  We introduce the relativistic version of the well-known Henon's isochrone
spherical models: static spherically symmetrical spacetimes in which all
bounded trajectories are isochrone in Henon's sense, i.e., their radial periods
do not depend on their angular momenta. Analogously to the Newtonian case,
these "isochrone spacetimes" have as particular cases the so-called Bertrand
spacetimes, in which all bounded trajectories are periodic. We propose a
procedure to generate isochrone spacetimes by means of an algebraic equation,
present explicitly several families of these spacetimes, and discuss briefly
their main properties. We identify, in particular, the family whose Newtonian
limit corresponds to the Henon's isochrone potentials and that could be
considered as the relativistic extension of the original Henon's proposal for
the study of globular clusters. Nevertheless, isochrone spacetimes generically
violate the weak energy condition and may exhibit naked singularities,
challenging their physical interpretation in the context of General Relativity.
