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General relativity tells us that what we perceive as gravity is curvature of space-time.

On the other hand (as I understand it) gravity can be understood as a force between objects which are exchanging (hypothetical) virtual particles called gravitons, similar to the way electromagnetic forces are due to objects exchanging virtual photons?

At least at first glance, the two concepts seem mutually exclusive. Is there a description of gravity which includes both, or is this contradiction one of the problems in combining GR with quantum mechanics?

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Here is just a small remark. It is possible to give a strict mathematical proof about the equivalence of these two pictures.

If you just start with the three (semi-experimental) facts: Lorentz invariance, $1/r$ long-range tail of gravitational force and its one-way action (attraction only) and the fact that the bending of light almost doesn't depend on its frequency and polarization, then you will find that these facts are compatible (in the large distance limit) only with the massless helicity $\pm2$ particle exchange. After that, it has been proved that special relativity and analytic properties of scattering amplitude lead to the equivalence principle [1,2]. This theorem is a pure analog of Gell-Mann-Low-Goldberger soft photon theorem, which claims that the power expansion of the amplitude of photon scattering by a hadron (with respect to photon frequency) does not depend on the spin or internal structure of the hadron (up to the second order). By considering multigraviton scattering amplitudes one can prove that the all local vertices for soft gravitons correspond to the expansion of the Einstein action.

It means that the exchange of helicity $\pm2$ massless particle unavoidably leads to the classical general relativity (the opposite statement is trivial).

This program was initiated by Steven Weinberg [1,2] and finished by Deser and Boulware [3]. You can find the complete consideration in their paper [3] with the title “Classical general relativity derived from quantum gravity”. This paper is a real masterpiece of clear physical explanation of this problem.

References

[1] S. Weinberg, Photons and gravitons in S-matrix theory: derivation of charge conservation and equality of gravitational and inertial mass, Phys. Rev. B135 (1964) 1049.

[2] S. Weinberg, Photons and gravitons in perturbation theory: derivation of Maxwell’s and Einstein’s equations, Phys. Rev. B138 (1965) 988.

[3] D. G. Boulware, S. Deser, Classical general relativity derived from quantum gravity, Ann. Phys. 89 (1975) 193.

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  • $\begingroup$ Hi @Grisha, That is a beautiful explanation. I tried to download the Boulware-Deser paper, but Elsevier won't let me, even with a university subscription ! So if you have a digital copy that you can pass on I would really appreciate it. Cheers, $\endgroup$
    – user346
    Commented Dec 11, 2010 at 19:42
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    $\begingroup$ @space_cadet Try here tinyurl.com/25or7oz By the way, there is a small inaccuracy in my answer. It is about Gell-Mann-Low-Goldberger theorem, about the spin more precisely. The theorem claims that the first two terms ($\propto\omega^{0}$ and $\propto\omega^{1}$) of the expansion depends only on the total electric charge and the anomalous magnetic moment of the hadron. It is interesting that there is no such correction as "anomalous moment" in quantum gravity. $\endgroup$
    – Grisha Kirilin
    Commented Dec 12, 2010 at 4:31
  • $\begingroup$ Thanks @Grisha. Don't you just love the internet?! As for the "anomalous moment" in quantum gravity, that is a very interesting observation and I'll get back to you if I have something useful to say about it! $\endgroup$
    – user346
    Commented Dec 12, 2010 at 5:06
  • $\begingroup$ @space_cadet It is very simple - you can't just construct the operator like $\sigma_{\mu\nu}F^{\mu\nu}$ because $\sigma_{\mu\nu}R^{\mu\nu}=0$ for Ricci tensor. The rigorous soft graviton theorem was established by Gross and Jackiw in Phys. Rev. 166 (1968) 1287. As far as I remember, they consider a graviton scattering by a spin-0 particle, but the theorem is valid for any spin because of the reason I mention above. $\endgroup$
    – Grisha Kirilin
    Commented Dec 12, 2010 at 12:51
  • $\begingroup$ I can't find a public copy of this paper, so please explain (in simple terms) how this theory accounts for time dilation. $\endgroup$
    – user32023
    Commented Apr 3, 2020 at 12:36
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Well, consider this: the same thing happens with electromagnetic forces. We can describe them as particles responding to the presence of electric and magnetic fields, or we can describe them as resulting from the exchange of virtual photons. Those views seem similarly incompatible, but nevertheless both theories (classical electrodynamics and quantum electrodynamics, respectively) give excellent predictions. We can't really say that one is more "right" than another; we just have to accept them both.

The situation with gravity is pretty much a direct analogy to electromagnetism. We can describe gravity as particles responding to the presence of spacetime curvature, or we can describe them as resulting from the exchange of virtual gravitons. As with EM, these views would correspond to classical gravity and quantum gravity, respectively. But the difference is that, although general relativity fills the role of the classical theory, we don't have a good quantum theory of gravity yet.

I wouldn't say that the field/particle duality is one of the problems that impedes the combination of quantum mechanics with GR. After all, we had no problem getting around the dual descriptions of electromagnetism. It's just the peculiar details of quantum gravity that make it a difficult theory to develop.

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  • $\begingroup$ I guess that makes sense... I understand wave-particle duality, and electromagnetic fields vs photons. So if gravity was just another field / virtual particle, I wouldn't be bothered. But changing the geometry of space-time seems like something more fundamental. Probably just a failure of my imagination :) $\endgroup$
    – dF_
    Commented Nov 9, 2010 at 20:22
  • $\begingroup$ Yeah, it is a little weird, but you have to get used to thinking of the geometry of spacetime itself as just another field. $\endgroup$
    – David Z
    Commented Nov 9, 2010 at 20:23
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    $\begingroup$ Exactly right. In fact it's worth emphasizing that in both cases (photons in QED and gravitons in quantum GR) the progenitors of these propagating excitations are present in the classical theory, as electromagnetic waves and gravity waves respectively. Just as the existence of electromagnetic waves leads to photons in the quantum version of the theory, the existence of gravitational waves leads to gravitons in quantum GR. $\endgroup$
    – Matt Reece
    Commented Nov 9, 2010 at 21:30
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    $\begingroup$ @Noldorin: Sure, but I mean that I understand the concept of wave-particle duality, and how to use it, not why things behave like that. $\endgroup$
    – dF_
    Commented Nov 9, 2010 at 21:52
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    $\begingroup$ I only see a problem with taking this analogy too far: equivalence principle (and hence space-time curvature) is an exact principle that is expected to hold at all energies, however EM fields are an approximation that breaks up at low-scales and is not reflected in the underlying theory $\endgroup$
    – lurscher
    Commented Feb 1, 2011 at 21:32
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Although the analogy between gravity and electromagnetism made by David is fine and self-suggesting, it must be cautiously added that there is no proof that gravity must look like exchange of gravitons at the microscopic level. We actually do not know what the microscopic picture of gravity is, and it might turn out to be very different from the familiar description in terms of particle-carriers of that force.

For example, earlier this year there was a preprint by Erik Verlinde suggesting that gravity might be an entropic force. It this is true, gravitons do not appear in this picture at all. This preprint is being actively discussed (more that 100 citations this year). However it also must be said that Verlinde's suggestion still remains a suggestion, not a theory, as it relies on some murky heuristic arguments, not a solid mathematical theory.

Update: the commenter below correctly points out that regardless of the microscopic theory the large-wavelength gravitational waves exist in any case and they can be quantized giving rise to gravitons. So, I guess my caution was misleading.

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    $\begingroup$ It doesn't matter what the unknown microphysics of gravity turns out to be. As a low-energy effective theory, it contains gravitons, which look like good local propagating degrees of freedom to high precision. Think, if you like, of phonons in a solid; they are UV-completed by the physics of atoms, which looks entirely different, but nonetheless they are good degrees of freedom for describing long-wavelength vibrations. $\endgroup$
    – Matt Reece
    Commented Nov 9, 2010 at 21:28
  • $\begingroup$ Well, your arguments make sense. Low-energy propagating modes indeed exist in any case, and you are saying we can always quantize them regardless of the UV-completion. Looks like you are right. $\endgroup$
    – Igor Ivanov
    Commented Nov 9, 2010 at 21:51

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