  The relative geodesic motion on $(1+3)$-dimensional anti-de Sitter spacetimes
is studied in terms of conserved quantities by adapting the Nachtmann boosting
method created initially for the de Sitter spacetimes. In this approach the
Lorentzian isometriy is derived, relating the coordinates of the local chart of
a fixed observer with the coordinates of a mobile chart considered as the rest
frame of a massive mobile object moving on a timelike anti-de Sitter geodesic.
The transformation of the conserved quantities is also investigated,
constructing thus a complete theory of the relative geodesic motion on anti-de
Sitter spacetimes. Some applications are discussed among them the problems of
twin paradox and Lorentz contraction are solved.
