Mathematics | Physical Sciences and Mathematics
The static electrogravitational equations are studied and it is shown that an aligned type D metric which has a Weyl-type relationship between the gravitational and electric potential has shearfree geodesic lines of force. All such fields are then found and turn out to be the fields of a charged sphere, charged infinite rod and charged infinite plate. A further solution is also found with shearing geodesic lines of force. This new solution can have m > |e| or m < |e|, but cannot be in the Majumdar-Papapetrou class (in which m = |e|). It is algebraically general and has flat equipotential surfaces.
Brendan S. Guilfoyle , "Weyl-type fields with geodesic lines of force", J. Math. Phys. 40, 2032-2045 (1999) https://doi.org/10.1063/1.532849