Geodesic Congruences and Their Deformations in Bertrand Space-times
  We study the energy conditions and geodesic deformations in Bertrand space-times. We show that these can be thought of as interesting physical space-times in certain regions of the underlying parameter space, where the weak and strong energy conditions hold. We further compute the ESR parameters and analyze them numerically. The focusing of radial time-like and radial null geodesics is shown explicitly, which verifies the Raychaudhuri equation. 
Prashant Kumar, Kaushik Bhattacharya, Tapobrata Sarkar
gr-qc hep-th