Why does Gravity distort space and time while the electromagnetic, strong, and weak forces do not?
Does this have to do with why Quantum Mechanics and Relativity are incompatible?
Although it would be more precise to say that gravity is the manifestation of the effect of curved space-time on moving bodies, and it is mass that curves the space-time, so prof. Rennie is correct about this, there are differences of opinion, at least, about the other aspects. It is not at all clear that mass is a kind of charge analogous to electric charge, although some researchers are trying to make this idea work and unify gravity with the Standard Model or QFT.
Be that as it may, what is clear is that gravity or curvature is different from electromagnetism or charm etc., for one thing, because gravity is not a force. Einstein, Schroedinger, other pioneers in GR were quite explicit about this. See gravity is not a force mantra, http://physics.stackexchange.com/a/18324/6432 for a discussion of this.
So there are major differences between gravity and the (other) fundamental forces, and this may well be the reason why gravity has not yet been successfully quantised.
But there are even more incompatibilities between the whole spirit of GR and the spirit of QM. J.S. Bell was quite concerned about the seemingly fundamental incompatibilities between relativity and quantum theory, too. For me, I would point out that in QM, the wave functions live on configuration space, which for, say, two particles, is six-dimensional, and also QM treats other dynamical variables such as spin as being equal in right, this makes the space even larger. Also QM treats momentum as just as valid a basis for coordinates as position, and this, too, is alien to the spirit of relativity, which treats the actual four-dimensional Riemannian manifold as basic.
For precisely the need to overcome this incompatibility, passing to Quantum Field Theory replaced these wave functions over configuration space with operator-valued functions on space-time. But although this kinda works to overcome the incompatibility of special relativity with QM, it makes the foundations of QFT much murkier (the role of probabilities, for instance the Born rule) and introduces infinities. Thus although it might be a way to reconcile QM and relativity theory, it is still more of an unfinished project and because of the unsatisfactory foundations of QFT (compared to the clear foundations of QM), one can still suspect there is a missing idea to really reconcile the two or even that somebody has to budge and concede something or there will be no treaty...
It's not really correct to say that gravity distorts space and time. It's mass that distorts spacetime, and we see the effect of this as gravity.
At the risk of over simplifying, for each of the forces there is a "charge" that causes the force. Ordinary electric charge causes the electromagnetic force while weak charge and colour charge cause the weak and strong force. The "charge" that causes the gravitational force is mass.
Actually the strong etc forces can distort spacetime. For example in a proton the quarks are held together by the strong force, and the amount of energy this produces does cause a gravitational force. If you measure the mass of a proton (which describes how much gravitational potential it generates) you find that it's much greater than if you add up the masses of the three quarks. The extra mass comes from the effect of the strong force. In principle this is also true of the weak and electromagnetic forces though I'm not sure offhand what systems you'd have to study to see this.
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