  In this paper we introduce a model of relativistic short wave-long wave
interaction where the short waves are described by the massless
$1+3$-dimensional Thirring model of nonlinear Dirac equation and the long waves
are described by the $1+3$-dimensional relativistic Euler equations. The
interaction coupling terms are modeled by a potential proportional to the
relativistic specific volume in the Dirac equation and an external force
proportional to the square modulus of the Dirac wave function in the
relativistic Euler equation. An important feature of the model is that the
Dirac equations are based on the Lagrangian coordinates of the relativistic
fluid flow. In particular, an important contribution of this paper is a clear
formulation of the relativistic Lagrangian transformation. This is done by
means of the introduction of natural auxiliary dependent variables, rendering
the discussion totally similar to the non-relativistic case. As far as the
authors know the definition of the Lagrangian transformation given in this
paper is new. Finally, we establish the short-time existence and uniqueness of
a smooth solution of the Cauchy problem for the regularized model. This follows
through the symmetrization of the relativistic Euler equation introduced by
Makino and Ukai (1995) and requires a slight extension of a well known theorem
of T.~Kato (1975) on quasi-linear symmetric hyperbolic systems.
