Exploring gravity theories with gravitational waves and compact objects

This thesis is devoted to the study of tests of General Relativitywhich could be performed using astrophysical observations of stars or compact objects. The thesis consists of two parts. In the first one, I have investigated how the future gravitational wave observations by the space-based detector LISA will permit mapping the spacetime of the supermassive black holes which are thought to reside in galactic centres. In particular, I have analysed the dynamics of a stellar black hole orbiting around a supermassive black hole and have investigated under which conditions the gravitational wave signal emitted by such a system can allow one to detect the presence of an accretion torus around the supermassive black hole. I have also studied the motion of a stellar black hole in the very strong field region of a nearly extreme supermassive black hole: contrary to our expectations and to suggestions present in the literature, we have found that although the motion presents peculiar characteristics, the emitted gravitational waves do not retain an observable imprint of the almost maximal rotation of the supermassive black hole. Also, I considered black hole binaries with arbitrary masses and spins. Although the coalescence of such systems can be studied only with numerical simulations, I have derived a compact analytic formula for the spin of the final remnant. This formula is in agreement with all the numerical simulations available to date.