Despite the fact that the Schwarzschild and Kerr solutions for the Einstein
equations, when written in standard Schwarzschild and Boyer-Lindquist
coordinates, present coordinate singularities, all numerical studies of
accretion flows onto collapsed objects have been widely using them over the
years. This approach introduces conceptual and practical complications in
places where a smooth solution should be guaranteed, i.e., at the gravitational
radius. In the present paper, we propose an alternative way of solving the
general relativistic hydrodynamic equations in background (fixed) black hole
spacetimes. We identify classes of coordinates in which the (possibly rotating)
black hole metric is free of coordinate singularities at the horizon,
independent of time, and admits a spacelike decomposition. In the spherically
symmetric, non-rotating case, we re-derive exact solutions for dust and perfect
fluid accretion in Eddington-Finkelstein coordinates, and compare with
numerical hydrodynamic integrations. We perform representative axisymmetric
computations. These demonstrations suggest that the use of those coordinate
systems carries significant improvements over the standard approach, especially
for higher dimensional studies.Comment: 10 pages, 4 postscript figures, accepted for publication in Phys.
Rev.
