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As far as I know, quantum gravity basically says that the graviton mediates the force of gravity. And general relativity says that gravitation is the result of masses warping spacetime, and thus gravity can't really be considered a "force", so to speak.

Now, is it because of this that quantum gravity theory contradicts relativity? That gravitons "carry" the force which relativity states isn't a force at all? Or am I misunderstanding something?

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First, equations (ie, firm predictions) are more important than words (ie, interpretation). In the regimes where both are valid and we think we understand them, GR and (perturbative) quantum gravity agree in their predictions (https://arxiv.org/abs/gr-qc/9405057). There are other examples of well-understood physics where the words of different frameworks are different but the equations are the same; for example the Lagrangian vs Hamiltonian vs Newtonian versions of classical mechanics, or the Schrodinger vs Heisenberg vs path integral versions of quantum mechanics.

Second, we don't know the final theory of quantum gravity so we can't fully answer this question, but somehow GR and perturbative quantum gravity should both fall out as different limits of this more complete and better understanding. This does happen in string theory; perturbative quantum gravity is an effective field theory that reproduces scattering amplitudes calculable in string theory, and Einstein's equations (coupled to a specific set of fields) arises as a low energy limit of string theory that can be computed using renormalization group methods and conformal invariance of the string worldsheet action (see, eg, Ch. 7 of David Tong's string theory notes https://www.damtp.cam.ac.uk/user/tong/string.html).

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