Name: Arun Mangalam Affiliation: Indian Institute of Astrophysics Conference ID : ASI2022_833 Title : Relativistic dynamics around black holes in galactic nuclei Authors : Arun Mangalam Abstract Type: Invited Abstract Category : General Relativity and Cosmology Abstract : Homoclinic orbits in black hole dynamics offer the kind of crucial signpost that demarcates physically distinct regions of the conservative and inspiral dynamics: bound from plunging, whirling from not-whirling, smooth from chaotic. They define salient details of black hole dynamics, and we have derived an exact parametric solution for homoclinic motion with several astrophysical applications. One of them is calculating Extreme-mass ratio inspirals that follow a series of Kerr orbits before plunging onto the supermassive black hole. We examine the relativistic dynamics of stellar clusters in galactic nuclei where stellar diffusion through gravitational encounters culminate in stellar capture or disruption in the vicinity of the central black hole. Using our fully relativistic models, we discuss the conditions for both events to make predictions for either case. We show that statistics of Tidal Disruption Events (TDEs) can build the black hole mass function (BHMF) using the inferred TDE rates from various surveys. We use our steady loss cone model for the theoretical capture rate of stars $\dot{N}_t$, the Schechter black hole mass function, the Faber-Jackson law, and the $M_\bullet -\sigma$ relation to calculate the TDE detection rate for various surveys. By comparing it with observed rates, we extract the Schechter parameters for galaxies hosting TDEs. We show that the rate tension between the observed TDE rate ($\sim 10^{-5}~{\rm yr^{-1}}$) and theory ($\sim 10^{-4}~{\rm yr^{-1}}$) can be explained by a statistical average of $\dot{N}_t$ over BHMFs. Anticipating the discovery of GW events by eLISA, we predict event rates of extreme mass ratio inspirals for compact objects orbiting the black hole.