Neutral perfect fluids of Majumdar-type in general relativity

We consider the extension of the Majumdar-type class of static solutions for the Einstein-Maxwell equations, proposed by Ida to include charged perfect fluid sources. We impose the equation of state ρ+3p = 0 and discuss spherically symmetric solutions for the linear potential equation satisfied by the metric. In this particular case the fluid charge density vanishes and we locate the arising neutral perfect fluid in the intermediate region defined by two thin shells with respective charges Q and −Q. With its innermost flat and external (Schwarzschild) asymptotically flat spacetime regions, the resultant condenser-like geometries resemble solutions discussed by Cohen and Cohen in a different context. We explore this relationship and point out an exotic gravitational property of our neutral perfect fluid. We mention possible continuations of this study to embrace non-spherically symmetric situations and higher dimensional spacetimes. Our previous contribution [1] was devoted to the study of static charged dust solutions of the Einstein-Maxwell equations. The derivation of the solutions was based on three main assumptions: (i) the spacetime metric has the conformastatic form ds = −V 2 dt + 1 V 2 hijdx dx, (1) where Latin indices indicate 1, 2, 3; (ii) the background three-dimensional metric hij is Euclidean; (iii) hij and V depend only on the space-like coordinates x 1, x2, x3. As a consequence, the corresponding static solutions of the Einstein-Maxwell-charged dust equations satisfied a linear relation between V and the Coulombian potential A0. We will refer to solutions of the field equations with such a linear relationship as being of Majumdar-type. See Guilfoyles’ paper [2] for a discussion of these geometries within the framework of the Weyl class of solutions [3]. More recent papers by Vogt and Letelier [4], Kleber et al. [5], Ivanov [6], and Wickramasuriya [7] deal with interesting physical and mathematical aspects of Majumdar-type solutions.