Shadow of nulllike and timelike naked singularities without photon spheres

In this paper, we derive general conditions for a shadow to occur, without a photon sphere in a spacetime, caused by central nulllike or timelike naked singularities. Using these conditions, we propose classes of spacetimes which have nulllike and timelike naked singularities at the center, and that fulfill these `shadow without photon sphere' conditions. Considering additional asymptotically flat conditions, we show that, for a fixed Schwarzschild mass, the timelike naked singularities can cast a shadow of a size that is equal to, greater or smaller, than the size of a black hole shadow. On the other hand, the size of shadow of a nulllike naked singularity is always less than that of a black hole. Such novel features of shadows of nulllike and timelike naked singularities in the absence of photon spheres may help us to distinguish between black holes and naked singularities observationally.
Comment: 10 pages, 10 figures