Optical reference geometry and inertial forces in Kerr–de Sitter spacetimes

Abstract

Optical reference geometry and the related concept of inertial forces are investigated in Kerr–de Sitter spacetimes. Properties of the inertial forces are summarized and their typical behaviour is illustrated. The intuitive ‘Newtonian’ application of the forces in the relativistic dynamics is demonstrated in the case of the test particle circular motion, static equilibrium positions and perfect fluid toroidal configurations. Features of the optical geometry are illustrated by the embedding diagrams of its equatorial plane. The embedding diagrams do not wholly cover the stationary regions of the spacetimes, therefore the limits of embeddability are established. The shape of the embedding diagrams is related to the behaviour of the centrifugal force and it is characterized by the number of turning points of the diagrams. Discussion of the number of embeddable photon circular orbits is also included and the typical embedding diagrams are constructed. The Kerr–de Sitter spacetimes are classified according to the properties of the inertial forces and embedding diagrams.