M ay 2 00 7 Can a charged dust ball be sent through the Reissner – Nordström wormhole ?

In a previous paper we formulated a set of necessary conditions for the spherically symmetric weakly charged dust to avoid Big Bang/Big Crunch, shell crossing and permanent central singularities. However, we did not discuss the properties of the energy density, some of which are surprising and seem not to have been known up to now. A singularity of infinite energy density does exist – it is a point singularity situated on the world line of the center of symmetry. The condition that no mass shell collapses to R = 0 if it had R > 0 initially thus turns out to be still insufficient for avoiding a singularity. Moreover, at the singularity the energy density is minus infinity, and the appearance of negative-energy-density regions turns out to be inevitable. We discuss various aspects of this property of our configuration. We also show that a permanently pulsating configuration, with the period of pulsation independent of mass, is possible only if there exists a permanent central singularity. 1 The problem In our previous paper [1] (Paper I) we formulated a set of necessary conditions for the spherically symmetric weakly charged dust to avoid Big Bang/Big Crunch, shell crossing and permanent central singularities. In fact, it had already been proven by Ori [2, 3] that weakly charged dust with no central singularity necessarily evolves toward a shell crossing. However, Ori assumed that the absolute value of the charge density (in geometric units) is strictly smaller than the mass density throughout the volume, including the center. His proof did not include the subcase when the values of these two densities at the center of symmetry are equal. We discussed precisely this subcase. We found that shell crossings are still unavoidable when the energy function E ≥ 0, but the conditions did not lead to a contradiction when E < 0. We also provided an example of a solution of these conditions, which was in fact incorrect (see below). However, we did not discuss the properties of the energy density. While we tried to construct a numerical example of a solution of those conditions, it turned out that this density was becoming negative in a vicinity of the ∗This research was supported by the Polish Ministry of Education and Science grant no 1 P03B 075 29.