Abstract
The problem of obtaining the gravitational field of static, axially symmetric, thin shells is elucidated. In particular, a clear distinction between global and local frames is made. An algorithm is given for obtaining the fields of disks. There are two significant gravitational potentials and . The potential is straightforwardly determined from the radial stresses by solving a two-dimensional potential problem. This potential is analytic everywhere except on the disk and, together with its stream function , can be used to generate a conformal transformation which brings the equation for into the form of Laplace's equation. This potential can then be found by solving a Neumann boundary-value problem. However, the surface in the new coordinate system is not a disk since is discontinuous across the disk. This is due to the fact that the Cauchy-Riemann equations imply that if the normal derivative of is discontinuous, then the tangential derivative of will be discontinuous.
- Received 29 July 1970
DOI:https://doi.org/10.1103/PhysRevD.2.2756
©1970 American Physical Society