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If we talk about a "quantum theory" of General Relativity, we know that the particle that mediates the gravitational force would be the so called Graviton, a massless particle with spin $2$.

I wonder if (and if it would work) there could be also a different theory like a "massive gravity", whose force is mediated by a massive boson (maybe not that massive too, even a small mass). What would it implicate?

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    $\begingroup$ I am not sure what you are talking about. We don't know anything about quantum gravity, not even if we need it. Theoreticians have proposed plenty of models for quantum gravity, many of which contain spin-2 gravitons, but that's not scientific knowledge, it's just "intellectual non-sense" as my old theory professor used to say, at least until the day someone actually measurer quanta or gravitation. $\endgroup$ – CuriousOne Mar 4 '16 at 23:36
  • $\begingroup$ @CuriousOne That is true! But since most of people tend to speak about gravitons and so on, I was wondering if something "massive" existed. It's just curiosity about speculations and models. $\endgroup$ – Les Adieux Mar 4 '16 at 23:37
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    $\begingroup$ Related: physics.stackexchange.com/q/5456/2451 , physics.stackexchange.com/q/22010/2451 , physics.stackexchange.com/q/39260/2451 and links therein. $\endgroup$ – Qmechanic Mar 4 '16 at 23:42
  • $\begingroup$ Until there is a measurement you can always propose anything, you could even have claimed that the Moon was made of cheese... right until Neil Armstrong stepped on it. $\endgroup$ – CuriousOne Mar 4 '16 at 23:49
  • $\begingroup$ @CuriousOne Well that actually would have been really unlikely! $\endgroup$ – Les Adieux Mar 4 '16 at 23:55
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The reason why the graviton is proposed to be massless is that the range the gravitational force acts on is infinite and falls as 1/r. The full derivation of this requires a lot of knowledge of relativistic quantum field theory but I will try to motivate this using basic quantum mechanics and special relativity principles.

The Heisengberg Uncertainty Principle tell us that

$$ \Delta E \Delta T ≥ \hbar $$

This then can be interpreted as follows: we can borrow energy $\Delta E$ from the universe as long as we give it back within time $\Delta T$.

From that follows $$\Delta T = \frac{\hbar}{mc^2}$$ where m is the mass of the particle that mediates the interaction, and c is the speed of light, the maximum speed this particle moves at. (We used $E = mc^2$ here)

The range of the force is then proportional to $\Delta T$ because it is the distance that our mediating particle can travel in this time.

Now we know that gravity acts on an infinite range, from that follows that $\Delta T = \infty $ and therefore, m = 0.

So, until we find something contradictory with that gravity acts on an infinite range, this forces us to choose massless gravitons when we talk about a gravity-mediating particle.

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    $\begingroup$ Interesting way to explain! If you can provide me some material about QFT I will read it! $\endgroup$ – Les Adieux Mar 4 '16 at 23:57
  • $\begingroup$ I would expect this to be in most QFT textbooks (e.g. Weinberg or Zee), usually in a chapter that talks about the Yukawa potential $\endgroup$ – hsnee Mar 5 '16 at 0:10
  • $\begingroup$ There have been serious searches for short ranged contributions to gravity (what the question implies). They were all unsuccessful.. $\endgroup$ – Lewis Miller Mar 5 '16 at 0:35
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It is well understood that if gravity is a 'long range force' its quanta's mass is zero. That is the graviton. Excuse the term in quotation marks, its the easiest way to say what is known. It really means the gravitational field, at infinity in aan asymptotic flat spacetime goes like 1/r

But the fact is that the graviton mass could indeed be greater than 0, but very very small, and all the observations and measurements ever done on gravity would not detect any deviation. The current limits on its mass from all the observations/measurements are that the mass must be smaller than 10 to the minus 22 ev. That's is extremely small. The most stringent limits have been set by astronomical observations. But we are soon to to be observing possible effects, and will be able to set the limit lower by orders of magnitude, if indeed it is 0. Those upcoming measurements will be from observations of gravitational waves in the space-based eLISA satellites that will constitute a 3 leg interferometer in space with interferometer led aisles of 1 million kilometers. It is to be launched in the next very few years. That greater spacing will increase the sensitivity and allow for much longer wavelengths to observe gravitational waves. It'll be a significant sensitivity improvement over the ground based 5 Kms leg sizes that did the first direct detection of gravitational waves in 2016, announced in February of this year.

The new limits will be based on looking for dispersion in observed gravitational waves - meaning slightly different velocities measured at different frequencies. It will be that sensitive. It'll also observe those waves from dual merging supermassive black holes much longer, up to months, so a lot of data to do statistical averaging over also.

Those measurements will be looking for plenty other possible deviations from general relativity, and it will do it in the realm of strong gravity where first post Newtonian approximations will not do. It'll also be able to detect higher multipole moments from those black holes in the merger phase and ringdown, and thus see both dynamic effects of the settling to the no hair Kerr black holes. It'll see any deviations from the no hair theorem, up to its sensitivity limit. It'll be able to check other gravity theories such as strong gravity. Some scalar massive spin 0 plus tHe spin 2 theories will also be able to be ruled out, or evidence found for that possibility. It'll be able to see cosmological gravitational waves, and any waves emitted by some Big Bang relics such as cosmic string, if any existed.

Over time there will be even bigger gravitational wave space based 'observatories' with longer legs and even better sensitivity

The point is that yes there are some theories that allow non zero mass gravitons and which have not been ruled out, and those will be explored, and if the mass is truly 0 the mass limits will be made more stringent. At the very least we will be observing a lot of the details of those waves front many astrophysical and cosmologically predicted or expected objects. Someone named some of those measurements as 'gravitational spectroscopy' (sorry I can remember who but it was in relation to the quasi-normal modes in black hole ringdown after merger.

Just like the neutrino is now believed to have a very small mass, we still don't know enough about gravity, nor quantum gravity, to know for sure, to full accuracy and theoretical consistency, for the graviton. Still, for now, all measurements and observations, and accepted theory, have found it to be a big zero.

Just tune in over time.

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