Does gravity exist between quarks in a neutron?
I mean, I've learned that gravity exists between objects that have mass/energy. But does this apply between quarks in a neutron?
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As far as we know the classical (i.e. non-quantum) laws of gravity apply at all length scales. There are theoretical reasons to suppose that the classical description fails at scales approaching a Planck length, but this is far, far smaller than the size of a neutron.
So inside a neutron we would expect the classical laws of gravity to apply, and in particular we would expect Newton's law to be good description - it doesn't seem likely we'd have to use general relativity. A simple calculation will show you that the gravitational energy is immeasurably small compared to the strong force binding energies, but in principle there will be some gravitational interactions.
But if you have an image of the neutron with the three quarks as little balls orbiting each other then you need to abandon that image. Firstly the quarks are delocalised so they exist as fuzzy objects distributed throughout the nucleus, and secondly 99% (ish) of the stuff inside a neutron is interaction energy not the three valence quarks. Exactly how you would calculate the gravitational interactions in such a system is unclear.