The mass of a proton has only been calculated to an accuracy of around 4% - (roughly 938 MeV/c2 or 1.672 × 10-27 kg). The constituent particles - quarks and gluons - which make up the proton have individual masses that add up to only around 1% or so of the measured mass (which can be determined accurately with specialised devices called Penning traps , see: Phys. Rev. Lett. 119, 033001 )
In order to explain the missing 99%, the constituent particles are thought to be 'vibrating' (and otherwise interacting) at near light-speed - which, because of relativistic effects, increases their apparent mass (plus the interaction-energy of the strong nuclear forces that hold the proton together).
The picture is further complicated by the fact that protons (and neutrons) have (on average) three quarks each. Because they are in constant 'motion', calculating the forces between them invokes the classic 'Three Body Problem'.
In order to simplify the calculations, researchers instead work on the basis that the quarks are in pairs - called 'diquarks'. This is a known oversimplification. Even so, the calculations are so complex that supercomputers are needed. And the accuracy of the calculations still has a 4% error margin.
1) Similar lack of precision affects calculations of the mass of neutrons.
2) Since most of the mass of matter comes from the protons and neutorns which it contains, it follows that around 99% of the mass of all matter is derived from relativistic (i.e. near lightspeed) effects.
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