Harald88, you are mistaken with your interpretation of the transverse Doppler effect. I strongly recommend that you withdraw your modification, because it most probably adds to confusion. You should first study Bradley's theorem and its meaning in optics of moving bodies. I recommend that you consult E. Herlt und N. Salié, Spezielle Relativitätstheorie, Akademie Verlag, Berlin 1978, Section 1.2: "Elektrodynamik und ausgezeichnetes Bezugssystem".