Gravitational lensing in the Kerr spacetime: an analytic approach for light and high-frequency gravitational waves

Abstract Astrophysical observations in the electromagnetic spectrum as well as detected gravitational wave signals indicate that the Kerr spacetime can be used to describe the spacetime around candidates for rotating black holes. While the geodesic structure of the Kerr spacetime is already well known for decades, using exact analytic solutions to the equations of motion for applications to astrophysical problems has only attracted attention relatively recently. In particular using the exact analytic solutions to investigate gravitational lensing of light and gravitational waves emitted by sources outside the accretion disk has only received limited attention so far. Therefore, the focus of this paper will be to address this point. For this purpose we assume that we have a standard observer in the domain of outer communication. We introduce a local orthonormal tetrad to relate the constants of motion of light rays and high-frequency gravitational waves detected by the observer to latitude–longitude coordinates on the observer’s celestial sphere. We show that in this parametrisation we can derive the radius coordinates of the photon orbits and their latitudinal projections onto the observer’s celestial sphere as functions of the celestial longitude. We use the latitude–longitude coordinates to classify the different types of motion, and solve the equations of motion analytically using elementary and Jacobi’s elliptic functions as well as Legendre’s elliptic integrals. We use the analytic solutions to write down an exact lens equation, and to calculate the redshift and the travel time. Here, this approach allows us to visualise them with much higher resolution than numerical techniques, in particular close to the boundary of the shadow. We discuss their sky projections for different stationary sources and the implications of the results in the context of conventional astronomical observations and the observation of high-frequency gravitational waves gravitationally lensed by supermassive black hole candidates.