Interior spacetimes sourced by stationary differentially rotating irrotational cylindrical fluids. III. Azimuthal pressure

In a recent series of papers, new exact analytical solutions to field equations of General Relativity representing interior spacetimes sourced by stationary rigidly rotating cylinders of fluids with various equations of state have been displayed. This work is currently extended to the case of differentially rotating irrotational fluids. The results are presented in a new series of papers considering, in turn, a perfect fluid source, arXiv:2305.11565 [gr-qc], as well as the three anisotropic pressure cases already studied in the rigidly rotating configuration. The axially directed pressure case has already been developed in arXiv:2307.07263. Here, a fluid with an azimuthally directed pressure is considered. A general method for generating the corresponding new mathematical solutions to the field equations when the ratio $h=$pressure/energy density varies with the radial coordinate is proposed, and a class of solutions exemplifying this recipe is derived. Then, the case where $h=const.$ is solved. It splits into two subclasses depending on the value of $h$. The mathematical and physical properties of these three classes are analyzed which provides some constraints on $h$, different for each class and subclass. Their matching to an exterior Lewis-Weyl vacuum and the conditions for avoiding an angular deficit are discussed. A comparison with the rigidly rotating fluid case is provided.