The relativistic kinematics underlying Poincare's ellipse
  We show that the kinematics underlying Poincare's elongated ellipse, immediately deduced from Lorentz Transformation (LT), is not the same as Einstein's kinematics. The two main differences are : On one hand the relationship of Lorentz contraction with LT, and on the other hand, the relationship of Doppler formula with wave four-vector. Einstein's wave four-vector (quantum theory of light, photon, transverse gauge) is not the same as Poincare's wave four-vector (classical theory of light, electromagnetic wave, Lorenz gauge) and therefore the Doppler formulas are not the same. Poincare's Doppler formula is not independent of the metric of space-time. We show that Poincare's metrics of space-time is based on a cosmological definition of distance and predicts an expansion of space for remote objects (Hubble), directly connected with Poincare's Doppler formula. 
Dr Yves Pierseaux