An example of induced centrifugal force in general relativity

Abstract

Thirring's solution¹ for a rotating spherical mass is cited often as an example which exhibits induced Coriolis and centrifugal forces in general relativity^2,3. Previously, it was pointed out⁴ that the ‘centrifugal terms’ proportional to the square of the angular velocity actually represented quadrupole effects. These effects arose as a consequence of the latitude-dependent velocity distribution which generated an axially symmetric (non-spherical) mass distribution. This raises the question: Are there solutions to Einstein's equations which exhibit induced centrifugal force? If no such solutions exist, then the concept of induced rotation of inertial frames breaks down. Such a lack of connection between rotating masses and induced inertial frame rotation would in turn imply a breakdown of the Machian^5,6 idea that an inertial frame is a system of reference in which the stress-energy of the universe is at rest on the average⁷. Here we demonstrate that there are solutions to Einstein's equations which exhibit induced centrifugal force (other solutions may also be of interest in this connection⁸).