Gravitational Collapse of Rotating Bodies

We investigate the effect of gravitational collapse of rotating bodies on the induced rotation of inertial frames. In particular, it is shown that the angular velocity of the inertial frames, within an adiabatically collapsing, slowly rotating mass shell supported by elastic stresses, approaches that of the shell as the shell radius approaches the Schwarzschild radius. Even when this relative angular velocity approaches zero, the angular momentum (a conserved quantity) does not vanish; it remains constant during the collapse. On the other hand, an observer within a slowly rotating dust shell (at the point of maximum expansion) does not see the angular velocity of the inertial frames approach that of the shell as the radius approaches the Schwarzschild radius. This difference between the two situations is shown to be in accordance with Mach's principle. The effect of rotation on gravitational collapse is also considered. This is done to shed some light on an important question in astrophysics: Does rotation stop collapse, or does collapse crush rotation? A spherical shell of dust supported by "centrifrigal forces" is considered. It is shown that rotation cannot stop collapse unless the shell radius is equal to or larger than (9/8)\ifmmode\times\else\texttimes\fi{}(Schwarzschild radius). This happens even though the velocity of the particles in the shell is allowed to approach that of light.