Truly naked spherically symmetric and distorted black holes

We demonstrate the existence of spherically symmetric truly naked black holes (TNBH) for which the Kretschmann scalar is finite on the horizon but some curvature components including those responsible for tidal forces as well as the energy density (rho) over bar measured by a free-falling observer are infinite. We choose a rather generic powerlike asymptotics for the metric functions and analyze possible types of a horizon depending on the behavior of curvature components in the free-falling frame. It is also shown in a general case of distorted black holes that (rho) over bar and tidal forces are either both finite or both infinite. The general approach developed in the article includes previously found examples and, in particular, TNBHs with an infinite area of a horizon. The fact that the detection of singularity depends on a frame may be relevant for a more accurate definition of the cosmic censorship conjecture. TNBHs may be considered as a new example of so-called nonscalar singularities for which the scalar curvature invariants are finite but some components of the Riemann tensor may diverge in certain frames.