Inertial forces and the foundations of optical geometry
Journal article, 2006
Assuming a general timelike congruence of worldlines as a reference frame, we derive a covariant general formalism of inertial forces in general relativity. Inspired by the works of Abramowicz et al (see e.g. Abramowicz and Lasota 1997 Class. Quantum Grav. 14 A23-30), we also study conformal rescalings of spacetime and investigate how these affect the inertial force formalism. While many ways of describing spatial curvature of a trajectory have been discussed in papers prior to this, one particular prescription (which differs from the standard projected curvature when the reference congruence is shearing), appears novel. For the particular case of a hypersurface-forming congruence, using a suitable rescaling of spacetime, we show that a geodesic photon always follows a line that is spatially straight with respect to the new curvature measure. This fact is intimately connected to Fermat's principle, and allows for a certain generalization of the optical geometry as will be further pursued in a companion paper (Jonsson and Westman 2006 Class. Quantum Grav. 23 61). For the particular case when the shear tensor vanishes, we present the inertial force equation in a three-dimensional form (using the bold-face vector notation), and note how similar it is to its Newtonian counterpart. From the spatial curvature measures that we introduce, we derive corresponding covariant differentiations of a vector defined along a spacetime trajectory. This allows us to connect the formalism of this paper to that of Jantzen and co-workers (see e.g. Bini et al 1997 Int. J. Mod. Phys. D 6 143-98).