  We are concerned with the nonlinear stability of vortex sheets for the
relativistic Euler equations in three-dimensional Minkowski spacetime. This is
a nonlinear hyperbolic problem with a characteristic free boundary. In this
paper, we introduce a new symmetrization by choosing appropriate functions as
primary unknowns. A necessary and sufficient condition for the weakly linear
stability of relativistic vortex sheets is obtained by analyzing the roots of
the Lopatinski\u{\i} determinant associated to the constant coefficient
linearized problem. Under this stability condition, we show that the variable
coefficient linearized problem obeys an energy estimate with a loss of
derivatives. The construction of certain weight functions plays a crucial role
in absorbing error terms caused by microlocalization. Based on the weakly
linear stability result, we establish the existence and nonlinear stability of
relativistic vortex sheets under small initial perturbations by a Nash--Moser
iteration scheme.
