Gravitational time delay effects by Kerr and Kerr-Newman black holes in strong field limits

We study the time delay between two relativistic images due to strong gravitational lensing of the light rays caused by the Kerr and Kerr-Newman black holes. The trajectories of the light rays are restricted on the equatorial plane. Using the known form of the deflection angle in the strong deflection limit (SDL) allows us to analytically develop the formalism for the travel time of the light from the distant source winding around the black hole several times and reaching the observer. We find that the black holes with higher mass or with spin of the extreme black hole potentially have a higher time delay. The effect of the charge of the black hole enhances the time delay between the images lying on the opposite side of the optical axis, resulting from the light rays when one light ray is in the direct orbit and the other is in the retrograde orbit. In contrary, when both light rays travel along either direct or retrograde orbits, giving the images on the same side of the optical axis, the charge effect reduces the time delay between them. We then examine the time delay observations due to the galactic and supermassive black holes, respectively.