Machian effects in compact, rapidly spinning shells

A mass shell with flat interior cannot rotate rigidly

“…Therefore, for a compact mass shell with M = 2R, and only in this case, the condition of interior flatness is realisable with rigid rotation (and furthermore, according to I, with exactly spherical shape and uniform mass density). This result is in agreement with earlier work of de la Cruz and Israel [4] and Orwig [ 5 ] , who investigated rotating mass shells with the restriction that the exterior geometry is given by Kerr's metric. They obtained the result that sphericity and rigid rotation of the shell as well as flat inside geometry can only be reached in the limit R + M/2.…”

A spinning shell around a Kerr black hole in a slow rotation approximation

“…This correlated to the result obtained by Cohen [8]. Orwig [9] and Pfister and Braun [10] further analyzed shells with a flat interior. They determined that the local inertial frames are "dragged" by the shell when the shell radius approaches the gravitational radius.…”

“…It is known that in the case of a flat interior, local inertial frames inside the shell rotate at the same angular velocity as the shell thus, being effectively "dragged" by the shell-when the shell radius R approaches the gravitational radius [7,9,10]. If the behavior of the local inertial frames is the same as in the case of the Kerr metric inside the shell, we would expect the angular velocity of the shell with respect to a distant observer to be…”

“…Equation (9) lead to five non-zero components of the energy-impulse tensor. Their explicit form is given in the Appendix.…”

“…Matching conditions for the gravitational field across the boundary were presented by Israel [13] and have been used in numerous studies (e.g., [7,9,14,15]). These conditions imply the existence of a thin material shell at the boundary and lead to a determination of the energy-impulse tensor distribution in such a shell.…”