  We present a method for general relativistic smoothed particle hydrodynamics
(GRSPH), based on an entropy-conservative form of the general relativistic
hydrodynamic equations for a perfect fluid. We aim to replace approximate
treatments of general relativity in current SPH simulations of tidal disruption
events and accretion discs. We develop an improved shock capturing formulation
that distinguishes between shock viscosity and conductivity in relativity. We
also describe a new Hamiltonian time integration algorithm for relativistic
orbital dynamics and GRSPH. Our method correctly captures both Einstein and
spin-induced precession around black holes. We benchmark our scheme in 1D and
3D against mildly and ultra relativistic shock tubes, exact solutions for
epicyclic and vertical oscillation frequencies, and Bondi accretion. We assume
fixed background metrics (Minkowski, Schwarzschild and Kerr in Cartesian
Boyer-Lindquist coordinates) but the method lays the foundation for future
direct coupling with numerical relativity.
