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 Centre S-stars. However, the evolution of orbits below the SB, e > e(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 an SBH including general relativistic corrections to the equations of motion and interactions with a large (a parts per thousand(3)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 and resonant relaxation, respectively. In addition, we find a new regime of 'anomalous relaxation'. 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 an SBH.