  The equations governing dissipative relativistic hydrodynamics are formulated
within the 3+1 approach for arbitrary spacetimes. Dissipation is accounted for
by applying the theory of extended causal thermodynamics (Israel-Stewart
theory). This description eliminates the causality violating infinite signal
speeds present in the conventional Navier-Stokes equation. As an example we
treat the astrophysically relevant case of stationary and axisymmetric
spacetimes, including the Kerr metric. The equations take a simpler form
whenever the inertia due to the dissipative contributions can be neglected.
