Eckart energy momentum tensor [@Eckart]. However, this formulation results to be affected by non-causal features, allowing the propagation of superluminar signals [@HL]. In order to amend such a non-physical behavior, a revised approach has been proposed by Israel [@Israel], solving the non-causality problem via the introduction of phenomenological relaxation times. Having in mind the basic role the Bianchi I cosmology has in understanding the generic behavior of an inhomogeneous Universe near the singularity, in [@BK_Bianchi] and [@BK_Israel] such a model is studied as sourced by a viscous fluid, in both the Eckart and in the Israel approach respectively. One of the most intriguing issue coming out from such a study must be undoubtedly identified in the possibility of a singularity, from which the Universe emerges with negligible energy density and then a process of matter creation takes place.\
In the present analysis, we face the study of this aforesaid peculiar solution in terms of an alternative causal regularization of the Eckart energy-momentum tensor, proposed by Lichnerowicz in [@LICH1]. Such a revised formulation is based on the introduction of the so-called *index of the fluid*, de facto a regulator scaling the four-velocity field, so defining a *dynamical velocity* of the fluid. 