  This article belongs to a series where the influence of anisotropic pressure
on the gravitational properties of rigidly rotating fluids is studied using new
exactsolutions of GR constructed for the purpose. For mathematical
simplification, stationarity and cylindrical symmetry implying three Killing
vectors are considered. Moreover, two pressure components are set to vanish in
turn. In Papers 1 and 2 the pressure is axially directed, while it is azimuthal
in Paper 3. In present Paper 4, a radially directed pressure is considered.
Since a generic differential equation, split into three parts, emerges from the
field equations, three different classes of solutions can be considered. Two
could only be partially integrated. The other one, that is fully integrated,
yields a set of solutions with negative pressure. Physical processes where a
negative pressure is encountered are depicted and give a rather solid
foundation to this class of solutions. Moreover, these fully integrated
solutions satisfy the axisymmetry condition while they do not verify the
so-called "regularity condition". But, since their Kretschmann scalar does not
diverge on the axis, this feature must be considered as reporting a mere
coordinate singularity. Finally, the matching of these solutions to an exterior
appropriate vacuum enforce other constraints on the two constant parameters
defining each solution in the class. The results displayed here deserve to be
interpreted in the light of those depicted in the other four papers in the
series.
