An experiment is described showing that a 36 GHz microwave signal received by rotating antennas or mirrors is not exhibiting the frequency shift ("transverse Doppler effect") predicted by the relativistic Doppler formula. The sensitivity of the apparatus used has been tested to be sufficient for detecting frequency shifts as small as 10/sup -3/ Hz which corresponds to the value of (v/c)/sup 2/ = 5.10/sup -14/ used in the transverse Doppler shift experiment reported here. From the observed absence of the transverse Doppler shift it is speculated that either the time dilation predicted by the standard theory of special relativity does not exist in reality or, if it does, is a phenomenon which does not depend on relative velocities but may be a function of absolute velocities in the fundamental frame of the isotropic microwave background radiation.