the Time Dilation equation, of the Lorentz-Einstein transformations, is a purely kinematical equation. if we apply it to the cosmic muon as shown, then we must assume that the Earth is moving at -0.998c relative to the muon. however, this would be a violation of the conservation of linear momentum for the Earth-Muon system. for the earth-muon system we have M(earth)*v(earth) = -m(muon)*v(muon). hence, v(earth)= -(m/M) * 0.998c. this is not equal to v(earth) = -0.998c. in fact the v(earth) is essentially 0. we cannot use v(earth) = -v(muon) = 0.998c in order to apply the time dilation equation. we have to use gravitational time dilation which is insignificant given the GF of the earth to increase the decay time of the cosmic muon to reach the earth surface.
