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

286 EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY
and one inhomogeneous Maxwell equation
Eq. (20) together with enforces the ansatz
with a dimensionless function Eq. (21) then results
in the form with These forms
constitute the dipole character of the (rotationally induced) magnetic
field. Eqs. (20) and (22) constitute two coupled ordinary differential
equations for the functions and In the interior these
equations simplify drastically, and give the results
with dimensionless constants and i.e. the interior stays flat, and
the magnetic field has (in Cartesian coordinates) only a (constant)
In the intermediate and exterior regions, due to
one integration of Eq. (20) is trivial:
with a dimensionless integration constant Insertion of (23) into (22),
together with outside the charged shell, results in
The general solution of Eq. (25) has the form
with dimensionless integration constants and and where
is a special inhomogeneous solution, and are independent
homogeneous solutions of (24). Luckily, there exist simple polynomial
solutions
and can then be found from by d’Alembert’s reduction procedure:
with