Exact Solutions and Scalar Fields in Gravity - Instituto Avanzado de ...

On Electromagnetic Thirring Problems 283
the potentials in the exterior region read:
In the region we set and The charge
parameter should have the same value as in the exterior region because
the shell at is uncharged. In order to enable continuous potentials
and at the constant from (3 is generally different
from 1, and a nontrivial time transformation is necessary in this
region. Then the potentials in the region read:
In the interior region we have, due to and
and the potentials
We now specify the parameter In our models, the function of
the inner shell is mainly to provide a charge, and not so much to provide
additional mass. It seems therefore reasonable to simplify our
models (in accordance with [2, 3] by setting the rest mass density of
the inner shell to zero. If we write Einstein’s field equations in the
form with being the electromagnetic energy–
momentum tensor, then in our two–shell models consists of two parts:
and and the are expressed by the
derivatives and in the form
Equivalent relations are valid at the position With the useful
abbreviations and
the condition leads to Herewith, the
continuity conditions for and at and fix the
constants as: