The multiple electromagnetic excitation with fast projectiles (heavy ions) is studied theoretically in the sudden approximation. Of special interest is the excitation of rotational states coupled to giant (dipole) vibrations. Closed form expressions are obtained for the excitation of a rigid rotor. The strong pulse of high energy equivalent photons in relativistic heavy ion collisions opens up new possibilities for nuclear structure studies, not possible e.g. with electron scatterirtg or nuclear Raman scattering. It is also pointed out that the "Brink-hypothesis" can be investigated in a new way by means of multiple electromagnetic excitation with relativistic heavy ions of low lying states coupled to the giant dipole mode.