  The relativity of simultaneity implies that the image of a Lorentz
transformed (LT) spherical (circular) wavefront is not a spherical (circular)
wavefront (Einstein 1905) but an ellipsoidal (elliptical) wavefront (Moreau,
Am.J.of Phys).We show firstly that the relativity of simultaneity leads to the
consequence that the image of a Lorentz transformed plane wavefront is a
tangent plane to an ellipsoid and not a tangent plane to a sphere (Einstein
1905). We deduce then a longitudinal component of the tangent vector to
Poincare's ellipse which is directly connected to the relativity of
simultaneity. We suggest finally that this violation of relativity of
simultaneity is related to Einstein's implicit choice of the (non relativistic)
transverse gauge in his theory of (rigid) wavefronts.
