Electrically charged fluids with pressure in Newtonian gravitation and general relativity indspacetime dimensions: Theorems and results for Weyl type systems

Abstract

Previous theorems concerning Weyl type systems, including Majumdar-Papapetrou systems, are generalized in two ways, namely, we take these theorems into $d$ spacetime dimensions ($d\ensuremath{\ge}4$), and we also consider the very interesting Weyl-Guilfoyle systems, i.e., general relativistic charged fluids with nonzero pressure. In particular within the Newton-Coulomb theory of charged gravitating fluids, a theorem by Bonnor (1980) in three-dimensional space is generalized to arbitrary $(d\ensuremath{-}1)>3$ space dimensions. Then, we prove a new theorem for charged gravitating fluid systems in which we find the condition that the charge density and the matter density should obey. Within general relativity coupled to charged dust fluids, a theorem by De and Raychaudhuri (1968) in four-dimensional spacetime is rendered into arbitrary $d>4$ dimensions. Then a theorem, new in $d=4$ and $d>4$ dimensions, for Weyl-Guilfoyle systems, is stated and proved, in which we find the condition that the charge density, the matter density, the pressure, and the electromagnetic energy density should obey. This theorem comprises, in particular cases, a theorem by Gautreau and Hoffman (1973) and results in four dimensions by Guilfoyle (1999). Upon connection of an interior charged solution to an exterior Tangherlini solution (i.e., a Reissner-Nordstr\"om solution in $d$ dimensions), one is able to give a general definition for gravitational mass for this kind of relativistic systems and find a mass relation with several quantities of the interior solution. It is also shown that for sources of finite extent the mass is identical to the Tolman mass.