Why is there a search for an exchange particle for gravity? If I understand correctly, according to Einstein's General Theory of Relativity, mass results in a distortion in space-time. In turn, the motion of the mass is affected by the distortion. A result of the interplay between mass and space-time is that the 'force' of gravity may be explained away. Masses are not subject to a force, but are merely following a 4-dimensional space-time geodesic; gravity is just geometry.
And yet physicists are searching for exchange particles for the force of gravity, and are trying to unify quantum mechanics with relativity, or to unify the weak/strong/electromagnetic forces with that of gravity.


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*What have I missed? Are these different communities of physicists? Does relativity explain only part of the story of masses acting under gravity?

*Is gravity a force or not? Is it only an apparent force or not?

*Can such an apparent force 'generate' exchange particles? Are the exchange particle and geometric models both different views of the same underlying truth?

*A side question might be: why can't the other forces be explained away similarly? Or is that what is happening with all this talk of small extra dimensions?
I'd appreciate any illumination on this matter, or suggested reading (preferably at the 'popular science' or undergraduate level).
 A: An addendum to the answers of Daniel Grumiller and sb1: 
The major difference of the gravitational field and other fields is that according to general relativity the gravitational field defines space and time and therefore defines the relation of events. It is true that it is possible to do an "arbitrary" split of a certain linear approximation of the gravitational field into a "flat background" and "waves" propagating on this background. In principle this kind of reasoning is a violation of the very idea that the gravitational field defines the background of spacetime in a holistic way, and it was the subject of a lot of discussions if this approximation is of any use.
This is considered to be settled by the observational evidence that bistar systems loose energy in exact the way that the "graviational wave approximation" predicts, as cited by Daniel Grumiller.
The existence of gravitons is a conjecture based on the assumption that gravitational waves exhibit the same quantum nature as classical waves, e.g. waves in classical electromagnetism. At the basis of this conjecture is the idea that it should be possible to split the gravitational field in a part defining the background, and then having gravitational waves propagating on this background and exhibiting the same wave-particle duality as other waves. It would then be possible to treat quantum gravitational effects in a semi-classical approximation. 
Since there is no observational evidence for this, this conjecture is still the subject of controversy.

Are these different communities of physicists?

Some have a strong believe in the existence of gravitons, some think that quantum gravity needs a bigger conceptual step than only gravitions, and some do both, so, yes, there are different communities believing different things.

Does relativity explain only part of the story of masses acting under gravity?

It explains everything in a classical setting with not too strong forces alright, but does not exlpain quantum effects or what happens when forces get so strong that singularities occur.

Is gravity a force or not? Is it only an apparent force or not?

It is a force, it is an apparent force in the sense that classical GR says that you feel it because you live in an accelerating reference frame. Both statements are valid in the classical setting and are independent of the quantum nature of gravity, and in particular of the existence of gravitons.

Can such an apparent force 'generate' exchange particles? Are the exchange particle and geometric models both different views of the same underlying truth?

Yes, see above (geometric model = classical setting, exchange particle = semi-classical approximation). 

why can't the other forces be explained away similarly? Or is that what is happening with all this talk of small extra dimensions?

The gravitational field is fundemantally different from other fields (see above), and this has no connection to extra dimensions.

I'd appreciate any illumination on this matter, or suggested reading (preferably at the 'popular science' or undergraduate level).

The problem is that if you are able to ask this question, you're already beyond the popular science level. I'd really like to recommend to you an introducory class on QFT and one on GR, there you'd get the best answer to your question :-)
A: General relativity is a purely classical theory means it does not take the uncertainty principle of quantum mechanics into account. Although it is a very good description of gravity in large scale it should not be a good description for very small scale since in those very small scale quantum effects can not be ignored. There are at least two instances, the big bang and black hole where GR predicts mathematical singularities. That means in those situations GR fails to be a good description of the gravitational interaction. According to quantum mechanics every classical field have a quantum analogue. The classical Field of Maxwell, called electromagnetic field, has a quantum analogue. We picture the electromagnetic interaction by messenger particles called virtual photons which has spin 1. This virtual photons are the particles of electromagnetic field. The disturbances in this field called electromagnetic waves, which according to quantum field theory are emission of real photons. There is no contradiction here. This is a basic feature of quantum theory. We call it wave particle duality. Similarly for gravitation the quantum analogue of the relevant classical field can be thought of as consisting of virtual gravitons of spin 2. Gravitational interactions are mediated by these virtual gravitons. When a real gravitons are emitted we call it gravitational waves. Gravitons are messenger particles of gravity.  Again there is no contradiction here.
A: Firstly I would like to question whether anyone is really experimentally searching for a graviton. The effect would be too weak to detect with current technology. There is the closely related concept of "gravitational wave" which is a "curvature wave" in General Relativity and that is being searched for.
There are probably two main reasons for expecting the graviton idea to have some merit.


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*The Gravitational Wave is a classical wave, and all other classical waves (like with Electromagnetism) have found a particle form in Quantum Theory and Quantum Field Theory. The familiar concept here is "wave particle duality". So the expectation is that under a theory of "Quantum Gravity" there would be a graviton. This point does not in itself prejudice the fundamental form of that Quantum Gravity theory: it might still be geometric in its foundations.

*The equations which describe quantum particles and fields when expressed in a notation called "2-spinor form" have a basic pattern and form. So an electron is described via Dirac equations as $\phi_A$. This relates to spin 1/2 which is all correct for the electron. Spin 1 particles/fields (e.g. the Photon) are described by $\phi_{AB}$ in this form of Maxwell's equations. And so the pattern continues with N labels A,B, ..., K describing a spin N/2 particle. Well Einstein's General Relativity equations when written in this spinor form have $\phi_{ABCD}$. So this suggests a spin 2 - even without any linearisation or quantisation, etc.
The big difference between Einstein's equations and these others of lower spin is that the Einstein equations are non-linear (representing space-time curvature) whereas these other equations are linear (essentially representing a linear underlying space). So one photon plus one photon equals two photons, but one graviton plus one graviton would not quite equal two gravitons.
The arena in which the debate continues is that of Quantum Gravity in which different camps take different views as to the nature of any underlying space. These debates are reflected in some of the other Stack questions.
A: General relativity, currently our best model of gravity, not only predicts the gravitational force as a geometric effect, it also predicts the existence of gravitons, the "exchange particle for gravity". 
A theory without gravitons or gravitational waves (I'll use them interchangably) could exist, but as we have already indirect evidence for gravitational waves from the Hulse-Taylor pulsar it seems unlikely that gravitons do not exist in Nature.
We shall see in the coming decade, since advanced LIGO is bound to find gravitational waves if they exist.
