  We extend the work of Ryan on mapping the spacetime of the central object of
an extreme mass-ratio inspiral (EMRI) by using gravitational waves (GWs)
emitted by the system, which may be observed in future missions such as LISA.
Whether the central object is a black hole or not can be probed by observing
the phasing of these waves, which carry information about its mass and spin
multipole moments. We go beyond the phase terms found by Ryan, which were
obtained in the quadrupolar approximation of the point-particle limit, and
derive terms up to the fifth post-Newtonian (PN) order. Since corrections due
to horizon absorption (i.e., if the central object is a black hole) and tidal
heating appear by that order, at 2.5PN and 5PN, respectively, we include them
here. Corrections due to the motion of the central object, which was addressed
only partially by Ryan, are included as well. Additionally, we obtain the
contribution of the higher order radiative multipole moments. For the tidal
interaction, our results have been derived in the approximation of the
Newtonian tidal field. Therefore, in the potential for tidal field only the
contribution due to the mass of the central object has been included as well.
Using these results we argue that it might be possible for LISA to probe if the
central object in an EMRI has a horizon or not. We also discuss how our results
can be used to test the No-hair theorem from the inspiral phase of such
systems.
