Nonlocal equation of state in anisotropic static fluid spheres in general relativity

We show that it is possible to obtain, at least certain regions within spherically symmetric static matter configurations, credible anisotropic fluids satisfying a nonlocal equation of state. This particular type of equation of state provides, at a given point, the radial pressure not only as a function of the density at that point, but its functional throughout the enclosed distribution. To establish the physical plausibility of the proposed family of solutions satisfying a nonlocal equation of state, we study the constraints imposed by the junction, energy, and some intuitive physical conditions. We show that these static fluids having this particular equation of state are "naturally" anisotropic in the sense that they satisfy, identically, the anisotropic Tolman-Oppenheimer-Volkov equation. We also show that it is possible to obtain physically plausible static anisotropic spherically symmetric matter configurations starting from known density profiles, and also for configurations where tangential pressures vanish. This very particular type of relativistic sphere with vanishing tangential stresses is inspired by some of the models proposed to describe extremely magnetized neutron stars (magnetars) during the transverse quantum collapse.