Isochrone spacetimes

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.