  We consider a first order formulation of relativistic fluids with bulk
viscosity based on a stress-energy tensor introduced by Lichnerowicz. Choosing
a barotropic equation of state, we show that this theory satisfies basic
physical requirements and, under the further assumption of vanishing vorticity,
that the equations of motion are causal, both in the case of a fixed background
and when the equations are coupled to Einstein's equations. Furthermore,
Lichnerowicz's proposal does not fit into the general framework of first order
theories studied by Hiscock and Lindblom, and hence their instability results
do not apply. These conclusions apply to the full-fledged non-linear theory,
without any equilibrium or near equilibrium assumptions. Similarities and
differences between the approach here explored and other theories of
relativistic viscosity, including the Mueller-Israel-Stewart formulation, are
addressed. Cosmological models based on the Lichnerowicz stress-energy tensor
are studied. Finally, it is also shown how the present model can be generalized
to a second order formulation.
