Relativistic invariance of the time-domain waveguide modes

The relativistic invariance of a complete set of time-domain modes under Lorentz transformation is proved. The time-domain modes are exhibited as the particular solutions to the system of Maxwell´s equations with time derivative. The waveguide surface has the properties of perfect electric conductor. The modal fields are presented via transverse-longitudinal decompositions. Every field component is a product of a modal amplitude depending on longitudinal coordinate and time, and an element of modal basis in the waveguide cross section. The modal basis is specified in a general form. Exact explicit solutions for the modal amplitudes are presented.