  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.
