  The Majumdar-Papapetrou system is the subset of the Einstein-Maxwell-charged
dust matter theory, when the charge of each particle is equal to its mass.
Solutions for this system are less difficult to find, in general one does not
need even to impose any spatial symmetry a priori. For instance, any number of
extreme Reissner-Nordstrom solutions (which in vacuum reduce to extreme
Reissner-Nordstrom black holes) located at will is a solution. In matter one
can also find solutions with some ease. Here we find an exact solution of the
Majumdar-Papapetrou system, a spherically symmetric charged thick shell, with
mass m, outer radius r_o, and inner radius r_i. This solution consists of three
regions, an inner Minkowski region, a middle region with extreme charged dust
matter, and an outer Reissner-Nordstrom region. The matching of the regions,
obeying the usual junction conditions for boundary surfaces, is continuous. For
vanishing inner radius, one obtains a Bonnor star, whereas for vanishing
thickness, one obtains an infinitesimally thin shell. For sufficiently high
mass of the thick shell or sufficiently small outer radius, it forms an extreme
Reissner-Nordstrom quasi-black hole, i.e., a star whose gravitational
properties are virtually indistinguishable from a true extreme black hole. This
quasi-black hole has no hair and has a naked horizon, meaning that the Riemann
tensor at the horizon on an infalling probe diverges. At the critical value,
when the mass is equal to the outer radius, m=r_o, there is no smooth manifold.
Above the critical value when m>r_o there is no solution, the shell collapses
into a singularity. Systems with m< r_o are neutrally stable. Many of these
properties are similar to those of gravitational monopoles.
