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I am wondering if the Higgs field is related to time dilation. Photons travel at the speed of light, but muons are very close but never at the speed of light. Would Muons be slowed by the interaction with Higgs field? If so, the bigger the mass, the lower the max speed it can travel. When explaining the relativity, one often quotes a rocket travelling near the speed of light. Maybe it is fundamentally impossible to reach such a speed.

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The Higgs field is not responsible for time dilation. Kinematic time dilation and gravitational time dilation would both happen even if there were no Higgs field.

An interaction between the muon field and the Higgs field (plus something called “spontaneous symmetry breaking” of the Higgs field, which causes it to have a nonzero value throughout the universe) makes muons have mass, and thus prevents them from traveling at the speed of light. However a muon, like any particle with mass, can travel arbitrarily close to the speed of light, given enough energy.

More massive particles do not have lower speed limits. Given sufficent energy, the only speed limit is $c$. If you have a fixed amount of energy to give to a particle, then a more massive particle will go slower than a less massive one.

It is fundamentally impossible for any particle with mass to actually reach the speed of light, because this would require it to have infinite energy.

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