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Mass curves spacetime creating a gravitational gradient/time dilation, and mass is mostly energy when considering the miniscule mass of the quarks themselves. This implies that the dilation is caused by the energy and not the mass of the subatomic particles. I've understood quarks as bouncing around inside the nucleon radius with the Gluons acting as springs. If the springs would be constantly expanding/contracting, wouldn't this mean there would also be a constantly changing potential in the color field/strong force, and thus a change in time dilation due to that energy?

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Nucleons are complicated quark-gluon systems. If you smear out the energy density and model a nucleon as a sphere of mass around $10^{-27}$ kilograms and a radius of around $10^{-15}$ meters, you will find that the gravitational time dilation, which differs from 1 by something on the order of $GM/c^2R\sim 10^{-39}$, is completely negligible.

If you think this model is too simple, then you need a theory of quantum gravity to do better.

If gravitational effects were significant in the dynamics of nucleons, we would have plenty of experimental evidence to guide us to a theory of quantum gravity, because we have tons of data on nuclear physics. One of the reasons that quantum gravity is a hard problem is that we have basically zero data about it.

The quarks in a nucleon move at relativistic speeds and their kinematic time dilation from Special Relativity is not negligible. But you asked about gravitational time dilation from General Relativity, and it is negligible.

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