Focusing versus defocusing properties of truly naked black holes
  We study the properties of the congruence of null geodesics propagating near the so-called truly naked horizons (TNH) - objects having finite Kretschmann scalar but with diverging tidal acceleration for freely falling observers. The expansion of outgoing rays near the future horizon always tends to vanish for the non-extremal case but may be non-zero for the distorted (ultra)extremal one. It tends to diverge for the ingoing ones if the the null energy condition (NEC) is satisfied in the vicinity of the horizon outside. However, it also tends to zero for NEC violating cases except the remote horizons. We also discuss the validity of test particle approximation for TNHs and find the sufficient condition for backreaction remaining small. 
Naresh Dadhich and Oleg B. Zaslavskii