Sharp bounds on the radius of relativistic charged spheres: Guilfoyle's stars saturate the Buchdahl-Andr\'easson bound

Buchdahl, by imposing a few physical assumptions on the matter, i.e., its density is a nonincreasing function of the radius and the fluid is a perfect fluid, and on the configuration, such as the exterior is the Schwarzschild solution, found that the radius $r_0$ to mass $m$ ratio of a star would obey the Buchdahl bound $r_0/m\geq9/4$. He noted that the bound was saturated by the Schwarzschild interior solution, the solution with $\rho_{\rm m}(r)= {\rm constant}$, where $\rho_{\rm m}(r)$ is the energy density of the matter at $r$, when the central central pressure blows to infinity. Generalizations of this bound have been studied. One generalization was given by Andr\'easson by including electrically charged matter and imposing that $p+2p_T \leq\rho_{\rm m}$, where $p$ is the radial pressure and $p_T$ the tangential pressure. His bound is given by $r_0/m\geq9/\left(1+\sqrt{1+3\,q^2/r_0^2}\right)^{2}$, the Buchdahl-Andr\'easson bound, with $q$ being the star's total electric charge. Following Andr\'easson's proof, the configuration that saturates the Buchdahl bound is an uncharged shell, rather than the Schwarzschild interior solution. By extension, the configurations that saturate the Buchdahl-Andr\'easson bound are charged shells. One expects then, in turn, that there should exist an electrically charged equivalent to the interior Schwarzschild limit. We find here that this equivalent is provided by the equation $\rho_{\rm m}(r) + {Q^2(r)}/ {\left(8\pi\,r^4\right)}= {\rm constant}$, where $Q(r)$ is the electric charge at $r$. This equation was put forward by Cooperstock and de la Cruz, and Florides, and realized in Guilfoyle's stars. When the central pressure goes to infinity Guilfoyle's stars are configurations that saturate the Buchdahl-Andr\'easson bound. It remains to find a proof in Buchdahl's manner such that these configurations are the limiting configurations of the bound.