Geodesic motion in Kerr's spacetime

This chapter studies the geodesic motion of massive and massless particles in Kerr&s;s spacetime. The study is restricted to the motion outside the outer horizon, since this is the region relevant for astrophysical observations. It is shown that, unlike the Scharzschild case, only the orbits on the equatorial plane are planar. The geodesic equations for the time coordinate and for the azimuthal, angular coordinate are found exploiting the constants of motion associated to the symmetries of the Kerr metric. Those for the radial and the remaining angular coordinate are found using the Hamilton-Jacobi approach, showing the existence of the Carter constant which allows one to find the general solution of the geodesic equations in a closed form. The structure of the potential for equatorial geodesics is studied in detail. The study is then specialized to the motion of massless particles on the equatorial plane, whereas timelike geodesics are discussed only qualitatively. Kepler&s;s third law is generalized to describe massive particles in circular orbit on the equatorial plane. The shadow of a Kerr black hole is briefly described. The chapter ends with the derivation of the process of energy extraction from a Kerr black hole (Penrose&s;s process) and with a brief discussion on superradiant scattering.