Sharp bounds on the radius of relativistic charged spheres: Guilfoyle's stars saturate the Buchdahl–Andréasson bound

E-ISSN： 1361-6382|32|13|135009-135017

ISSN： 0264-9381

Source： Classical and Quantum Gravity, Vol.32, Iss.13, 2015-07, pp. : 135009-135017

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Abstract

Buchdahl, by imposing a few reasonable physical assumptions on 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 r₀ to mass m ratio of a star would obey the bound &${{r}_{0}}/mgeqslant 9/4$;, the Buchdahl bound. He also noted that the bound was saturated by the Schwarzschild interior solution, i.e., 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 in various forms have been studied. An important generalization was given by Andréasson, by including electrically charged matter and imposing a different set of conditions, namely, &$p+2{{p}_{T}}leqslant {{rho }_{{rm m}}}$;, where p is the radial pressure and p_T the tangential pressure. His bound is sharp and given by &${{r}_{0}}/mgeqslant 9/{{(1+sqrt{1+3;{{q}^{2}}/r_{0}^{2}})}^{2}}$;, the Buchdahl–Andréasson bound, with q being the total electric charge of the star. For q = 0 one recovers the Buchdahl bound. However, following Andréasson'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 electrically charged Buchdahl–Andréasson bound are charged shells. One could expect 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 &${{rho }_{{rm m}}}(r)+{{Q}^{2}}(r)/(8pi ;{{r}^{4}})={rm constant}$;, 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 also saturate the Buchdahl–Andréasson bound. A proof in Buchdahl's manner, such that these configurations are the limiting configurations of the bound, remains to be found.