved locally but not globally, in a graviton theory, local energy conservation implies global energy conservation.

3. In GR, there is no quantum tunneling through singularities, in a graviton theory, there is.

4. In GR, gravitational field energy is not an input into the stress-energy tensor even though gravitational field energy does influence the predictions of GR by a more indirect route in the field equations; in a graviton theory, the strength with which a graviton couples to any other particle with mass-energy is universal and so gravitons self-interact with each other with a a coupling strength proportionate to their mass-energy as an input on the same footing as any particle with mass-energy. In my view, the claim that these different mechanism produce exactly equivalent phenomenology has not been rigorously proven (in part because of the extreme mathematical difficulties associated with doing calculations involving a massless spin-2 graviton for which the usual tricks of Standard Model quantum mechanical calculations don't work).

5. GR is deterministic, a graviton theory is stochastic.

6. In GR, the cosmological constant is easily added as part of the field equations as an integration constant. The cosmological constant is not nearly so natural an addition to a graviton theory.

I'm sure that there are other technical distinctions as well, but the point is that there are basic, undeniable, qualitative differences between GR using the field equations, and a graviton theory.

For the most part, they are identical in phenomenology. For example, massless gravitons and gravitational effects in the GR field equations both propagate at the speed of light and both give rise to gravitational wave phenomena. But, they aren't truly identical and the ways in which a graviton based quantum gravity differs from GR are critical to reconciling the Standard Model and GR, which is why developing a theory of quantum gravity is such a major objective.

andrew: could you give sources or arguments for your six points?

Mitchell,

Just a short note --- for a reference, I always recommend the MTW book to particle physicists who studied GR from the Feynman's book. The latter just doesn't do justice to GR, despite being written by Feynman.

Also, regarding Andrew's points (1), (2) and (4), it is all a consequence that in the graviton theory the general metric of curved spacetime is being rewritten as a spin-two field on top of the flat Minkowski metric. The presence of the Minkowski metric gives rise to all those spurious things like localized gravitational energy, global conservation law and graviton stress-energy. All those things are undefinable in curved spacetime.

Regarding (3) and (5), GR is a classical theory, not quantized. OTOH, the graviton theory can be quantized, but is nonrenormalizable, and thus ill-defined.

Finally, (6) is also very interesting, since when you add a nonzero CC into GR, flat Minkowski spacetime fails to be a vacuum solution of the theory, so it doesn't even make sense to rewrite the metric as a flat metric plus a spin-two field.

In general, forcing GR into the formalism of perturbative QFT in flat spacetime is a disastrous idea, and it doesn't really work. The fact that Feynman pushed for that approach is responsible for much of misunderstanding of GR by particle physicists. Feynman even gave up on the idea eventually, but his "Feynman's Lectures on Gravitation" is still doing the damage, IMO.

HTH, :-)

Marko

@ vmarko,

I don't think it's fair to blame "Feynman's Lectures on Gravitation" for the early attempts to quantize GR using the approach that led to the successful development of QED. Reality is that many obstacles to the implementation of this approach were either unknown or partially known in 1962.

Sure, i'm not doing the blame-game here. Feynman certainly thought he was doing a good thing, and I agree that back then most of the issues with the perturbative gravity approach were not known.

However, ever since then, many other (much better and more informative) books have been written about GR (like the MTW book), and Feynman's book should have been universally obsoleted by now. But the problem lies in sociology of particle physicists, who consider Feynman a last-word on every topic, including the topics for which he was certainly not good enough. Even today, a junior hep-th student picks up Feynman's book, reads it, and knowing that Feynman was a genius, believes that he has learned all there is to say about GR. And nobody informs the student that he chose a wrong book to learn from. That's the damage I was talking about.

Best :-)

Marko

This discussion in the comment section, of classical GR vs the corpuscular quantum description in terms of graviton looks surreal to me. GR is just a classical theory, therefore just an approximation (an extremely good one in most of the cases) of a quantum theory. My understanding is that it's not true, as stated by andrew, that GR is just fine, precisely because it's not a quantum theory. (For example, the Einstein equations, as they are in GR, are just meaningless since the T_{\mu\nu} is an operator in QM, one should amend the theory and explain to take expectaion values in certain limits...)

Similarly, matter fields are quantum and the states of matter can be entangled, while they are sourcing a classical gravitational field??

In contrast, the underlying theory obeying quantum mechanics can be certainly formulated at low-energy, as a quantum EFT, in terms of corpuscular gravitons (and not just in flat space, although in completely generic non-particularly symmetric backgrounds may be hard). Moreover, string theory provides an (the?) example that such low-energy quantum EFT can be extended to a quantum theory valid to all energy scales. Since this corpuscular quantum theory of gravity reproduces GR at low-energy, within the accuracy of any current and past (and most likely near future) experiment (and it is a quantum one), it's a better theory than GR itself. Of course, there are challenges to this picture, such as the cosmological constant problem, which ironically emerges not in the UV, but rather at large distances in the deep IR. But they are not real mathematical contradictions of the quantum gravity theory. I think that not finding a graviton would be a more revolutionary discovery that finding one experimentally.

@Jester, have you noticed this recent paper by Chamseddine and Mukhanov?

https://arxiv.org/abs/1805.06283

https://arxiv.org/abs/1805.06598

They claim to have a theory of massive gravity, free of ghosts and with mimetic dark matter. I am incompetent to give an opinion, but massive gravity may not be dead after all...

@Anonymous of May 19th, the arguments Jester was blogging about don't dependent on the specific model of massive gravity. In Minkowski space they all have 5 degress of freedom once the ghost is removed, and the problematic ones are still the remaining scalar and the vector ones. This remains true for Chamseddine and Mukhanov too. Note, however, that Chamseddine and Mukhanov claim to have a strong coupling coupling scale that goes to zero as $m^{1/2}$ (in contrast to $\sim m^{2/3}$ as in dRGT-massive gravity) so perhaps 'quantitatively' the bounds from positivity on their case are weaker.

Actually, the model of Chamseddine and Mukhanov violates Lorentz symmetry, so the arguments described in this post do not apply to it. I can't say whether or not this proposal makes any sense - one would need to get an opinion from an expert in this kind of exotic theories.

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