  We investigate the question of how an observer in 4D perceives the
five-dimensional geodesic motion. We consider the interpretation of null and
non-null bulk geodesics in the context of brane theory, space-time-matter
theory (STM) and other non-compact approaches. We develop a "frame-invariant"
formalism that allows the computation of the rest mass and its variation as
observed in 4D. We find the appropriate expression for the four-acceleration
and thus obtain the extra force observed in 4D. Our formulae extend and
generalize all previous results in the literature. An important result here is
that the extra force in brane-world models with ${\bf Z}_{2}$-symmetry is
continuous and well defined across the brane. This is because the momentum
component along the extra dimension is discontinuous across the brane, which
effectively compensates the discontinuity of the extrinsic curvature. We show
that brane theory and STM produce identical interpretation of the bulk geodesic
motion. This holds for null and non-null bulk geodesics. Thus, experiments with
test particles are unable to distinguish whether our universe is described by
the brane world scenario or by STM. However, they do discriminate between the
brane/STM scenario and other non-compact approaches. Among them the canonical
and embedding approaches, which we examine in detail here.
