Spinning Shell as a Source of the Kerr Metric

A method is described which develops the interior field and the physical properties of a slowly spinning, nearly spherical thin mass shell as a power series in the angular velocity $\omega$, on the presumption that the exterior field is the Kerr metric. Results are worked out explicitly, correct to the third order in $\omega$. The ellipticity of the shell is arbitrarily assignable to within quantities of order ${\omega }^2$. If it is suitably prescribed, the interior field tends to flatness, in the limit where the radius of the shell approaches the gravitational radius, and the extended inertial frames thus defined in the interior are dragged around rigidly by the shell in "Machian" fashion. This amplifies a previous first-order result due to Brill and Cohen.