Relativistic smooth particle hydrodynamics on a given background spacetime

We review the derivation of fixed-metric, relativistic smooth particle hydrodynamics (SPH) from the Lagrangian of an ideal fluid. Combining the Euler-Lagrange equations with the first law of thermodynamics, we explicitly derive evolution equations for the canonical momentum and energy. This new set of SPH equations also accounts for corrective terms that result from derivatives of the SPH smoothing kernel and that are called 'grad-h' terms in non-relativistic SPH. The new equations differ from earlier formulations with respect to these corrective terms and the symmetries in the SPH particle indices while being identical in gravitational terms.