Time delay due to spin inside a rotating shell

In a previous paper, we have shown that when different images of the same source are observed at a far point by gravitational lensing, there may be an appreciable time delay between these images due to the spin of the central body. In this paper, we derive the time delay in the travel time of massless particles, e.g., photons, propagating inside a massive rotating shell. We analyse this time delay, due to the spin of a shell, in the case when two or more images of the same source, whose photons travel inside a rotating massive shell, are observed at a far point by gravitational lensing. We show that there may be an appreciable time delay due to the spin of the shell. In particular, we consider some astrophysical configurations: a typical rotating galaxy and a typical rotating cluster, or super-cluster of galaxies; this effect might be large enough to be detected at Earth. Thus, this phenomenon should be taken into account in the modelling of the time delay of different images from gravitational lensing and might be detectable in some gravitational-lensing images.