Slow flows of a relativistic perfect fluid in a static gravitational field

Relativistic hydrodynamics of an isentropic fluid in a gravitational field is considered as a particular example from the family of Lagrangian hydrodynamic-type systems which possess an infinite set of integrals of motion due to the symmetry of the Lagrangian with respect to the relabeling of fluid particle labels. Flows with fixed topology of the vorticity are investigated in the quasistatic regime, when deviations of the space-time metric and the density of the fluid from the corresponding equilibrium configuration are negligibly small. On the basis of the variational principle for frozen-in vortex line dynamics, the equation of motion for a thin relativistic vortex filament is derived in the local induction approximation.