Bertrand spacetimes

Abstract

The author determines all spherically symmetric and static Lorentzian spacetimes in which any bounded trajectory is periodic. The author calls such spacetimes 'Bertrand spacetimes', thereby referring to Bertrand's classical theorem which deals with the analogous situation in Newtonian mechanics. Any Bertrand spacetime is characterized by a rational number beta which gives the apsidal angle of all bounded trajectories in fractions of pi . Whereas Newtonian Bertrand potentials exist only for beta =1 (Kepler potential) and for beta =2 (harmonic oscillator potential), general relativity allows one to construct Bertrand spacetimes for any beta . An asymptotically Minkowskian Bertrand spacetime, however, must have beta =1, and a Bertrand spacetime with regular centre must have beta =2. Whereas all asymptotically Minkowskian Bertrand spacetimes violate the weak energy condition, some Bertrand spacetimes with regular centre admit an interpretation as charged perfect fluid solution of Einstein's field equation.