I want to clarfiy some ideas, so let us assume an inital observer measures a 4 velocity of an object. ##\vec{u} = (c\gamma_u, \gamma_uu^x, \gamma_uu^y, \gamma_uu^z)##. When we calculate the magnitude of the 4-velocity we get,
$$\vec{u} \cdot \vec{u} = -c^2\gamma_u^2 + \gamma_u^2U^2$$ where...