Relativistic Invariance of the Phase of a Spherical Wave, relativistic   Doppler Formula and Poincare's expansion of space
  Recently Einstein's invariance of the phase of a plane wave (1905) has been described as "questionable" (Huang). Another definition of this phase, taking into account a "relativistically induced optical anisotropy" for isotropic medium in moving, has been proposed (Gjurkinovski). We suggest (logically) to determine this "relativistically induced effect" if the isotropic medium is the vacuum. We prove that the basic Lorentz invariant, in vacuum, is not the phase of a plane wave but the phase of a spherical wave. According to Poincare an isotropic spherical wave is not LTed (Lorentz transformed) into an isotropic spherical wave (Einstein 1905) but LTed into an anisotropic ellipsoidal wave (relativity of simultaneity). Poincare's ellipsoidal wavefront (1906) is an {equiphase} surface. The Lorenz gauge is connected with the invariance of the phase of a spherical wave and the transverse gauge with Einstein's invariant. We deduce from Poincare's invariant a relativistic Doppler formula which is unseparable from Poincare's theory of expansion of space and therefore the measurements of Hubble. 
Yves Pierseaux