Twin Paradox, Lorentz Transformation, Relativity.
transformation x = γ*(x_1+v*t_1), again for x_1=0, we verify that x = γ*v*t_1 and passing to the differentials we obtain the following relation dx = γ(t_1)*v _1(t_1)*dt_1. The last equation is correct, in fact in the meantime dx_1 = v_1(t_1)*dt_1; for the frame S_1 the infinitesimal distance dx is