  Everyday experience with centrifugal forces has always guided thinking on the
close relationship between gravitational forces and accelerated systems of
reference. Once spatial gravitational forces and accelerations are introduced
into general relativity through a splitting of spacetime into space-plus-time
associated with a family of test observers, one may further split the local
rest space of those observers with respect to the direction of relative motion
of a test particle world line in order to define longitudinal and transverse
accelerations as well. The intrinsic covariant derivative (induced connection)
along such a world line is the appropriate mathematical tool to analyze this
problem, and by modifying this operator to correspond to the observer
measurements, one understands more clearly the work of Abramowicz et al who
define an ``optical centrifugal force'' in static axisymmetric spacetimes and
attempt to generalize it and other inertial forces to arbitrary spacetimes. In
a companion article the application of this framework to some familiar
stationary axisymmetric spacetimes helps give a more intuitive picture of their
rotational features including spin precession effects, and puts related work of
de Felice and others on circular orbits in black hole spacetimes into a more
general context.
