Gravitational Field of Shells and Disks in General Relativity

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 $\lambda$ and $\phi$. The potential $\lambda$ 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 $\overline{z}$, can be used to generate a conformal transformation which brings the equation for $\phi$ 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 $\overline{z}$ is discontinuous across the disk. This is due to the fact that the Cauchy-Riemann equations imply that if the normal derivative of $\overline{\rho }$ is discontinuous, then the tangential derivative of $\overline{z}$ will be discontinuous.