Null hypersurface caustics, closed null curves, and super entropy

Recently it was discovered that null hypersurfaces can develop caustics outside the event horizon of super-entropic Kerr-AdS black holes, in contrast to usual case. In this work we explore a few more examples hole spacetimes which such exterior develop. If closed curve is present, e.g., case Taub-NUT and "transunital" spacetimes, then coincides with hypersurface caustic (NHC) minimal separation parameter. Thus spacetime on verge forming timelike curves could caustic. Known holes also have NHC, although are free null/timelike curves. Nevertheless relationship between causal curves, super-entropy not straightforward. This best illustrated BTZ string, for some choices warp factor extra dimension value charge, be super-entropic. However, even those admit NHC horizon.