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If matter feels gravitation by way of absorbing gravitons, how is it that gravity passes through matter unfettered?

For instance during a solar eclipse, should not the Moon block at least some of the gravitational force acting on Earth due to the Sun?

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First let us put it clear that gravitons are hypothetical particles. Elementary particles are described as quantum excitations of fields pervading space and up to now there is no quantum theory for gravity. However the idea of gravitons looks very appealing so let us assume they exist.

There is a quite spread out idea that Quantum Field Theory (QFT) explains forces as the exchange of mediating particles, the virtual particles. This is not quite true, virtual particles are not particles. This can be seen from a relativistic, quantum and even semi-classical point of view - they do not satisfy the relativistic relation between energy and momentum, $E^2=|\vec p|^2+m^2$, do not have creation and annihilation operators and they are not solutions to a wave equation. QFT explains particles interaction as mediated by fields and that includes some perturbations of that fields which people prefer to call virtual particles. Of course, a quantized gravity field would have gravitons as their associated particles, but those would not be the mediators of gravitational interaction.

With those ideas in mind we can understand that graviton field pervades space regardless whether the moon is in between Sun and Earth or not. There is a difference in the two cases though. The Sun-Earth interaction when Moon is not present is different to the Sun-Earth interaction in the presence of the Moon. This is due to the fact that gravity is non-linear, it does not satisfy the Superposition Principle. However, the difference must be small due to the small mass of the Moon.

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