Constraints on Extra Time Dimensions

We discuss phenomenology of extra time dimensions in a scenario where the standard model particles are localized in ``our'' time, whereas gravity can propagate in all time dimensions. For an odd number of extra times, at small distances, the (real part of the) Newtonian potential is ``screened'' by tachyonic Kaluza-Klein gravitons. In general, the gravitational self-energy of objects acquires an imaginary part. This complexity may either be interpreted as an amplitude for the disappearance into ``nothing'', leading to the causality and probability violation in low-energy processes, or as an artifact of the fictitious decay into the unphysical negative energy tachyons. The former case would put the most severe phenomenological restrictions on such theories. In the latter case the size of extra times may be within the reach of the proposed gravitational experiments. In such a case these experiments should observe that the strength of the Newtonian gravity diminishes at short distances.