Relativistic dynamics of stars near a supermassive black hole
  General relativistic precession limits the ability of gravitational
encounters to increase the eccentricity $e$ of orbits near a supermassive black
hole (SBH). This "Schwarzschild barrier" (SB) has been shown to play an
important role in the orbital evolution of stars like the galactic center
S-stars. However, the evolution of orbits below the SB, $e>e_\mathrm{SB}$, is
not well understood; the main current limitation is the computational
complexity of detailed simulations. Here we present an $N$-body algorithm that
allows us to efficiently integrate orbits of test stars around a SBH including
general relativistic corrections to the equations of motion and interactions
with a large ($\gtrsim 10^3$) number of field stars. We apply our algorithm to
the S-stars and extract diffusion coefficients describing the evolution in
angular momentum $L$. We identify three angular momentum regimes, in which the
diffusion coefficients depend in functionally different ways on $L$. Regimes of
lowest and highest $L$ are well-described in terms of non-resonant relaxation
(NRR) and resonant relaxation (RR), respectively. In addition, we find a new
regime of "anomalous relaxation" (AR). We present analytic expressions, in
terms of physical parameters, that describe the diffusion coefficients in all
three regimes, and propose a new, empirical criterion for the location of the
SB in terms of the $L$-dependence of the diffusion coefficients. Subsequently
we apply our results to obtain the steady-state distribution of angular
momentum for orbits near a SBH.
