  We prove that Einstein's equations coupled to equations of
Israel-Stewart-type, describing the dynamics of a relativistic fluid with bulk
viscosity and nonzero baryon charge (without shear viscosity or baryon
diffusion) dynamically coupled to gravity, are causal in the full nonlinear
regime. We also show that these equations can be written as a first-order
symmetric hyperbolic system, implying local existence and uniqueness of
solutions to the equations of motion. We use an arbitrary equation of state and
do not make any simplifying symmetry or near-equilibrium assumption, requiring
only physically natural conditions on the fields. These results pave the way
for the inclusion of bulk viscosity effects in simulations of
gravitational-wave signals coming from neutron star mergers.
