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Recently Einstein's invariance of the phase of a plane wave (1905) has beendescribed as "questionable" (Huang). Another definition of this phase, takinginto account a "relativistically induced optical anisotropy" for isotropicmedium in moving, has been proposed (Gjurkinovski). We suggest (logically) todetermine this "relativistically induced effect" if the isotropic medium is thevacuum. We prove that the basic Lorentz invariant, in vacuum, is not the phaseof a plane wave but the phase of a spherical wave. According to Poincare anisotropic spherical wave is not LTed (Lorentz transformed) into an isotropicspherical 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 thephase of a spherical wave and the transverse gauge with Einstein's invariant.We deduce from Poincare's invariant a relativistic Doppler formula which isunseparable from Poincare's theory of expansion of space and therefore themeasurements of Hubble.

Relativistic Invariance of the Phase of a Spherical Wave, relativistic Doppler Formula and Poincare's expansion of space