  There are many ways of embedding a 4D spacetime in a given higher-dimensional
manifold while, satisfying the field equations. In this work we extend and
generalize a recent paper by Mashhoon and Wesson ({\it Gen. Rel. Gravit.} {\bf
39}, 1403(2007)) by showing different ways of embedding a solution of the 4D
Einstein equations, in vacuum with a cosmological constant, in a Ricci-flat, as
well as in an anti-de Sitter, 5D manifold. These embeddings lead to different
physics in 4D. In particular, to non-equivalent cosmological terms as functions
of the extra coordinate. We study the motion of test particles for different
embeddings and show that there is a complete equivalence between several
definitions for the effective mass of test particles measured in 4D, obtained
from different theoretical approaches like the Hamilton-Jacobi formalism and
the principle of least action. For the case under consideration, we find that
the effective mass observed in 4D is the same regardless of whether we consider
null or non-null motion in 5D.
