Induction of correct centrifugal force in a rotating mass shell

Abstract

Mach's idea of relativity of rotation is confirmed for a shell-type model of the Universe by showing that flat geometry in rotating coordinates, realising correct Coriolis and centrifugal forces, can be continuously connected through a rotating mass shell with not exactly spherical shape and latitude-dependent mass density to an asymptotically Minkowskian outside metric. The corresponding solutions of Einstein's field equations are given to second order in the angular velocity omega but it is plausible that the problem has a solution to any order of omega .

Slowly rotating thin shell gravastars

“…De la Cruz and Israel [20] and Pfister and Braun [21] obtained slowly rotating thin shells very similar to those discussed in this subsection. To investigate what is a possible source of the Kerr spacetime, de la Cruz and Israel obtained slowly rotating thin shells.…”

“…This is another interesting and fundamental situation apart from the gravastar. Therefore, similar but not the same situations have been so far considered by several authors, e.g., [19,20,21,22]. The quadrupole metric perturbations for the interior spacetime of the thin shell, given in (16) and (17), vanish in the limit of L → ∞.…”

“…They also showed that even in the Newtonian limit (M/R → 0), the shape of their slowly rotating thin shells is prolate. The common features of the slowly rotating thin shells obtained by De la Cruz and Israel [20] and Pfister and Braun [21] are that the thin shell matter stress is described by anisotropic pressure.…”

The History of the So-Called Lense–thirring Effect

“…An extension of the Brill and Cohen results to higher orders in ω, and especially the long-standing problem of the induction of a correct centrifugal force by rotating masses, had to wait for another 19 years for a solution (Pfister, & Braun, 1985). The solution is based on two "new" observations which could and should have been made already in Thirring's time, but which, for unexplicable reasons, were overlooked by all authors before 1985: a) Any physically realistic, rotating body will suffer a centrifugal deformation in orders ω 2 and higher, and cannot be expected to keep its spherical shape.…”

“…According to (Pfister, & Braun, 1985), in order ω 2 the shell geometry is given by r S = R(1 + ω 2 c 2 P 2 (cos θ)), with a constant c 2 , and with corresponding corrections in higher (even) orders ω 2n . Furthermore, it turns out (Pfister, & Braun, 1986) that in order ω 3 the flatness of the interior space-time can only be maintained if the shell material rotates differentially, ω S = ω(1 + ω 2 e 2 P 2 (cos θ)), with a constant e 2 , and with corresponding corrections in higher (odd) orders ω 2n+1 .…”